blender-archive/source/blender/geometry/intern/uv_parametrizer.cc

/* SPDX-License-Identifier: GPL-2.0-or-later */

/** \file
 * \ingroup eduv
 */

#include "GEO_uv_parametrizer.h"

#include "MEM_guardedalloc.h"

#include "BLI_boxpack_2d.h"
#include "BLI_convexhull_2d.h"
#include "BLI_ghash.h"
#include "BLI_heap.h"
#include "BLI_memarena.h"
#include "BLI_polyfill_2d.h"
#include "BLI_polyfill_2d_beautify.h"
#include "BLI_rand.h"

#include "eigen_capi.h"

/* Utils */

#define param_assert(condition) \
  if (!(condition)) { /*printf("Assertion %s:%d\n", __FILE__, __LINE__); abort();*/ \
  } \
  (void)0
#define param_warning(message) \
  {/*printf("Warning %s:%d: %s\n", __FILE__, __LINE__, message);*/}(void)0

/* Special Purpose Hash */

typedef uintptr_t PHashKey;

typedef struct PHashLink {
  struct PHashLink *next;
  PHashKey key;
} PHashLink;

typedef struct PHash {
  PHashLink **list;
  PHashLink **buckets;
  int size, cursize, cursize_id;
} PHash;

/* Simplices */

struct PVert {
  struct PVert *nextlink;

  union PVertUnion {
    PHashKey key;       /* Construct. */
    int id;             /* ABF/LSCM matrix index. */
    HeapNode *heaplink; /* Edge collapsing. */
  } u;

  struct PEdge *edge;
  float co[3];
  float uv[2];
  uint flag;
};

struct PEdge {
  struct PEdge *nextlink;

  union PEdgeUnion {
    PHashKey key;               /* Construct. */
    int id;                     /* ABF matrix index. */
    HeapNode *heaplink;         /* Fill holes. */
    struct PEdge *nextcollapse; /* Simplification. */
  } u;

  struct PVert *vert;
  struct PEdge *pair;
  struct PEdge *next;
  struct PFace *face;
  float *orig_uv, old_uv[2];
  uint flag;
};

struct PFace {
  struct PFace *nextlink;

  union PFaceUnion {
    PHashKey key; /* Construct. */
    int chart;    /* Construct splitting. */
    float area3d; /* Stretch. */
    int id;       /* ABF matrix index. */
  } u;

  struct PEdge *edge;
  uint flag;
};

enum PVertFlag {
  PVERT_PIN = 1,
  PVERT_SELECT = 2,
  PVERT_INTERIOR = 4,
  PVERT_COLLAPSE = 8,
  PVERT_SPLIT = 16,
};

enum PEdgeFlag {
  PEDGE_SEAM = 1,
  PEDGE_VERTEX_SPLIT = 2,
  PEDGE_PIN = 4,
  PEDGE_SELECT = 8,
  PEDGE_DONE = 16,
  PEDGE_FILLED = 32,
  PEDGE_COLLAPSE = 64,
  PEDGE_COLLAPSE_EDGE = 128,
  PEDGE_COLLAPSE_PAIR = 256,
};

/* for flipping faces */
#define PEDGE_VERTEX_FLAGS (PEDGE_PIN)

enum PFaceFlag {
  PFACE_CONNECTED = 1,
  PFACE_FILLED = 2,
  PFACE_COLLAPSE = 4,
};

/* Chart */

struct PChart {
  PVert *verts;
  PEdge *edges;
  PFace *faces;
  int nverts, nedges, nfaces;
  int nboundaries;

  PVert *collapsed_verts;
  PEdge *collapsed_edges;
  PFace *collapsed_faces;

  union PChartUnion {
    struct PChartLscm {
      LinearSolver *context;
      float *abf_alpha;
      PVert *pin1, *pin2;
      PVert *single_pin;
      float single_pin_area;
      float single_pin_uv[2];
    } lscm;
    struct PChartPack {
      float rescale, area;
      float size[2];
      float origin[2];
    } pack;
  } u;

  bool has_pins;
};

enum PHandleState {
  PHANDLE_STATE_ALLOCATED,
  PHANDLE_STATE_CONSTRUCTED,
  PHANDLE_STATE_LSCM,
  PHANDLE_STATE_STRETCH,
};

struct ParamHandle {
  enum PHandleState state;
  MemArena *arena;
  MemArena *polyfill_arena;
  Heap *polyfill_heap;

  PChart *construction_chart;
  PHash *hash_verts;
  PHash *hash_edges;
  PHash *hash_faces;

  struct GHash *pin_hash;
  int unique_pin_count;

  PChart **charts;
  int ncharts;

  float aspx, aspy;

  RNG *rng;
  float blend;
};

/* PHash
 * - special purpose hash that keeps all its elements in a single linked list.
 * - after construction, this hash is thrown away, and the list remains.
 * - removing elements is not possible efficiently.
 */

static int PHashSizes[] = {
    1,       3,       5,       11,      17,       37,       67,       131,       257,       521,
    1031,    2053,    4099,    8209,    16411,    32771,    65537,    131101,    262147,    524309,
    1048583, 2097169, 4194319, 8388617, 16777259, 33554467, 67108879, 134217757, 268435459,
};

#define PHASH_hash(ph, item) (((uintptr_t)(item)) % ((uint)(ph)->cursize))
#define PHASH_edge(v1, v2) (((v1) < (v2)) ? ((v1)*39) ^ ((v2)*31) : ((v1)*31) ^ ((v2)*39))

static PHash *phash_new(PHashLink **list, int sizehint)
{
  PHash *ph = (PHash *)MEM_callocN(sizeof(PHash), "PHash");
  ph->size = 0;
  ph->cursize_id = 0;
  ph->list = list;

  while (PHashSizes[ph->cursize_id] < sizehint) {
    ph->cursize_id++;
  }

  ph->cursize = PHashSizes[ph->cursize_id];
  ph->buckets = (PHashLink **)MEM_callocN(ph->cursize * sizeof(*ph->buckets), "PHashBuckets");

  return ph;
}

static void phash_delete(PHash *ph)
{
  if (ph) {
    MEM_SAFE_FREE(ph->buckets);
    MEM_freeN(ph);
  }
}

static int phash_size(PHash *ph)
{
  return ph->size;
}

static void phash_insert(PHash *ph, PHashLink *link)
{
  int size = ph->cursize;
  uintptr_t hash = PHASH_hash(ph, link->key);
  PHashLink *lookup = ph->buckets[hash];

  if (lookup == nullptr) {
    /* insert in front of the list */
    ph->buckets[hash] = link;
    link->next = *(ph->list);
    *(ph->list) = link;
  }
  else {
    /* insert after existing element */
    link->next = lookup->next;
    lookup->next = link;
  }

  ph->size++;

  if (ph->size > (size * 3)) {
    PHashLink *next = nullptr, *first = *(ph->list);

    ph->cursize = PHashSizes[++ph->cursize_id];
    MEM_freeN(ph->buckets);
    ph->buckets = (PHashLink **)MEM_callocN(ph->cursize * sizeof(*ph->buckets), "PHashBuckets");
    ph->size = 0;
    *(ph->list) = nullptr;

    for (link = first; link; link = next) {
      next = link->next;
      phash_insert(ph, link);
    }
  }
}

static PHashLink *phash_lookup(PHash *ph, PHashKey key)
{
  PHashLink *link;
  uintptr_t hash = PHASH_hash(ph, key);

  for (link = ph->buckets[hash]; link; link = link->next) {
    if (link->key == key) {
      return link;
    }
    if (PHASH_hash(ph, link->key) != hash) {
      return nullptr;
    }
  }

  return link;
}

static PHashLink *phash_next(PHash *ph, PHashKey key, PHashLink *link)
{
  uintptr_t hash = PHASH_hash(ph, key);

  for (link = link->next; link; link = link->next) {
    if (link->key == key) {
      return link;
    }
    if (PHASH_hash(ph, link->key) != hash) {
      return nullptr;
    }
  }

  return link;
}

/* Geometry */

static float p_vec_angle(const float v1[3], const float v2[3], const float v3[3])
{
  return angle_v3v3v3(v1, v2, v3);
}
static float p_vec2_angle(const float v1[2], const float v2[2], const float v3[2])
{
  return angle_v2v2v2(v1, v2, v3);
}
static void p_triangle_angles(
    const float v1[3], const float v2[3], const float v3[3], float *r_a1, float *r_a2, float *r_a3)
{
  *r_a1 = p_vec_angle(v3, v1, v2);
  *r_a2 = p_vec_angle(v1, v2, v3);
  *r_a3 = (float)M_PI - *r_a2 - *r_a1;
}

static void p_face_angles(PFace *f, float *r_a1, float *r_a2, float *r_a3)
{
  PEdge *e1 = f->edge, *e2 = e1->next, *e3 = e2->next;
  PVert *v1 = e1->vert, *v2 = e2->vert, *v3 = e3->vert;

  p_triangle_angles(v1->co, v2->co, v3->co, r_a1, r_a2, r_a3);
}

static float p_face_area(PFace *f)
{
  PEdge *e1 = f->edge, *e2 = e1->next, *e3 = e2->next;
  PVert *v1 = e1->vert, *v2 = e2->vert, *v3 = e3->vert;

  return area_tri_v3(v1->co, v2->co, v3->co);
}

static float p_area_signed(const float v1[2], const float v2[2], const float v3[2])
{
  return 0.5f * (((v2[0] - v1[0]) * (v3[1] - v1[1])) - ((v3[0] - v1[0]) * (v2[1] - v1[1])));
}

static float p_face_uv_area_signed(PFace *f)
{
  PEdge *e1 = f->edge, *e2 = e1->next, *e3 = e2->next;
  PVert *v1 = e1->vert, *v2 = e2->vert, *v3 = e3->vert;

  return 0.5f * (((v2->uv[0] - v1->uv[0]) * (v3->uv[1] - v1->uv[1])) -
                 ((v3->uv[0] - v1->uv[0]) * (v2->uv[1] - v1->uv[1])));
}

static float p_edge_length(PEdge *e)
{
  return len_v3v3(e->vert->co, e->next->vert->co);
}

static float p_edge_uv_length(PEdge *e)
{
  return len_v2v2(e->vert->uv, e->next->vert->uv);
}

static void p_chart_uv_bbox(PChart *chart, float minv[2], float maxv[2])
{
  PVert *v;

  INIT_MINMAX2(minv, maxv);

  for (v = chart->verts; v; v = v->nextlink) {
    minmax_v2v2_v2(minv, maxv, v->uv);
  }
}

static float p_chart_uv_area(PChart *chart)
{
  float area = 0.0f;

  for (PFace *f = chart->faces; f; f = f->nextlink) {
    area += fabsf(p_face_uv_area_signed(f));
  }

  return area;
}

static void p_chart_uv_scale(PChart *chart, float scale)
{
  PVert *v;

  for (v = chart->verts; v; v = v->nextlink) {
    v->uv[0] *= scale;
    v->uv[1] *= scale;
  }
}

static void p_chart_uv_scale_xy(PChart *chart, float x, float y)
{
  PVert *v;

  for (v = chart->verts; v; v = v->nextlink) {
    v->uv[0] *= x;
    v->uv[1] *= y;
  }
}

static void p_chart_uv_translate(PChart *chart, const float trans[2])
{
  PVert *v;

  for (v = chart->verts; v; v = v->nextlink) {
    v->uv[0] += trans[0];
    v->uv[1] += trans[1];
  }
}

static void p_chart_uv_transform(PChart *chart, const float mat[2][2])
{
  PVert *v;

  for (v = chart->verts; v; v = v->nextlink) {
    mul_m2_v2(mat, v->uv);
  }
}

static void p_chart_uv_to_array(PChart *chart, float (*points)[2])
{
  PVert *v;
  uint i = 0;

  for (v = chart->verts; v; v = v->nextlink) {
    copy_v2_v2(points[i++], v->uv);
  }
}

static bool p_intersect_line_2d_dir(const float v1[2],
                                    const float dir1[2],
                                    const float v2[2],
                                    const float dir2[2],
                                    float r_isect[2])
{
  float lmbda, div;

  div = dir2[0] * dir1[1] - dir2[1] * dir1[0];

  if (div == 0.0f) {
    return false;
  }

  lmbda = ((v1[1] - v2[1]) * dir1[0] - (v1[0] - v2[0]) * dir1[1]) / div;
  r_isect[0] = v1[0] + lmbda * dir2[0];
  r_isect[1] = v1[1] + lmbda * dir2[1];

  return true;
}

/* Topological Utilities */

static PEdge *p_wheel_edge_next(PEdge *e)
{
  return e->next->next->pair;
}

static PEdge *p_wheel_edge_prev(PEdge *e)
{
  return (e->pair) ? e->pair->next : nullptr;
}

static PEdge *p_boundary_edge_next(PEdge *e)
{
  return e->next->vert->edge;
}

static PEdge *p_boundary_edge_prev(PEdge *e)
{
  PEdge *we = e, *last;

  do {
    last = we;
    we = p_wheel_edge_next(we);
  } while (we && (we != e));

  return last->next->next;
}

static bool p_vert_interior(PVert *v)
{
  return v->edge->pair;
}

static void p_face_flip(PFace *f)
{
  PEdge *e1 = f->edge, *e2 = e1->next, *e3 = e2->next;
  PVert *v1 = e1->vert, *v2 = e2->vert, *v3 = e3->vert;
  int f1 = e1->flag, f2 = e2->flag, f3 = e3->flag;
  float *orig_uv1 = e1->orig_uv, *orig_uv2 = e2->orig_uv, *orig_uv3 = e3->orig_uv;

  e1->vert = v2;
  e1->next = e3;
  e1->orig_uv = orig_uv2;
  e1->flag = (f1 & ~PEDGE_VERTEX_FLAGS) | (f2 & PEDGE_VERTEX_FLAGS);

  e2->vert = v3;
  e2->next = e1;
  e2->orig_uv = orig_uv3;
  e2->flag = (f2 & ~PEDGE_VERTEX_FLAGS) | (f3 & PEDGE_VERTEX_FLAGS);

  e3->vert = v1;
  e3->next = e2;
  e3->orig_uv = orig_uv1;
  e3->flag = (f3 & ~PEDGE_VERTEX_FLAGS) | (f1 & PEDGE_VERTEX_FLAGS);
}

#if 0
static void p_chart_topological_sanity_check(PChart *chart)
{
  PVert *v;
  PEdge *e;

  for (v = chart->verts; v; v = v->nextlink) {
    GEO_uv_parametrizer_test_equals_ptr("v->edge->vert", v, v->edge->vert);
  }

  for (e = chart->edges; e; e = e->nextlink) {
    if (e->pair) {
      GEO_uv_parametrizer_test_equals_ptr("e->pair->pair", e, e->pair->pair);
      GEO_uv_parametrizer_test_equals_ptr("pair->vert", e->vert, e->pair->next->vert);
      GEO_uv_parametrizer_test_equals_ptr("pair->next->vert", e->next->vert, e->pair->vert);
    }
  }
}
#endif

/* Loading / Flushing */

static void p_vert_load_pin_select_uvs(ParamHandle *handle, PVert *v)
{
  PEdge *e;
  int nedges = 0, npins = 0;
  float pinuv[2];

  v->uv[0] = v->uv[1] = 0.0f;
  pinuv[0] = pinuv[1] = 0.0f;
  e = v->edge;
  do {
    if (e->orig_uv) {
      if (e->flag & PEDGE_SELECT) {
        v->flag |= PVERT_SELECT;
      }

      if (e->flag & PEDGE_PIN) {
        pinuv[0] += e->orig_uv[0] * handle->aspx;
        pinuv[1] += e->orig_uv[1] * handle->aspy;
        npins++;
      }
      else {
        v->uv[0] += e->orig_uv[0] * handle->aspx;
        v->uv[1] += e->orig_uv[1] * handle->aspy;
      }

      nedges++;
    }

    e = p_wheel_edge_next(e);
  } while (e && e != (v->edge));

  if (npins > 0) {
    v->uv[0] = pinuv[0] / npins;
    v->uv[1] = pinuv[1] / npins;
    v->flag |= PVERT_PIN;
  }
  else if (nedges > 0) {
    v->uv[0] /= nedges;
    v->uv[1] /= nedges;
  }
}

static void p_flush_uvs(ParamHandle *handle, PChart *chart)
{
  PEdge *e;

  for (e = chart->edges; e; e = e->nextlink) {
    if (e->orig_uv) {
      e->orig_uv[0] = e->vert->uv[0] / handle->aspx;
      e->orig_uv[1] = e->vert->uv[1] / handle->aspy;
    }
  }
}

static void p_flush_uvs_blend(ParamHandle *handle, PChart *chart, float blend)
{
  PEdge *e;
  float invblend = 1.0f - blend;

  for (e = chart->edges; e; e = e->nextlink) {
    if (e->orig_uv) {
      e->orig_uv[0] = blend * e->old_uv[0] + invblend * e->vert->uv[0] / handle->aspx;
      e->orig_uv[1] = blend * e->old_uv[1] + invblend * e->vert->uv[1] / handle->aspy;
    }
  }
}

static void p_face_backup_uvs(PFace *f)
{
  PEdge *e1 = f->edge, *e2 = e1->next, *e3 = e2->next;

  if (e1->orig_uv) {
    e1->old_uv[0] = e1->orig_uv[0];
    e1->old_uv[1] = e1->orig_uv[1];
  }
  if (e2->orig_uv) {
    e2->old_uv[0] = e2->orig_uv[0];
    e2->old_uv[1] = e2->orig_uv[1];
  }
  if (e3->orig_uv) {
    e3->old_uv[0] = e3->orig_uv[0];
    e3->old_uv[1] = e3->orig_uv[1];
  }
}

static void p_face_restore_uvs(PFace *f)
{
  PEdge *e1 = f->edge, *e2 = e1->next, *e3 = e2->next;

  if (e1->orig_uv) {
    e1->orig_uv[0] = e1->old_uv[0];
    e1->orig_uv[1] = e1->old_uv[1];
  }
  if (e2->orig_uv) {
    e2->orig_uv[0] = e2->old_uv[0];
    e2->orig_uv[1] = e2->old_uv[1];
  }
  if (e3->orig_uv) {
    e3->orig_uv[0] = e3->old_uv[0];
    e3->orig_uv[1] = e3->old_uv[1];
  }
}

/* Construction (use only during construction, relies on u.key being set */

static PVert *p_vert_add(ParamHandle *handle, PHashKey key, const float co[3], PEdge *e)
{
  PVert *v = (PVert *)BLI_memarena_alloc(handle->arena, sizeof(*v));
  copy_v3_v3(v->co, co);

  /* Sanity check, a single nan/inf point causes the entire result to be invalid.
   * Note that values within the calculation may _become_ non-finite,
   * so the rest of the code still needs to take this possibility into account. */
  for (int i = 0; i < 3; i++) {
    if (UNLIKELY(!isfinite(v->co[i]))) {
      v->co[i] = 0.0f;
    }
  }

  v->u.key = key;
  v->edge = e;
  v->flag = 0;

  phash_insert(handle->hash_verts, (PHashLink *)v);

  return v;
}

static PVert *p_vert_lookup(ParamHandle *handle, PHashKey key, const float co[3], PEdge *e)
{
  PVert *v = (PVert *)phash_lookup(handle->hash_verts, key);

  if (v) {
    return v;
  }
  return p_vert_add(handle, key, co, e);
}

static PVert *p_vert_copy(ParamHandle *handle, PVert *v)
{
  PVert *nv = (PVert *)BLI_memarena_alloc(handle->arena, sizeof(*nv));

  copy_v3_v3(nv->co, v->co);
  nv->uv[0] = v->uv[0];
  nv->uv[1] = v->uv[1];
  nv->u.key = v->u.key;
  nv->edge = v->edge;
  nv->flag = v->flag;

  return nv;
}

static PEdge *p_edge_lookup(ParamHandle *handle, const PHashKey *vkeys)
{
  PHashKey key = PHASH_edge(vkeys[0], vkeys[1]);
  PEdge *e = (PEdge *)phash_lookup(handle->hash_edges, key);

  while (e) {
    if ((e->vert->u.key == vkeys[0]) && (e->next->vert->u.key == vkeys[1])) {
      return e;
    }
    if ((e->vert->u.key == vkeys[1]) && (e->next->vert->u.key == vkeys[0])) {
      return e;
    }

    e = (PEdge *)phash_next(handle->hash_edges, key, (PHashLink *)e);
  }

  return nullptr;
}

static int p_face_exists(ParamHandle *handle, const ParamKey *pvkeys, int i1, int i2, int i3)
{
  PHashKey *vkeys = (PHashKey *)pvkeys;
  PHashKey key = PHASH_edge(vkeys[i1], vkeys[i2]);
  PEdge *e = (PEdge *)phash_lookup(handle->hash_edges, key);

  while (e) {
    if ((e->vert->u.key == vkeys[i1]) && (e->next->vert->u.key == vkeys[i2])) {
      if (e->next->next->vert->u.key == vkeys[i3]) {
        return true;
      }
    }
    else if ((e->vert->u.key == vkeys[i2]) && (e->next->vert->u.key == vkeys[i1])) {
      if (e->next->next->vert->u.key == vkeys[i3]) {
        return true;
      }
    }

    e = (PEdge *)phash_next(handle->hash_edges, key, (PHashLink *)e);
  }

  return false;
}

static bool p_edge_implicit_seam(PEdge *e, PEdge *ep)
{
  float *uv1, *uv2, *uvp1, *uvp2;
  float limit[2];

  limit[0] = 0.00001;
  limit[1] = 0.00001;

  uv1 = e->orig_uv;
  uv2 = e->next->orig_uv;

  if (e->vert->u.key == ep->vert->u.key) {
    uvp1 = ep->orig_uv;
    uvp2 = ep->next->orig_uv;
  }
  else {
    uvp1 = ep->next->orig_uv;
    uvp2 = ep->orig_uv;
  }

  if ((fabsf(uv1[0] - uvp1[0]) > limit[0]) || (fabsf(uv1[1] - uvp1[1]) > limit[1])) {
    e->flag |= PEDGE_SEAM;
    ep->flag |= PEDGE_SEAM;
    return true;
  }
  if ((fabsf(uv2[0] - uvp2[0]) > limit[0]) || (fabsf(uv2[1] - uvp2[1]) > limit[1])) {
    e->flag |= PEDGE_SEAM;
    ep->flag |= PEDGE_SEAM;
    return true;
  }

  return false;
}

static bool p_edge_has_pair(ParamHandle *handle, PEdge *e, bool topology_from_uvs, PEdge **r_pair)
{
  PHashKey key;
  PEdge *pe;
  PVert *v1, *v2;
  PHashKey key1 = e->vert->u.key;
  PHashKey key2 = e->next->vert->u.key;

  if (e->flag & PEDGE_SEAM) {
    return false;
  }

  key = PHASH_edge(key1, key2);
  pe = (PEdge *)phash_lookup(handle->hash_edges, key);
  *r_pair = nullptr;

  while (pe) {
    if (pe != e) {
      v1 = pe->vert;
      v2 = pe->next->vert;

      if (((v1->u.key == key1) && (v2->u.key == key2)) ||
          ((v1->u.key == key2) && (v2->u.key == key1))) {

        /* don't connect seams and t-junctions */
        if ((pe->flag & PEDGE_SEAM) || *r_pair ||
            (topology_from_uvs && p_edge_implicit_seam(e, pe))) {
          *r_pair = nullptr;
          return false;
        }

        *r_pair = pe;
      }
    }

    pe = (PEdge *)phash_next(handle->hash_edges, key, (PHashLink *)pe);
  }

  if (*r_pair && (e->vert == (*r_pair)->vert)) {
    if ((*r_pair)->next->pair || (*r_pair)->next->next->pair) {
      /* non unfoldable, maybe mobius ring or klein bottle */
      *r_pair = nullptr;
      return false;
    }
  }

  return (*r_pair != nullptr);
}

static bool p_edge_connect_pair(ParamHandle *handle,
                                PEdge *e,
                                bool topology_from_uvs,
                                PEdge ***stack)
{
  PEdge *pair = nullptr;

  if (!e->pair && p_edge_has_pair(handle, e, topology_from_uvs, &pair)) {
    if (e->vert == pair->vert) {
      p_face_flip(pair->face);
    }

    e->pair = pair;
    pair->pair = e;

    if (!(pair->face->flag & PFACE_CONNECTED)) {
      **stack = pair;
      (*stack)++;
    }
  }

  return (e->pair != nullptr);
}

static int p_connect_pairs(ParamHandle *handle, bool topology_from_uvs)
{
  PEdge **stackbase = (PEdge **)MEM_mallocN(sizeof(*stackbase) * phash_size(handle->hash_faces),
                                            "Pstackbase");
  PEdge **stack = stackbase;
  PFace *f, *first;
  PEdge *e, *e1, *e2;
  PChart *chart = handle->construction_chart;
  int ncharts = 0;

  /* Connect pairs, count edges, set vertex-edge pointer to a pair-less edge. */
  for (first = chart->faces; first; first = first->nextlink) {
    if (first->flag & PFACE_CONNECTED) {
      continue;
    }

    *stack = first->edge;
    stack++;

    while (stack != stackbase) {
      stack--;
      e = *stack;
      e1 = e->next;
      e2 = e1->next;

      f = e->face;
      f->flag |= PFACE_CONNECTED;

      /* assign verts to charts so we can sort them later */
      f->u.chart = ncharts;

      if (!p_edge_connect_pair(handle, e, topology_from_uvs, &stack)) {
        e->vert->edge = e;
      }
      if (!p_edge_connect_pair(handle, e1, topology_from_uvs, &stack)) {
        e1->vert->edge = e1;
      }
      if (!p_edge_connect_pair(handle, e2, topology_from_uvs, &stack)) {
        e2->vert->edge = e2;
      }
    }

    ncharts++;
  }

  MEM_freeN(stackbase);

  return ncharts;
}

static void p_split_vert(ParamHandle *handle, PChart *chart, PEdge *e)
{
  PEdge *we, *lastwe = nullptr;
  PVert *v = e->vert;
  bool copy = true;

  if (e->flag & PEDGE_PIN) {
    chart->has_pins = true;
  }

  if (e->flag & PEDGE_VERTEX_SPLIT) {
    return;
  }

  /* rewind to start */
  lastwe = e;
  for (we = p_wheel_edge_prev(e); we && (we != e); we = p_wheel_edge_prev(we)) {
    lastwe = we;
  }

  /* go over all edges in wheel */
  for (we = lastwe; we; we = p_wheel_edge_next(we)) {
    if (we->flag & PEDGE_VERTEX_SPLIT) {
      break;
    }

    we->flag |= PEDGE_VERTEX_SPLIT;

    if (we == v->edge) {
      /* found it, no need to copy */
      copy = false;
      v->nextlink = chart->verts;
      chart->verts = v;
      chart->nverts++;
    }
  }

  if (copy) {
    /* not found, copying */
    v->flag |= PVERT_SPLIT;
    v = p_vert_copy(handle, v);
    v->flag |= PVERT_SPLIT;

    v->nextlink = chart->verts;
    chart->verts = v;
    chart->nverts++;

    v->edge = lastwe;

    we = lastwe;
    do {
      we->vert = v;
      we = p_wheel_edge_next(we);
    } while (we && (we != lastwe));
  }
}

static PChart **p_split_charts(ParamHandle *handle, PChart *chart, int ncharts)
{
  PChart **charts = (PChart **)MEM_callocN(sizeof(*charts) * ncharts, "PCharts");

  for (int i = 0; i < ncharts; i++) {
    charts[i] = (PChart *)MEM_callocN(sizeof(*chart), "PChart");
  }

  PFace *f = chart->faces;
  while (f) {
    PEdge *e1 = f->edge, *e2 = e1->next, *e3 = e2->next;
    PFace *nextf = f->nextlink;

    PChart *nchart = charts[f->u.chart];

    f->nextlink = nchart->faces;
    nchart->faces = f;
    e1->nextlink = nchart->edges;
    nchart->edges = e1;
    e2->nextlink = nchart->edges;
    nchart->edges = e2;
    e3->nextlink = nchart->edges;
    nchart->edges = e3;

    nchart->nfaces++;
    nchart->nedges += 3;

    p_split_vert(handle, nchart, e1);
    p_split_vert(handle, nchart, e2);
    p_split_vert(handle, nchart, e3);

    f = nextf;
  }

  return charts;
}

static PFace *p_face_add(ParamHandle *handle)
{
  PFace *f;
  PEdge *e1, *e2, *e3;

  /* allocate */
  f = (PFace *)BLI_memarena_alloc(handle->arena, sizeof(*f));
  f->flag = 0; /* init ! */

  e1 = (PEdge *)BLI_memarena_alloc(handle->arena, sizeof(*e1));
  e2 = (PEdge *)BLI_memarena_alloc(handle->arena, sizeof(*e2));
  e3 = (PEdge *)BLI_memarena_alloc(handle->arena, sizeof(*e3));

  /* set up edges */
  f->edge = e1;
  e1->face = e2->face = e3->face = f;

  e1->next = e2;
  e2->next = e3;
  e3->next = e1;

  e1->pair = nullptr;
  e2->pair = nullptr;
  e3->pair = nullptr;

  e1->flag = 0;
  e2->flag = 0;
  e3->flag = 0;

  return f;
}

static PFace *p_face_add_construct(ParamHandle *handle,
                                   ParamKey key,
                                   const ParamKey *vkeys,
                                   const float **co,
                                   float **uv,
                                   int i1,
                                   int i2,
                                   int i3,
                                   const bool *pin,
                                   const bool *select)
{
  PFace *f = p_face_add(handle);
  PEdge *e1 = f->edge, *e2 = e1->next, *e3 = e2->next;

  e1->vert = p_vert_lookup(handle, vkeys[i1], co[i1], e1);
  e2->vert = p_vert_lookup(handle, vkeys[i2], co[i2], e2);
  e3->vert = p_vert_lookup(handle, vkeys[i3], co[i3], e3);

  e1->orig_uv = uv[i1];
  e2->orig_uv = uv[i2];
  e3->orig_uv = uv[i3];

  if (pin) {
    if (pin[i1]) {
      e1->flag |= PEDGE_PIN;
    }
    if (pin[i2]) {
      e2->flag |= PEDGE_PIN;
    }
    if (pin[i3]) {
      e3->flag |= PEDGE_PIN;
    }
  }

  if (select) {
    if (select[i1]) {
      e1->flag |= PEDGE_SELECT;
    }
    if (select[i2]) {
      e2->flag |= PEDGE_SELECT;
    }
    if (select[i3]) {
      e3->flag |= PEDGE_SELECT;
    }
  }

  f->u.key = key;
  phash_insert(handle->hash_faces, (PHashLink *)f);

  e1->u.key = PHASH_edge(vkeys[i1], vkeys[i2]);
  e2->u.key = PHASH_edge(vkeys[i2], vkeys[i3]);
  e3->u.key = PHASH_edge(vkeys[i3], vkeys[i1]);

  phash_insert(handle->hash_edges, (PHashLink *)e1);
  phash_insert(handle->hash_edges, (PHashLink *)e2);
  phash_insert(handle->hash_edges, (PHashLink *)e3);

  return f;
}

static PFace *p_face_add_fill(ParamHandle *handle, PChart *chart, PVert *v1, PVert *v2, PVert *v3)
{
  PFace *f = p_face_add(handle);
  PEdge *e1 = f->edge, *e2 = e1->next, *e3 = e2->next;

  e1->vert = v1;
  e2->vert = v2;
  e3->vert = v3;

  e1->orig_uv = e2->orig_uv = e3->orig_uv = nullptr;

  f->nextlink = chart->faces;
  chart->faces = f;
  e1->nextlink = chart->edges;
  chart->edges = e1;
  e2->nextlink = chart->edges;
  chart->edges = e2;
  e3->nextlink = chart->edges;
  chart->edges = e3;

  chart->nfaces++;
  chart->nedges += 3;

  return f;
}

static bool p_quad_split_direction(ParamHandle *handle, const float **co, const ParamKey *vkeys)
{
  /* Slight bias to prefer one edge over the other in case they are equal, so
   * that in symmetric models we choose the same split direction instead of
   * depending on floating point errors to decide. */
  float bias = 1.0f + 1e-6f;
  float fac = len_v3v3(co[0], co[2]) * bias - len_v3v3(co[1], co[3]);
  bool dir = (fac <= 0.0f);

  /* The face exists check is there because of a special case:
   * when two quads share three vertices, they can each be split into two triangles,
   * resulting in two identical triangles. For example in Suzanne's nose. */
  if (dir) {
    if (p_face_exists(handle, vkeys, 0, 1, 2) || p_face_exists(handle, vkeys, 0, 2, 3)) {
      return !dir;
    }
  }
  else {
    if (p_face_exists(handle, vkeys, 0, 1, 3) || p_face_exists(handle, vkeys, 1, 2, 3)) {
      return !dir;
    }
  }

  return dir;
}

/* Construction: boundary filling */

static void p_chart_boundaries(PChart *chart, PEdge **r_outer)
{
  PEdge *e, *be;
  float len, maxlen = -1.0;

  chart->nboundaries = 0;
  if (r_outer) {
    *r_outer = nullptr;
  }

  for (e = chart->edges; e; e = e->nextlink) {
    if (e->pair || (e->flag & PEDGE_DONE)) {
      continue;
    }

    chart->nboundaries++;

    len = 0.0f;

    be = e;
    do {
      be->flag |= PEDGE_DONE;
      len += p_edge_length(be);
      be = be->next->vert->edge;
    } while (be != e);

    if (r_outer && (len > maxlen)) {
      *r_outer = e;
      maxlen = len;
    }
  }

  for (e = chart->edges; e; e = e->nextlink) {
    e->flag &= ~PEDGE_DONE;
  }
}

static float p_edge_boundary_angle(PEdge *e)
{
  PEdge *we;
  PVert *v, *v1, *v2;
  float angle;
  int n = 0;

  v = e->vert;

  /* concave angle check -- could be better */
  angle = M_PI;

  we = v->edge;
  do {
    v1 = we->next->vert;
    v2 = we->next->next->vert;
    angle -= p_vec_angle(v1->co, v->co, v2->co);

    we = we->next->next->pair;
    n++;
  } while (we && (we != v->edge));

  return angle;
}

static void p_chart_fill_boundary(ParamHandle *handle, PChart *chart, PEdge *be, int nedges)
{
  PEdge *e, *e1, *e2;

  PFace *f;
  struct Heap *heap = BLI_heap_new();
  float angle;

  e = be;
  do {
    angle = p_edge_boundary_angle(e);
    e->u.heaplink = BLI_heap_insert(heap, angle, e);

    e = p_boundary_edge_next(e);
  } while (e != be);

  if (nedges == 2) {
    /* no real boundary, but an isolated seam */
    e = be->next->vert->edge;
    e->pair = be;
    be->pair = e;

    BLI_heap_remove(heap, e->u.heaplink);
    BLI_heap_remove(heap, be->u.heaplink);
  }
  else {
    while (nedges > 2) {
      PEdge *ne, *ne1, *ne2;

      e = (PEdge *)BLI_heap_pop_min(heap);

      e1 = p_boundary_edge_prev(e);
      e2 = p_boundary_edge_next(e);

      BLI_heap_remove(heap, e1->u.heaplink);
      BLI_heap_remove(heap, e2->u.heaplink);
      e->u.heaplink = e1->u.heaplink = e2->u.heaplink = nullptr;

      e->flag |= PEDGE_FILLED;
      e1->flag |= PEDGE_FILLED;

      f = p_face_add_fill(handle, chart, e->vert, e1->vert, e2->vert);
      f->flag |= PFACE_FILLED;

      ne = f->edge->next->next;
      ne1 = f->edge;
      ne2 = f->edge->next;

      ne->flag = ne1->flag = ne2->flag = PEDGE_FILLED;

      e->pair = ne;
      ne->pair = e;
      e1->pair = ne1;
      ne1->pair = e1;

      ne->vert = e2->vert;
      ne1->vert = e->vert;
      ne2->vert = e1->vert;

      if (nedges == 3) {
        e2->pair = ne2;
        ne2->pair = e2;
      }
      else {
        ne2->vert->edge = ne2;

        ne2->u.heaplink = BLI_heap_insert(heap, p_edge_boundary_angle(ne2), ne2);
        e2->u.heaplink = BLI_heap_insert(heap, p_edge_boundary_angle(e2), e2);
      }

      nedges--;
    }
  }

  BLI_heap_free(heap, nullptr);
}

static void p_chart_fill_boundaries(ParamHandle *handle, PChart *chart, PEdge *outer)
{
  PEdge *e, *be; /* *enext - as yet unused */
  int nedges;

  for (e = chart->edges; e; e = e->nextlink) {
    /* enext = e->nextlink; - as yet unused */

    if (e->pair || (e->flag & PEDGE_FILLED)) {
      continue;
    }

    nedges = 0;
    be = e;
    do {
      be->flag |= PEDGE_FILLED;
      be = be->next->vert->edge;
      nedges++;
    } while (be != e);

    if (e != outer) {
      p_chart_fill_boundary(handle, chart, e, nedges);
    }
  }
}

#if 0
/* Polygon kernel for inserting uv's non overlapping */

static int p_polygon_point_in(const float cp1[2], const float cp2[2], const float p[2])
{
  if ((cp1[0] == p[0]) && (cp1[1] == p[1])) {
    return 2;
  }
  else if ((cp2[0] == p[0]) && (cp2[1] == p[1])) {
    return 3;
  }
  else {
    return (p_area_signed(cp1, cp2, p) >= 0.0f);
  }
}

static void p_polygon_kernel_clip(float (*oldpoints)[2],
                                  int noldpoints,
                                  float (*newpoints)[2],
                                  int *r_nnewpoints,
                                  const float cp1[2],
                                  const float cp2[2])
{
  float *p2, *p1, isect[2];
  int i, p2in, p1in;

  p1 = oldpoints[noldpoints - 1];
  p1in = p_polygon_point_in(cp1, cp2, p1);
  *r_nnewpoints = 0;

  for (i = 0; i < noldpoints; i++) {
    p2 = oldpoints[i];
    p2in = p_polygon_point_in(cp1, cp2, p2);

    if ((p2in >= 2) || (p1in && p2in)) {
      newpoints[*r_nnewpoints][0] = p2[0];
      newpoints[*r_nnewpoints][1] = p2[1];
      (*r_nnewpoints)++;
    }
    else if (p1in && !p2in) {
      if (p1in != 3) {
        p_intersect_line_2d(p1, p2, cp1, cp2, isect);
        newpoints[*r_nnewpoints][0] = isect[0];
        newpoints[*r_nnewpoints][1] = isect[1];
        (*r_nnewpoints)++;
      }
    }
    else if (!p1in && p2in) {
      p_intersect_line_2d(p1, p2, cp1, cp2, isect);
      newpoints[*r_nnewpoints][0] = isect[0];
      newpoints[*r_nnewpoints][1] = isect[1];
      (*r_nnewpoints)++;

      newpoints[*r_nnewpoints][0] = p2[0];
      newpoints[*r_nnewpoints][1] = p2[1];
      (*r_nnewpoints)++;
    }

    p1in = p2in;
    p1 = p2;
  }
}

static void p_polygon_kernel_center(float (*points)[2], int npoints, float *center)
{
  int i, size, nnewpoints = npoints;
  float(*oldpoints)[2], (*newpoints)[2], *p1, *p2;

  size = npoints * 3;
  oldpoints = MEM_mallocN(sizeof(float[2]) * size, "PPolygonOldPoints");
  newpoints = MEM_mallocN(sizeof(float[2]) * size, "PPolygonNewPoints");

  memcpy(oldpoints, points, sizeof(float[2]) * npoints);

  for (i = 0; i < npoints; i++) {
    p1 = points[i];
    p2 = points[(i + 1) % npoints];
    p_polygon_kernel_clip(oldpoints, nnewpoints, newpoints, &nnewpoints, p1, p2);

    if (nnewpoints == 0) {
      /* degenerate case, use center of original polygon */
      memcpy(oldpoints, points, sizeof(float[2]) * npoints);
      nnewpoints = npoints;
      break;
    }
    else if (nnewpoints == 1) {
      /* degenerate case, use remaining point */
      center[0] = newpoints[0][0];
      center[1] = newpoints[0][1];

      MEM_freeN(oldpoints);
      MEM_freeN(newpoints);

      return;
    }

    if (nnewpoints * 2 > size) {
      size *= 2;
      MEM_freeN(oldpoints);
      oldpoints = MEM_mallocN(sizeof(float[2]) * size, "oldpoints");
      memcpy(oldpoints, newpoints, sizeof(float[2]) * nnewpoints);
      MEM_freeN(newpoints);
      newpoints = MEM_mallocN(sizeof(float[2]) * size, "newpoints");
    }
    else {
      float(*sw_points)[2] = oldpoints;
      oldpoints = newpoints;
      newpoints = sw_points;
    }
  }

  center[0] = center[1] = 0.0f;

  for (i = 0; i < nnewpoints; i++) {
    center[0] += oldpoints[i][0];
    center[1] += oldpoints[i][1];
  }

  center[0] /= nnewpoints;
  center[1] /= nnewpoints;

  MEM_freeN(oldpoints);
  MEM_freeN(newpoints);
}
#endif

#if 0
/* Edge Collapser */

int NCOLLAPSE = 1;
int NCOLLAPSEX = 0;

static float p_vert_cotan(const float v1[3], const float v2[3], const float v3[3])
{
  float a[3], b[3], c[3], clen;

  sub_v3_v3v3(a, v2, v1);
  sub_v3_v3v3(b, v3, v1);
  cross_v3_v3v3(c, a, b);

  clen = len_v3(c);

  if (clen == 0.0f) {
    return 0.0f;
  }

  return dot_v3v3(a, b) / clen;
}

static bool p_vert_flipped_wheel_triangle(PVert *v)
{
  PEdge *e = v->edge;

  do {
    if (p_face_uv_area_signed(e->face) < 0.0f) {
      return true;
    }

    e = p_wheel_edge_next(e);
  } while (e && (e != v->edge));

  return false;
}

static bool p_vert_map_harmonic_weights(PVert *v)
{
  float weightsum, positionsum[2], olduv[2];

  weightsum = 0.0f;
  positionsum[0] = positionsum[1] = 0.0f;

  if (p_vert_interior(v)) {
    PEdge *e = v->edge;

    do {
      float t1, t2, weight;
      PVert *v1, *v2;

      v1 = e->next->vert;
      v2 = e->next->next->vert;
      t1 = p_vert_cotan(v2->co, e->vert->co, v1->co);

      v1 = e->pair->next->vert;
      v2 = e->pair->next->next->vert;
      t2 = p_vert_cotan(v2->co, e->pair->vert->co, v1->co);

      weight = 0.5f * (t1 + t2);
      weightsum += weight;
      positionsum[0] += weight * e->pair->vert->uv[0];
      positionsum[1] += weight * e->pair->vert->uv[1];

      e = p_wheel_edge_next(e);
    } while (e && (e != v->edge));
  }
  else {
    PEdge *e = v->edge;

    do {
      float t1, t2;
      PVert *v1, *v2;

      v2 = e->next->vert;
      v1 = e->next->next->vert;

      t1 = p_vert_cotan(v1->co, v->co, v2->co);
      t2 = p_vert_cotan(v2->co, v->co, v1->co);

      weightsum += t1 + t2;
      positionsum[0] += (v2->uv[1] - v1->uv[1]) + (t1 * v2->uv[0] + t2 * v1->uv[0]);
      positionsum[1] += (v1->uv[0] - v2->uv[0]) + (t1 * v2->uv[1] + t2 * v1->uv[1]);

      e = p_wheel_edge_next(e);
    } while (e && (e != v->edge));
  }

  if (weightsum != 0.0f) {
    weightsum = 1.0f / weightsum;
    positionsum[0] *= weightsum;
    positionsum[1] *= weightsum;
  }

  olduv[0] = v->uv[0];
  olduv[1] = v->uv[1];
  v->uv[0] = positionsum[0];
  v->uv[1] = positionsum[1];

  if (p_vert_flipped_wheel_triangle(v)) {
    v->uv[0] = olduv[0];
    v->uv[1] = olduv[1];

    return false;
  }

  return true;
}

static void p_vert_harmonic_insert(PVert *v)
{
  PEdge *e;

  if (!p_vert_map_harmonic_weights(v)) {
    /* do polygon kernel center insertion: this is quite slow, but should
     * only be needed for 0.01 % of verts or so, when insert with harmonic
     * weights fails */

    int npoints = 0, i;
    float(*points)[2];

    e = v->edge;
    do {
      npoints++;
      e = p_wheel_edge_next(e);
    } while (e && (e != v->edge));

    if (e == nullptr) {
      npoints++;
    }

    points = MEM_mallocN(sizeof(float[2]) * npoints, "PHarmonicPoints");

    e = v->edge;
    i = 0;
    do {
      PEdge *nexte = p_wheel_edge_next(e);

      points[i][0] = e->next->vert->uv[0];
      points[i][1] = e->next->vert->uv[1];

      if (nexte == nullptr) {
        i++;
        points[i][0] = e->next->next->vert->uv[0];
        points[i][1] = e->next->next->vert->uv[1];
        break;
      }

      e = nexte;
      i++;
    } while (e != v->edge);

    p_polygon_kernel_center(points, npoints, v->uv);

    MEM_freeN(points);
  }

  e = v->edge;
  do {
    if (!(e->next->vert->flag & PVERT_PIN)) {
      p_vert_map_harmonic_weights(e->next->vert);
    }
    e = p_wheel_edge_next(e);
  } while (e && (e != v->edge));

  p_vert_map_harmonic_weights(v);
}

static void p_vert_fix_edge_pointer(PVert *v)
{
  PEdge *start = v->edge;

  /* set v->edge pointer to the edge with no pair, if there is one */
  while (v->edge->pair) {
    v->edge = p_wheel_edge_prev(v->edge);

    if (v->edge == start) {
      break;
    }
  }
}

static void p_collapsing_verts(PEdge *edge, PEdge *pair, PVert **r_newv, PVert **r_keepv)
{
  /* the two vertices that are involved in the collapse */
  if (edge) {
    *r_newv = edge->vert;
    *r_keepv = edge->next->vert;
  }
  else {
    *r_newv = pair->next->vert;
    *r_keepv = pair->vert;
  }
}

static void p_collapse_edge(PEdge *edge, PEdge *pair)
{
  PVert *oldv, *keepv;
  PEdge *e;

  p_collapsing_verts(edge, pair, &oldv, &keepv);

  /* change e->vert pointers from old vertex to the target vertex */
  e = oldv->edge;
  do {
    if ((e != edge) && !(pair && pair->next == e)) {
      e->vert = keepv;
    }

    e = p_wheel_edge_next(e);
  } while (e && (e != oldv->edge));

  /* set keepv->edge pointer */
  if ((edge && (keepv->edge == edge->next)) || (keepv->edge == pair)) {
    if (edge && edge->next->pair) {
      keepv->edge = edge->next->pair->next;
    }
    else if (pair && pair->next->next->pair) {
      keepv->edge = pair->next->next->pair;
    }
    else if (edge && edge->next->next->pair) {
      keepv->edge = edge->next->next->pair;
    }
    else {
      keepv->edge = pair->next->pair->next;
    }
  }

  /* update pairs and v->edge pointers */
  if (edge) {
    PEdge *e1 = edge->next, *e2 = e1->next;

    if (e1->pair) {
      e1->pair->pair = e2->pair;
    }

    if (e2->pair) {
      e2->pair->pair = e1->pair;
      e2->vert->edge = p_wheel_edge_prev(e2);
    }
    else {
      e2->vert->edge = p_wheel_edge_next(e2);
    }

    p_vert_fix_edge_pointer(e2->vert);
  }

  if (pair) {
    PEdge *e1 = pair->next, *e2 = e1->next;

    if (e1->pair) {
      e1->pair->pair = e2->pair;
    }

    if (e2->pair) {
      e2->pair->pair = e1->pair;
      e2->vert->edge = p_wheel_edge_prev(e2);
    }
    else {
      e2->vert->edge = p_wheel_edge_next(e2);
    }

    p_vert_fix_edge_pointer(e2->vert);
  }

  p_vert_fix_edge_pointer(keepv);

  /* mark for move to collapsed list later */
  oldv->flag |= PVERT_COLLAPSE;

  if (edge) {
    PFace *f = edge->face;
    PEdge *e1 = edge->next, *e2 = e1->next;

    f->flag |= PFACE_COLLAPSE;
    edge->flag |= PEDGE_COLLAPSE;
    e1->flag |= PEDGE_COLLAPSE;
    e2->flag |= PEDGE_COLLAPSE;
  }

  if (pair) {
    PFace *f = pair->face;
    PEdge *e1 = pair->next, *e2 = e1->next;

    f->flag |= PFACE_COLLAPSE;
    pair->flag |= PEDGE_COLLAPSE;
    e1->flag |= PEDGE_COLLAPSE;
    e2->flag |= PEDGE_COLLAPSE;
  }
}

static void p_split_vertex(PEdge *edge, PEdge *pair)
{
  PVert *newv, *keepv;
  PEdge *e;

  p_collapsing_verts(edge, pair, &newv, &keepv);

  /* update edge pairs */
  if (edge) {
    PEdge *e1 = edge->next, *e2 = e1->next;

    if (e1->pair) {
      e1->pair->pair = e1;
    }
    if (e2->pair) {
      e2->pair->pair = e2;
    }

    e2->vert->edge = e2;
    p_vert_fix_edge_pointer(e2->vert);
    keepv->edge = e1;
  }

  if (pair) {
    PEdge *e1 = pair->next, *e2 = e1->next;

    if (e1->pair) {
      e1->pair->pair = e1;
    }
    if (e2->pair) {
      e2->pair->pair = e2;
    }

    e2->vert->edge = e2;
    p_vert_fix_edge_pointer(e2->vert);
    keepv->edge = pair;
  }

  p_vert_fix_edge_pointer(keepv);

  /* set e->vert pointers to restored vertex */
  e = newv->edge;
  do {
    e->vert = newv;
    e = p_wheel_edge_next(e);
  } while (e && (e != newv->edge));
}

static bool p_collapse_allowed_topologic(PEdge *edge, PEdge *pair)
{
  PVert *oldv, *keepv;

  p_collapsing_verts(edge, pair, &oldv, &keepv);

  /* boundary edges */
  if (!edge || !pair) {
    /* avoid collapsing chart into an edge */
    if (edge && !edge->next->pair && !edge->next->next->pair) {
      return false;
    }
    else if (pair && !pair->next->pair && !pair->next->next->pair) {
      return false;
    }
  }
  /* avoid merging two boundaries (oldv and keepv are on the 'other side' of
   * the chart) */
  else if (!p_vert_interior(oldv) && !p_vert_interior(keepv)) {
    return false;
  }

  return true;
}

static bool p_collapse_normal_flipped(float *v1, float *v2, float *vold, float *vnew)
{
  float nold[3], nnew[3], sub1[3], sub2[3];

  sub_v3_v3v3(sub1, vold, v1);
  sub_v3_v3v3(sub2, vold, v2);
  cross_v3_v3v3(nold, sub1, sub2);

  sub_v3_v3v3(sub1, vnew, v1);
  sub_v3_v3v3(sub2, vnew, v2);
  cross_v3_v3v3(nnew, sub1, sub2);

  return (dot_v3v3(nold, nnew) <= 0.0f);
}

static bool p_collapse_allowed_geometric(PEdge *edge, PEdge *pair)
{
  PVert *oldv, *keepv;
  PEdge *e;
  float angulardefect, angle;

  p_collapsing_verts(edge, pair, &oldv, &keepv);

  angulardefect = 2 * M_PI;

  e = oldv->edge;
  do {
    float a[3], b[3], minangle, maxangle;
    PEdge *e1 = e->next, *e2 = e1->next;
    PVert *v1 = e1->vert, *v2 = e2->vert;
    int i;

    angle = p_vec_angle(v1->co, oldv->co, v2->co);
    angulardefect -= angle;

    /* skip collapsing faces */
    if (v1 == keepv || v2 == keepv) {
      e = p_wheel_edge_next(e);
      continue;
    }

    if (p_collapse_normal_flipped(v1->co, v2->co, oldv->co, keepv->co)) {
      return false;
    }

    a[0] = angle;
    a[1] = p_vec_angle(v2->co, v1->co, oldv->co);
    a[2] = M_PI - a[0] - a[1];

    b[0] = p_vec_angle(v1->co, keepv->co, v2->co);
    b[1] = p_vec_angle(v2->co, v1->co, keepv->co);
    b[2] = M_PI - b[0] - b[1];

    /* ABF criterion 1: avoid sharp and obtuse angles. */
    minangle = 15.0f * M_PI / 180.0f;
    maxangle = M_PI - minangle;

    for (i = 0; i < 3; i++) {
      if ((b[i] < a[i]) && (b[i] < minangle)) {
        return false;
      }
      else if ((b[i] > a[i]) && (b[i] > maxangle)) {
        return false;
      }
    }

    e = p_wheel_edge_next(e);
  } while (e && (e != oldv->edge));

  if (p_vert_interior(oldv)) {
    /* HLSCM criterion: angular defect smaller than threshold. */
    if (fabsf(angulardefect) > (float)(M_PI * 30.0 / 180.0)) {
      return false;
    }
  }
  else {
    PVert *v1 = p_boundary_edge_next(oldv->edge)->vert;
    PVert *v2 = p_boundary_edge_prev(oldv->edge)->vert;

    /* ABF++ criterion 2: avoid collapsing verts inwards. */
    if (p_vert_interior(keepv)) {
      return false;
    }

    /* Don't collapse significant boundary changes. */
    angle = p_vec_angle(v1->co, oldv->co, v2->co);
    if (angle < (M_PI * 160.0 / 180.0)) {
      return false;
    }
  }

  return true;
}

static bool p_collapse_allowed(PEdge *edge, PEdge *pair)
{
  PVert *oldv, *keepv;

  p_collapsing_verts(edge, pair, &oldv, &keepv);

  if (oldv->flag & PVERT_PIN) {
    return false;
  }

  return (p_collapse_allowed_topologic(edge, pair) && p_collapse_allowed_geometric(edge, pair));
}

static float p_collapse_cost(PEdge *edge, PEdge *pair)
{
  /* based on volume and boundary optimization from:
   * "Fast and Memory Efficient Polygonal Simplification" P. Lindstrom, G. Turk */

  PVert *oldv, *keepv;
  PEdge *e;
  PFace *oldf1, *oldf2;
  float volumecost = 0.0f, areacost = 0.0f, edgevec[3], cost, weight, elen;
  float shapecost = 0.0f;
  float shapeold = 0.0f, shapenew = 0.0f;
  int nshapeold = 0, nshapenew = 0;

  p_collapsing_verts(edge, pair, &oldv, &keepv);
  oldf1 = (edge) ? edge->face : nullptr;
  oldf2 = (pair) ? pair->face : nullptr;

  sub_v3_v3v3(edgevec, keepv->co, oldv->co);

  e = oldv->edge;
  do {
    float a1, a2, a3;
    float *co1 = e->next->vert->co;
    float *co2 = e->next->next->vert->co;

    if (!ELEM(e->face, oldf1, oldf2)) {
      float tetrav2[3], tetrav3[3];

      /* tetrahedron volume = (1/3!)*|a.(b x c)| */
      sub_v3_v3v3(tetrav2, co1, oldv->co);
      sub_v3_v3v3(tetrav3, co2, oldv->co);
      volumecost += fabsf(volume_tri_tetrahedron_signed_v3(tetrav2, tetrav3, edgevec));

#  if 0
      shapecost += dot_v3v3(co1, keepv->co);

      if (p_wheel_edge_next(e) == nullptr) {
        shapecost += dot_v3v3(co2, keepv->co);
      }
#  endif

      p_triangle_angles(oldv->co, co1, co2, &a1, &a2, &a3);
      a1 = a1 - M_PI / 3.0;
      a2 = a2 - M_PI / 3.0;
      a3 = a3 - M_PI / 3.0;
      shapeold = (a1 * a1 + a2 * a2 + a3 * a3) / (M_PI_2 * M_PI_2);

      nshapeold++;
    }
    else {
      p_triangle_angles(keepv->co, co1, co2, &a1, &a2, &a3);
      a1 = a1 - M_PI / 3.0;
      a2 = a2 - M_PI / 3.0;
      a3 = a3 - M_PI / 3.0;
      shapenew = (a1 * a1 + a2 * a2 + a3 * a3) / (M_PI_2 * M_PI_2);

      nshapenew++;
    }

    e = p_wheel_edge_next(e);
  } while (e && (e != oldv->edge));

  if (!p_vert_interior(oldv)) {
    PVert *v1 = p_boundary_edge_prev(oldv->edge)->vert;
    PVert *v2 = p_boundary_edge_next(oldv->edge)->vert;

    areacost = area_tri_v3(oldv->co, v1->co, v2->co);
  }

  elen = len_v3(edgevec);
  weight = 1.0f; /* 0.2f */
  cost = weight * volumecost * volumecost + elen * elen * areacost * areacost;
#  if 0
  cost += shapecost;
#  else
  shapeold /= nshapeold;
  shapenew /= nshapenew;
  shapecost = (shapeold + 0.00001) / (shapenew + 0.00001);

  cost *= shapecost;
#  endif

  return cost;
}

static void p_collapse_cost_vertex(PVert *vert, float *r_mincost, PEdge **r_mine)
{
  PEdge *e, *enext, *pair;

  *r_mine = nullptr;
  *r_mincost = 0.0f;
  e = vert->edge;
  do {
    if (p_collapse_allowed(e, e->pair)) {
      float cost = p_collapse_cost(e, e->pair);

      if ((*r_mine == nullptr) || (cost < *r_mincost)) {
        *r_mincost = cost;
        *r_mine = e;
      }
    }

    enext = p_wheel_edge_next(e);

    if (enext == nullptr) {
      /* The other boundary edge, where we only have the pair half-edge. */
      pair = e->next->next;

      if (p_collapse_allowed(nullptr, pair)) {
        float cost = p_collapse_cost(nullptr, pair);

        if ((*r_mine == nullptr) || (cost < *r_mincost)) {
          *r_mincost = cost;
          *r_mine = pair;
        }
      }

      break;
    }

    e = enext;
  } while (e != vert->edge);
}

static void p_chart_post_collapse_flush(PChart *chart, PEdge *collapsed)
{
  /* Move to `collapsed_*`. */

  PVert *v, *nextv = nullptr, *verts = chart->verts;
  PEdge *e, *nexte = nullptr, *edges = chart->edges, *laste = nullptr;
  PFace *f, *nextf = nullptr, *faces = chart->faces;

  chart->verts = chart->collapsed_verts = nullptr;
  chart->edges = chart->collapsed_edges = nullptr;
  chart->faces = chart->collapsed_faces = nullptr;

  chart->nverts = chart->nedges = chart->nfaces = 0;

  for (v = verts; v; v = nextv) {
    nextv = v->nextlink;

    if (v->flag & PVERT_COLLAPSE) {
      v->nextlink = chart->collapsed_verts;
      chart->collapsed_verts = v;
    }
    else {
      v->nextlink = chart->verts;
      chart->verts = v;
      chart->nverts++;
    }
  }

  for (e = edges; e; e = nexte) {
    nexte = e->nextlink;

    if (!collapsed || !(e->flag & PEDGE_COLLAPSE_EDGE)) {
      if (e->flag & PEDGE_COLLAPSE) {
        e->nextlink = chart->collapsed_edges;
        chart->collapsed_edges = e;
      }
      else {
        e->nextlink = chart->edges;
        chart->edges = e;
        chart->nedges++;
      }
    }
  }

  /* these are added last so they can be popped of in the right order
   * for splitting */
  for (e = collapsed; e; e = e->nextlink) {
    e->nextlink = e->u.nextcollapse;
    laste = e;
  }
  if (laste) {
    laste->nextlink = chart->collapsed_edges;
    chart->collapsed_edges = collapsed;
  }

  for (f = faces; f; f = nextf) {
    nextf = f->nextlink;

    if (f->flag & PFACE_COLLAPSE) {
      f->nextlink = chart->collapsed_faces;
      chart->collapsed_faces = f;
    }
    else {
      f->nextlink = chart->faces;
      chart->faces = f;
      chart->nfaces++;
    }
  }
}

static void p_chart_post_split_flush(PChart *chart)
{
  /* Move from `collapsed_*`. */

  PVert *v, *nextv = nullptr;
  PEdge *e, *nexte = nullptr;
  PFace *f, *nextf = nullptr;

  for (v = chart->collapsed_verts; v; v = nextv) {
    nextv = v->nextlink;
    v->nextlink = chart->verts;
    chart->verts = v;
    chart->nverts++;
  }

  for (e = chart->collapsed_edges; e; e = nexte) {
    nexte = e->nextlink;
    e->nextlink = chart->edges;
    chart->edges = e;
    chart->nedges++;
  }

  for (f = chart->collapsed_faces; f; f = nextf) {
    nextf = f->nextlink;
    f->nextlink = chart->faces;
    chart->faces = f;
    chart->nfaces++;
  }

  chart->collapsed_verts = nullptr;
  chart->collapsed_edges = nullptr;
  chart->collapsed_faces = nullptr;
}

static void p_chart_simplify_compute(PChart *chart)
{
  /* Computes a list of edge collapses / vertex splits. The collapsed
   * simplices go in the `chart->collapsed_*` lists, The original and
   * collapsed may then be view as stacks, where the next collapse/split
   * is at the top of the respective lists. */

  Heap *heap = BLI_heap_new();
  PVert *v, **wheelverts;
  PEdge *collapsededges = nullptr, *e;
  int nwheelverts, i, ncollapsed = 0;

  wheelverts = MEM_mallocN(sizeof(PVert *) * chart->nverts, "PChartWheelVerts");

  /* insert all potential collapses into heap */
  for (v = chart->verts; v; v = v->nextlink) {
    float cost;
    PEdge *e = nullptr;

    p_collapse_cost_vertex(v, &cost, &e);

    if (e) {
      v->u.heaplink = BLI_heap_insert(heap, cost, e);
    }
    else {
      v->u.heaplink = nullptr;
    }
  }

  for (e = chart->edges; e; e = e->nextlink) {
    e->u.nextcollapse = nullptr;
  }

  /* pop edge collapse out of heap one by one */
  while (!BLI_heap_is_empty(heap)) {
    if (ncollapsed == NCOLLAPSE) {
      break;
    }

    HeapNode *link = BLI_heap_top(heap);
    PEdge *edge = (PEdge *)BLI_heap_pop_min(heap), *pair = edge->pair;
    PVert *oldv, *keepv;
    PEdge *wheele, *nexte;

    /* remember the edges we collapsed */
    edge->u.nextcollapse = collapsededges;
    collapsededges = edge;

    if (edge->vert->u.heaplink != link) {
      edge->flag |= (PEDGE_COLLAPSE_EDGE | PEDGE_COLLAPSE_PAIR);
      edge->next->vert->u.heaplink = nullptr;
      SWAP(PEdge *, edge, pair);
    }
    else {
      edge->flag |= PEDGE_COLLAPSE_EDGE;
      edge->vert->u.heaplink = nullptr;
    }

    p_collapsing_verts(edge, pair, &oldv, &keepv);

    /* gather all wheel verts and remember them before collapse */
    nwheelverts = 0;
    wheele = oldv->edge;

    do {
      wheelverts[nwheelverts++] = wheele->next->vert;
      nexte = p_wheel_edge_next(wheele);

      if (nexte == nullptr) {
        wheelverts[nwheelverts++] = wheele->next->next->vert;
      }

      wheele = nexte;
    } while (wheele && (wheele != oldv->edge));

    /* collapse */
    p_collapse_edge(edge, pair);

    for (i = 0; i < nwheelverts; i++) {
      float cost;
      PEdge *collapse = nullptr;

      v = wheelverts[i];

      if (v->u.heaplink) {
        BLI_heap_remove(heap, v->u.heaplink);
        v->u.heaplink = nullptr;
      }

      p_collapse_cost_vertex(v, &cost, &collapse);

      if (collapse) {
        v->u.heaplink = BLI_heap_insert(heap, cost, collapse);
      }
    }

    ncollapsed++;
  }

  MEM_freeN(wheelverts);
  BLI_heap_free(heap, nullptr);

  p_chart_post_collapse_flush(chart, collapsededges);
}

static void p_chart_complexify(PChart *chart)
{
  PEdge *e, *pair, *edge;
  PVert *newv, *keepv;
  int x = 0;

  for (e = chart->collapsed_edges; e; e = e->nextlink) {
    if (!(e->flag & PEDGE_COLLAPSE_EDGE)) {
      break;
    }

    edge = e;
    pair = e->pair;

    if (edge->flag & PEDGE_COLLAPSE_PAIR) {
      SWAP(PEdge *, edge, pair);
    }

    p_split_vertex(edge, pair);
    p_collapsing_verts(edge, pair, &newv, &keepv);

    if (x >= NCOLLAPSEX) {
      newv->uv[0] = keepv->uv[0];
      newv->uv[1] = keepv->uv[1];
    }
    else {
      p_vert_harmonic_insert(newv);
      x++;
    }
  }

  p_chart_post_split_flush(chart);
}

#  if 0
static void p_chart_simplify(PChart *chart)
{
  /* Not implemented, needs proper reordering in split_flush. */
}
#  endif
#endif

/* ABF */

#define ABF_MAX_ITER 20

using PAbfSystem = struct PAbfSystem {
  int ninterior, nfaces, nangles;
  float *alpha, *beta, *sine, *cosine, *weight;
  float *bAlpha, *bTriangle, *bInterior;
  float *lambdaTriangle, *lambdaPlanar, *lambdaLength;
  float (*J2dt)[3], *bstar, *dstar;
  float minangle, maxangle;
};

static void p_abf_setup_system(PAbfSystem *sys)
{
  int i;

  sys->alpha = (float *)MEM_mallocN(sizeof(float) * sys->nangles, "ABFalpha");
  sys->beta = (float *)MEM_mallocN(sizeof(float) * sys->nangles, "ABFbeta");
  sys->sine = (float *)MEM_mallocN(sizeof(float) * sys->nangles, "ABFsine");
  sys->cosine = (float *)MEM_mallocN(sizeof(float) * sys->nangles, "ABFcosine");
  sys->weight = (float *)MEM_mallocN(sizeof(float) * sys->nangles, "ABFweight");

  sys->bAlpha = (float *)MEM_mallocN(sizeof(float) * sys->nangles, "ABFbalpha");
  sys->bTriangle = (float *)MEM_mallocN(sizeof(float) * sys->nfaces, "ABFbtriangle");
  sys->bInterior = (float *)MEM_mallocN(sizeof(float[2]) * sys->ninterior, "ABFbinterior");

  sys->lambdaTriangle = (float *)MEM_callocN(sizeof(float) * sys->nfaces, "ABFlambdatri");
  sys->lambdaPlanar = (float *)MEM_callocN(sizeof(float) * sys->ninterior, "ABFlamdaplane");
  sys->lambdaLength = (float *)MEM_mallocN(sizeof(float) * sys->ninterior, "ABFlambdalen");

  sys->J2dt = static_cast<float(*)[3]>(MEM_mallocN(sizeof(float) * sys->nangles * 3, "ABFj2dt"));
  sys->bstar = (float *)MEM_mallocN(sizeof(float) * sys->nfaces, "ABFbstar");
  sys->dstar = (float *)MEM_mallocN(sizeof(float) * sys->nfaces, "ABFdstar");

  for (i = 0; i < sys->ninterior; i++) {
    sys->lambdaLength[i] = 1.0;
  }

  sys->minangle = 1.0 * M_PI / 180.0;
  sys->maxangle = (float)M_PI - sys->minangle;
}

static void p_abf_free_system(PAbfSystem *sys)
{
  MEM_freeN(sys->alpha);
  MEM_freeN(sys->beta);
  MEM_freeN(sys->sine);
  MEM_freeN(sys->cosine);
  MEM_freeN(sys->weight);
  MEM_freeN(sys->bAlpha);
  MEM_freeN(sys->bTriangle);
  MEM_freeN(sys->bInterior);
  MEM_freeN(sys->lambdaTriangle);
  MEM_freeN(sys->lambdaPlanar);
  MEM_freeN(sys->lambdaLength);
  MEM_freeN(sys->J2dt);
  MEM_freeN(sys->bstar);
  MEM_freeN(sys->dstar);
}

static void p_abf_compute_sines(PAbfSystem *sys)
{
  int i;
  float *sine = sys->sine, *cosine = sys->cosine, *alpha = sys->alpha;

  for (i = 0; i < sys->nangles; i++, sine++, cosine++, alpha++) {
    *sine = sinf(*alpha);
    *cosine = cosf(*alpha);
  }
}

static float p_abf_compute_sin_product(PAbfSystem *sys, PVert *v, int aid)
{
  PEdge *e, *e1, *e2;
  float sin1, sin2;

  sin1 = sin2 = 1.0;

  e = v->edge;
  do {
    e1 = e->next;
    e2 = e->next->next;

    if (aid == e1->u.id) {
      /* we are computing a derivative for this angle,
       * so we use cos and drop the other part */
      sin1 *= sys->cosine[e1->u.id];
      sin2 = 0.0;
    }
    else {
      sin1 *= sys->sine[e1->u.id];
    }

    if (aid == e2->u.id) {
      /* see above */
      sin1 = 0.0;
      sin2 *= sys->cosine[e2->u.id];
    }
    else {
      sin2 *= sys->sine[e2->u.id];
    }

    e = e->next->next->pair;
  } while (e && (e != v->edge));

  return (sin1 - sin2);
}

static float p_abf_compute_grad_alpha(PAbfSystem *sys, PFace *f, PEdge *e)
{
  PVert *v = e->vert, *v1 = e->next->vert, *v2 = e->next->next->vert;
  float deriv;

  deriv = (sys->alpha[e->u.id] - sys->beta[e->u.id]) * sys->weight[e->u.id];
  deriv += sys->lambdaTriangle[f->u.id];

  if (v->flag & PVERT_INTERIOR) {
    deriv += sys->lambdaPlanar[v->u.id];
  }

  if (v1->flag & PVERT_INTERIOR) {
    float product = p_abf_compute_sin_product(sys, v1, e->u.id);
    deriv += sys->lambdaLength[v1->u.id] * product;
  }

  if (v2->flag & PVERT_INTERIOR) {
    float product = p_abf_compute_sin_product(sys, v2, e->u.id);
    deriv += sys->lambdaLength[v2->u.id] * product;
  }

  return deriv;
}

static float p_abf_compute_gradient(PAbfSystem *sys, PChart *chart)
{
  PFace *f;
  PEdge *e;
  PVert *v;
  float norm = 0.0;

  for (f = chart->faces; f; f = f->nextlink) {
    PEdge *e1 = f->edge, *e2 = e1->next, *e3 = e2->next;
    float gtriangle, galpha1, galpha2, galpha3;

    galpha1 = p_abf_compute_grad_alpha(sys, f, e1);
    galpha2 = p_abf_compute_grad_alpha(sys, f, e2);
    galpha3 = p_abf_compute_grad_alpha(sys, f, e3);

    sys->bAlpha[e1->u.id] = -galpha1;
    sys->bAlpha[e2->u.id] = -galpha2;
    sys->bAlpha[e3->u.id] = -galpha3;

    norm += galpha1 * galpha1 + galpha2 * galpha2 + galpha3 * galpha3;

    gtriangle = sys->alpha[e1->u.id] + sys->alpha[e2->u.id] + sys->alpha[e3->u.id] - (float)M_PI;
    sys->bTriangle[f->u.id] = -gtriangle;
    norm += gtriangle * gtriangle;
  }

  for (v = chart->verts; v; v = v->nextlink) {
    if (v->flag & PVERT_INTERIOR) {
      float gplanar = -2 * M_PI, glength;

      e = v->edge;
      do {
        gplanar += sys->alpha[e->u.id];
        e = e->next->next->pair;
      } while (e && (e != v->edge));

      sys->bInterior[v->u.id] = -gplanar;
      norm += gplanar * gplanar;

      glength = p_abf_compute_sin_product(sys, v, -1);
      sys->bInterior[sys->ninterior + v->u.id] = -glength;
      norm += glength * glength;
    }
  }

  return norm;
}

static bool p_abf_matrix_invert(PAbfSystem *sys, PChart *chart)
{
  int ninterior = sys->ninterior;
  int nvar = 2 * ninterior;
  LinearSolver *context = EIG_linear_solver_new(0, nvar, 1);

  for (int i = 0; i < nvar; i++) {
    EIG_linear_solver_right_hand_side_add(context, 0, i, sys->bInterior[i]);
  }

  for (PFace *f = chart->faces; f; f = f->nextlink) {
    float wi1, wi2, wi3, b, si, beta[3], j2[3][3], W[3][3];
    float row1[6], row2[6], row3[6];
    int vid[6];
    PEdge *e1 = f->edge, *e2 = e1->next, *e3 = e2->next;
    PVert *v1 = e1->vert, *v2 = e2->vert, *v3 = e3->vert;

    wi1 = 1.0f / sys->weight[e1->u.id];
    wi2 = 1.0f / sys->weight[e2->u.id];
    wi3 = 1.0f / sys->weight[e3->u.id];

    /* bstar1 = (J1*dInv*bAlpha - bTriangle) */
    b = sys->bAlpha[e1->u.id] * wi1;
    b += sys->bAlpha[e2->u.id] * wi2;
    b += sys->bAlpha[e3->u.id] * wi3;
    b -= sys->bTriangle[f->u.id];

    /* si = J1*d*J1t */
    si = 1.0f / (wi1 + wi2 + wi3);

    /* J1t*si*bstar1 - bAlpha */
    beta[0] = b * si - sys->bAlpha[e1->u.id];
    beta[1] = b * si - sys->bAlpha[e2->u.id];
    beta[2] = b * si - sys->bAlpha[e3->u.id];

    /* use this later for computing other lambda's */
    sys->bstar[f->u.id] = b;
    sys->dstar[f->u.id] = si;

    /* set matrix */
    W[0][0] = si - sys->weight[e1->u.id];
    W[0][1] = si;
    W[0][2] = si;
    W[1][0] = si;
    W[1][1] = si - sys->weight[e2->u.id];
    W[1][2] = si;
    W[2][0] = si;
    W[2][1] = si;
    W[2][2] = si - sys->weight[e3->u.id];

    vid[0] = vid[1] = vid[2] = vid[3] = vid[4] = vid[5] = -1;

    if (v1->flag & PVERT_INTERIOR) {
      vid[0] = v1->u.id;
      vid[3] = ninterior + v1->u.id;

      sys->J2dt[e1->u.id][0] = j2[0][0] = 1.0f * wi1;
      sys->J2dt[e2->u.id][0] = j2[1][0] = p_abf_compute_sin_product(sys, v1, e2->u.id) * wi2;
      sys->J2dt[e3->u.id][0] = j2[2][0] = p_abf_compute_sin_product(sys, v1, e3->u.id) * wi3;

      EIG_linear_solver_right_hand_side_add(context, 0, v1->u.id, j2[0][0] * beta[0]);
      EIG_linear_solver_right_hand_side_add(
          context, 0, ninterior + v1->u.id, j2[1][0] * beta[1] + j2[2][0] * beta[2]);

      row1[0] = j2[0][0] * W[0][0];
      row2[0] = j2[0][0] * W[1][0];
      row3[0] = j2[0][0] * W[2][0];

      row1[3] = j2[1][0] * W[0][1] + j2[2][0] * W[0][2];
      row2[3] = j2[1][0] * W[1][1] + j2[2][0] * W[1][2];
      row3[3] = j2[1][0] * W[2][1] + j2[2][0] * W[2][2];
    }

    if (v2->flag & PVERT_INTERIOR) {
      vid[1] = v2->u.id;
      vid[4] = ninterior + v2->u.id;

      sys->J2dt[e1->u.id][1] = j2[0][1] = p_abf_compute_sin_product(sys, v2, e1->u.id) * wi1;
      sys->J2dt[e2->u.id][1] = j2[1][1] = 1.0f * wi2;
      sys->J2dt[e3->u.id][1] = j2[2][1] = p_abf_compute_sin_product(sys, v2, e3->u.id) * wi3;

      EIG_linear_solver_right_hand_side_add(context, 0, v2->u.id, j2[1][1] * beta[1]);
      EIG_linear_solver_right_hand_side_add(
          context, 0, ninterior + v2->u.id, j2[0][1] * beta[0] + j2[2][1] * beta[2]);

      row1[1] = j2[1][1] * W[0][1];
      row2[1] = j2[1][1] * W[1][1];
      row3[1] = j2[1][1] * W[2][1];

      row1[4] = j2[0][1] * W[0][0] + j2[2][1] * W[0][2];
      row2[4] = j2[0][1] * W[1][0] + j2[2][1] * W[1][2];
      row3[4] = j2[0][1] * W[2][0] + j2[2][1] * W[2][2];
    }

    if (v3->flag & PVERT_INTERIOR) {
      vid[2] = v3->u.id;
      vid[5] = ninterior + v3->u.id;

      sys->J2dt[e1->u.id][2] = j2[0][2] = p_abf_compute_sin_product(sys, v3, e1->u.id) * wi1;
      sys->J2dt[e2->u.id][2] = j2[1][2] = p_abf_compute_sin_product(sys, v3, e2->u.id) * wi2;
      sys->J2dt[e3->u.id][2] = j2[2][2] = 1.0f * wi3;

      EIG_linear_solver_right_hand_side_add(context, 0, v3->u.id, j2[2][2] * beta[2]);
      EIG_linear_solver_right_hand_side_add(
          context, 0, ninterior + v3->u.id, j2[0][2] * beta[0] + j2[1][2] * beta[1]);

      row1[2] = j2[2][2] * W[0][2];
      row2[2] = j2[2][2] * W[1][2];
      row3[2] = j2[2][2] * W[2][2];

      row1[5] = j2[0][2] * W[0][0] + j2[1][2] * W[0][1];
      row2[5] = j2[0][2] * W[1][0] + j2[1][2] * W[1][1];
      row3[5] = j2[0][2] * W[2][0] + j2[1][2] * W[2][1];
    }

    for (int i = 0; i < 3; i++) {
      int r = vid[i];

      if (r == -1) {
        continue;
      }

      for (int j = 0; j < 6; j++) {
        int c = vid[j];

        if (c == -1) {
          continue;
        }

        if (i == 0) {
          EIG_linear_solver_matrix_add(context, r, c, j2[0][i] * row1[j]);
        }
        else {
          EIG_linear_solver_matrix_add(context, r + ninterior, c, j2[0][i] * row1[j]);
        }

        if (i == 1) {
          EIG_linear_solver_matrix_add(context, r, c, j2[1][i] * row2[j]);
        }
        else {
          EIG_linear_solver_matrix_add(context, r + ninterior, c, j2[1][i] * row2[j]);
        }

        if (i == 2) {
          EIG_linear_solver_matrix_add(context, r, c, j2[2][i] * row3[j]);
        }
        else {
          EIG_linear_solver_matrix_add(context, r + ninterior, c, j2[2][i] * row3[j]);
        }
      }
    }
  }

  bool success = EIG_linear_solver_solve(context);

  if (success) {
    for (PFace *f = chart->faces; f; f = f->nextlink) {
      float dlambda1, pre[3], dalpha;
      PEdge *e1 = f->edge, *e2 = e1->next, *e3 = e2->next;
      PVert *v1 = e1->vert, *v2 = e2->vert, *v3 = e3->vert;

      pre[0] = pre[1] = pre[2] = 0.0;

      if (v1->flag & PVERT_INTERIOR) {
        float x = EIG_linear_solver_variable_get(context, 0, v1->u.id);
        float x2 = EIG_linear_solver_variable_get(context, 0, ninterior + v1->u.id);
        pre[0] += sys->J2dt[e1->u.id][0] * x;
        pre[1] += sys->J2dt[e2->u.id][0] * x2;
        pre[2] += sys->J2dt[e3->u.id][0] * x2;
      }

      if (v2->flag & PVERT_INTERIOR) {
        float x = EIG_linear_solver_variable_get(context, 0, v2->u.id);
        float x2 = EIG_linear_solver_variable_get(context, 0, ninterior + v2->u.id);
        pre[0] += sys->J2dt[e1->u.id][1] * x2;
        pre[1] += sys->J2dt[e2->u.id][1] * x;
        pre[2] += sys->J2dt[e3->u.id][1] * x2;
      }

      if (v3->flag & PVERT_INTERIOR) {
        float x = EIG_linear_solver_variable_get(context, 0, v3->u.id);
        float x2 = EIG_linear_solver_variable_get(context, 0, ninterior + v3->u.id);
        pre[0] += sys->J2dt[e1->u.id][2] * x2;
        pre[1] += sys->J2dt[e2->u.id][2] * x2;
        pre[2] += sys->J2dt[e3->u.id][2] * x;
      }

      dlambda1 = pre[0] + pre[1] + pre[2];
      dlambda1 = sys->dstar[f->u.id] * (sys->bstar[f->u.id] - dlambda1);

      sys->lambdaTriangle[f->u.id] += dlambda1;

      dalpha = (sys->bAlpha[e1->u.id] - dlambda1);
      sys->alpha[e1->u.id] += dalpha / sys->weight[e1->u.id] - pre[0];

      dalpha = (sys->bAlpha[e2->u.id] - dlambda1);
      sys->alpha[e2->u.id] += dalpha / sys->weight[e2->u.id] - pre[1];

      dalpha = (sys->bAlpha[e3->u.id] - dlambda1);
      sys->alpha[e3->u.id] += dalpha / sys->weight[e3->u.id] - pre[2];

      /* clamp */
      PEdge *e = f->edge;
      do {
        if (sys->alpha[e->u.id] > (float)M_PI) {
          sys->alpha[e->u.id] = (float)M_PI;
        }
        else if (sys->alpha[e->u.id] < 0.0f) {
          sys->alpha[e->u.id] = 0.0f;
        }
      } while (e != f->edge);
    }

    for (int i = 0; i < ninterior; i++) {
      sys->lambdaPlanar[i] += (float)EIG_linear_solver_variable_get(context, 0, i);
      sys->lambdaLength[i] += (float)EIG_linear_solver_variable_get(context, 0, ninterior + i);
    }
  }

  EIG_linear_solver_delete(context);

  return success;
}

static bool p_chart_abf_solve(PChart *chart)
{
  PVert *v;
  PFace *f;
  PEdge *e, *e1, *e2, *e3;
  PAbfSystem sys;
  int i;
  float /* lastnorm, */ /* UNUSED */ limit = (chart->nfaces > 100) ? 1.0f : 0.001f;

  /* setup id's */
  sys.ninterior = sys.nfaces = sys.nangles = 0;

  for (v = chart->verts; v; v = v->nextlink) {
    if (p_vert_interior(v)) {
      v->flag |= PVERT_INTERIOR;
      v->u.id = sys.ninterior++;
    }
    else {
      v->flag &= ~PVERT_INTERIOR;
    }
  }

  for (f = chart->faces; f; f = f->nextlink) {
    e1 = f->edge;
    e2 = e1->next;
    e3 = e2->next;
    f->u.id = sys.nfaces++;

    /* angle id's are conveniently stored in half edges */
    e1->u.id = sys.nangles++;
    e2->u.id = sys.nangles++;
    e3->u.id = sys.nangles++;
  }

  p_abf_setup_system(&sys);

  /* compute initial angles */
  for (f = chart->faces; f; f = f->nextlink) {
    float a1, a2, a3;

    e1 = f->edge;
    e2 = e1->next;
    e3 = e2->next;
    p_face_angles(f, &a1, &a2, &a3);

    if (a1 < sys.minangle) {
      a1 = sys.minangle;
    }
    else if (a1 > sys.maxangle) {
      a1 = sys.maxangle;
    }
    if (a2 < sys.minangle) {
      a2 = sys.minangle;
    }
    else if (a2 > sys.maxangle) {
      a2 = sys.maxangle;
    }
    if (a3 < sys.minangle) {
      a3 = sys.minangle;
    }
    else if (a3 > sys.maxangle) {
      a3 = sys.maxangle;
    }

    sys.alpha[e1->u.id] = sys.beta[e1->u.id] = a1;
    sys.alpha[e2->u.id] = sys.beta[e2->u.id] = a2;
    sys.alpha[e3->u.id] = sys.beta[e3->u.id] = a3;

    sys.weight[e1->u.id] = 2.0f / (a1 * a1);
    sys.weight[e2->u.id] = 2.0f / (a2 * a2);
    sys.weight[e3->u.id] = 2.0f / (a3 * a3);
  }

  for (v = chart->verts; v; v = v->nextlink) {
    if (v->flag & PVERT_INTERIOR) {
      float anglesum = 0.0, scale;

      e = v->edge;
      do {
        anglesum += sys.beta[e->u.id];
        e = e->next->next->pair;
      } while (e && (e != v->edge));

      scale = (anglesum == 0.0f) ? 0.0f : 2.0f * (float)M_PI / anglesum;

      e = v->edge;
      do {
        sys.beta[e->u.id] = sys.alpha[e->u.id] = sys.beta[e->u.id] * scale;
        e = e->next->next->pair;
      } while (e && (e != v->edge));
    }
  }

  if (sys.ninterior > 0) {
    p_abf_compute_sines(&sys);

    /* iteration */
    /* lastnorm = 1e10; */ /* UNUSED */

    for (i = 0; i < ABF_MAX_ITER; i++) {
      float norm = p_abf_compute_gradient(&sys, chart);

      /* lastnorm = norm; */ /* UNUSED */

      if (norm < limit) {
        break;
      }

      if (!p_abf_matrix_invert(&sys, chart)) {
        param_warning("ABF failed to invert matrix");
        p_abf_free_system(&sys);
        return false;
      }

      p_abf_compute_sines(&sys);
    }

    if (i == ABF_MAX_ITER) {
      param_warning("ABF maximum iterations reached");
      p_abf_free_system(&sys);
      return false;
    }
  }

  chart->u.lscm.abf_alpha = (float *)MEM_dupallocN(sys.alpha);
  p_abf_free_system(&sys);

  return true;
}

/* Least Squares Conformal Maps */

static void p_chart_pin_positions(PChart *chart, PVert **pin1, PVert **pin2)
{
  if (!*pin1 || !*pin2 || *pin1 == *pin2) {
    /* degenerate case */
    PFace *f = chart->faces;
    *pin1 = f->edge->vert;
    *pin2 = f->edge->next->vert;

    (*pin1)->uv[0] = 0.0f;
    (*pin1)->uv[1] = 0.5f;
    (*pin2)->uv[0] = 1.0f;
    (*pin2)->uv[1] = 0.5f;
  }
  else {
    int diru, dirv, dirx, diry;
    float sub[3];

    sub_v3_v3v3(sub, (*pin1)->co, (*pin2)->co);
    sub[0] = fabsf(sub[0]);
    sub[1] = fabsf(sub[1]);
    sub[2] = fabsf(sub[2]);

    if ((sub[0] > sub[1]) && (sub[0] > sub[2])) {
      dirx = 0;
      diry = (sub[1] > sub[2]) ? 1 : 2;
    }
    else if ((sub[1] > sub[0]) && (sub[1] > sub[2])) {
      dirx = 1;
      diry = (sub[0] > sub[2]) ? 0 : 2;
    }
    else {
      dirx = 2;
      diry = (sub[0] > sub[1]) ? 0 : 1;
    }

    if (dirx == 2) {
      diru = 1;
      dirv = 0;
    }
    else {
      diru = 0;
      dirv = 1;
    }

    (*pin1)->uv[diru] = (*pin1)->co[dirx];
    (*pin1)->uv[dirv] = (*pin1)->co[diry];
    (*pin2)->uv[diru] = (*pin2)->co[dirx];
    (*pin2)->uv[dirv] = (*pin2)->co[diry];
  }
}

static bool p_chart_symmetry_pins(PChart *chart, PEdge *outer, PVert **pin1, PVert **pin2)
{
  PEdge *be, *lastbe = nullptr, *maxe1 = nullptr, *maxe2 = nullptr, *be1, *be2;
  PEdge *cure = nullptr, *firste1 = nullptr, *firste2 = nullptr, *nextbe;
  float maxlen = 0.0f, curlen = 0.0f, totlen = 0.0f, firstlen = 0.0f;
  float len1, len2;

  /* find longest series of verts split in the chart itself, these are
   * marked during construction */
  be = outer;
  lastbe = p_boundary_edge_prev(be);
  do {
    float len = p_edge_length(be);
    totlen += len;

    nextbe = p_boundary_edge_next(be);

    if ((be->vert->flag & PVERT_SPLIT) ||
        (lastbe->vert->flag & nextbe->vert->flag & PVERT_SPLIT)) {
      if (!cure) {
        if (be == outer) {
          firste1 = be;
        }
        cure = be;
      }
      else {
        curlen += p_edge_length(lastbe);
      }
    }
    else if (cure) {
      if (curlen > maxlen) {
        maxlen = curlen;
        maxe1 = cure;
        maxe2 = lastbe;
      }

      if (firste1 == cure) {
        firstlen = curlen;
        firste2 = lastbe;
      }

      curlen = 0.0f;
      cure = nullptr;
    }

    lastbe = be;
    be = nextbe;
  } while (be != outer);

  /* make sure we also count a series of splits over the starting point */
  if (cure && (cure != outer)) {
    firstlen += curlen + p_edge_length(be);

    if (firstlen > maxlen) {
      maxlen = firstlen;
      maxe1 = cure;
      maxe2 = firste2;
    }
  }

  if (!maxe1 || !maxe2 || (maxlen < 0.5f * totlen)) {
    return false;
  }

  /* find pin1 in the split vertices */
  be1 = maxe1;
  be2 = maxe2;
  len1 = 0.0f;
  len2 = 0.0f;

  do {
    if (len1 < len2) {
      len1 += p_edge_length(be1);
      be1 = p_boundary_edge_next(be1);
    }
    else {
      be2 = p_boundary_edge_prev(be2);
      len2 += p_edge_length(be2);
    }
  } while (be1 != be2);

  *pin1 = be1->vert;

  /* find pin2 outside the split vertices */
  be1 = maxe1;
  be2 = maxe2;
  len1 = 0.0f;
  len2 = 0.0f;

  do {
    if (len1 < len2) {
      be1 = p_boundary_edge_prev(be1);
      len1 += p_edge_length(be1);
    }
    else {
      len2 += p_edge_length(be2);
      be2 = p_boundary_edge_next(be2);
    }
  } while (be1 != be2);

  *pin2 = be1->vert;

  p_chart_pin_positions(chart, pin1, pin2);

  return !equals_v3v3((*pin1)->co, (*pin2)->co);
}

static void p_chart_extrema_verts(PChart *chart, PVert **pin1, PVert **pin2)
{
  float minv[3], maxv[3], dirlen;
  PVert *v, *minvert[3], *maxvert[3];
  int i, dir;

  /* find minimum and maximum verts over x/y/z axes */
  minv[0] = minv[1] = minv[2] = 1e20;
  maxv[0] = maxv[1] = maxv[2] = -1e20;

  minvert[0] = minvert[1] = minvert[2] = nullptr;
  maxvert[0] = maxvert[1] = maxvert[2] = nullptr;

  for (v = chart->verts; v; v = v->nextlink) {
    for (i = 0; i < 3; i++) {
      if (v->co[i] < minv[i]) {
        minv[i] = v->co[i];
        minvert[i] = v;
      }
      if (v->co[i] > maxv[i]) {
        maxv[i] = v->co[i];
        maxvert[i] = v;
      }
    }
  }

  /* find axes with longest distance */
  dir = 0;
  dirlen = -1.0;

  for (i = 0; i < 3; i++) {
    if (maxv[i] - minv[i] > dirlen) {
      dir = i;
      dirlen = maxv[i] - minv[i];
    }
  }

  *pin1 = minvert[dir];
  *pin2 = maxvert[dir];

  p_chart_pin_positions(chart, pin1, pin2);
}

static void p_chart_lscm_load_solution(PChart *chart)
{
  LinearSolver *context = chart->u.lscm.context;
  PVert *v;

  for (v = chart->verts; v; v = v->nextlink) {
    v->uv[0] = EIG_linear_solver_variable_get(context, 0, 2 * v->u.id);
    v->uv[1] = EIG_linear_solver_variable_get(context, 0, 2 * v->u.id + 1);
  }
}

static void p_chart_lscm_begin(PChart *chart, bool live, bool abf)
{
  PVert *v, *pin1, *pin2;
  bool select = false, deselect = false;
  int npins = 0, id = 0;

  /* give vertices matrix indices and count pins */
  for (v = chart->verts; v; v = v->nextlink) {
    if (v->flag & PVERT_PIN) {
      npins++;
      if (v->flag & PVERT_SELECT) {
        select = true;
      }
    }

    if (!(v->flag & PVERT_SELECT)) {
      deselect = true;
    }
  }

  if ((live && (!select || !deselect))) {
    chart->u.lscm.context = nullptr;
  }
  else {
#if 0
    p_chart_simplify_compute(chart);
    p_chart_topological_sanity_check(chart);
#endif

    if (npins == 1) {
      chart->u.lscm.single_pin_area = p_chart_uv_area(chart);
      for (v = chart->verts; v; v = v->nextlink) {
        if (v->flag & PVERT_PIN) {
          chart->u.lscm.single_pin = v;
          break;
        }
      }
    }

    if (abf) {
      if (!p_chart_abf_solve(chart)) {
        param_warning("ABF solving failed: falling back to LSCM.\n");
      }
    }

    if (npins <= 1) {
      /* No pins, let's find some ourself. */
      PEdge *outer;

      p_chart_boundaries(chart, &outer);

      /* Outer can be NULL with non-finite coords. */
      if (!(outer && p_chart_symmetry_pins(chart, outer, &pin1, &pin2))) {
        p_chart_extrema_verts(chart, &pin1, &pin2);
      }

      chart->u.lscm.pin1 = pin1;
      chart->u.lscm.pin2 = pin2;
    }

    for (v = chart->verts; v; v = v->nextlink) {
      v->u.id = id++;
    }

    chart->u.lscm.context = EIG_linear_least_squares_solver_new(
        2 * chart->nfaces, 2 * chart->nverts, 1);
  }
}

static bool p_chart_lscm_solve(ParamHandle *handle, PChart *chart)
{
  LinearSolver *context = chart->u.lscm.context;
  PVert *v, *pin1 = chart->u.lscm.pin1, *pin2 = chart->u.lscm.pin2;
  PFace *f;
  const float *alpha = chart->u.lscm.abf_alpha;
  float area_pinned_up, area_pinned_down;
  bool flip_faces;
  int row;

#if 0
  /* TODO: make loading pins work for simplify/complexify. */
#endif

  for (v = chart->verts; v; v = v->nextlink) {
    if (v->flag & PVERT_PIN) {
      p_vert_load_pin_select_uvs(handle, v); /* reload for live */
    }
  }

  if (chart->u.lscm.single_pin) {
    /* If only one pin, save area and pin for transform later. */
    copy_v2_v2(chart->u.lscm.single_pin_uv, chart->u.lscm.single_pin->uv);
  }

  if (chart->u.lscm.pin1) {
    EIG_linear_solver_variable_lock(context, 2 * pin1->u.id);
    EIG_linear_solver_variable_lock(context, 2 * pin1->u.id + 1);
    EIG_linear_solver_variable_lock(context, 2 * pin2->u.id);
    EIG_linear_solver_variable_lock(context, 2 * pin2->u.id + 1);

    EIG_linear_solver_variable_set(context, 0, 2 * pin1->u.id, pin1->uv[0]);
    EIG_linear_solver_variable_set(context, 0, 2 * pin1->u.id + 1, pin1->uv[1]);
    EIG_linear_solver_variable_set(context, 0, 2 * pin2->u.id, pin2->uv[0]);
    EIG_linear_solver_variable_set(context, 0, 2 * pin2->u.id + 1, pin2->uv[1]);
  }
  else {
    /* set and lock the pins */
    for (v = chart->verts; v; v = v->nextlink) {
      if (v->flag & PVERT_PIN) {
        EIG_linear_solver_variable_lock(context, 2 * v->u.id);
        EIG_linear_solver_variable_lock(context, 2 * v->u.id + 1);

        EIG_linear_solver_variable_set(context, 0, 2 * v->u.id, v->uv[0]);
        EIG_linear_solver_variable_set(context, 0, 2 * v->u.id + 1, v->uv[1]);
      }
    }
  }

  /* detect up direction based on pinned vertices */
  area_pinned_up = 0.0f;
  area_pinned_down = 0.0f;

  for (f = chart->faces; f; f = f->nextlink) {
    PEdge *e1 = f->edge, *e2 = e1->next, *e3 = e2->next;
    PVert *v1 = e1->vert, *v2 = e2->vert, *v3 = e3->vert;

    if ((v1->flag & PVERT_PIN) && (v2->flag & PVERT_PIN) && (v3->flag & PVERT_PIN)) {
      float area = p_face_uv_area_signed(f);

      if (area > 0.0f) {
        area_pinned_up += area;
      }
      else {
        area_pinned_down -= area;
      }
    }
  }

  flip_faces = (area_pinned_down > area_pinned_up);

  /* construct matrix */

  row = 0;
  for (f = chart->faces; f; f = f->nextlink) {
    PEdge *e1 = f->edge, *e2 = e1->next, *e3 = e2->next;
    PVert *v1 = e1->vert, *v2 = e2->vert, *v3 = e3->vert;
    float a1, a2, a3, ratio, cosine, sine;
    float sina1, sina2, sina3, sinmax;

    if (alpha) {
      /* use abf angles if passed on */
      a1 = *(alpha++);
      a2 = *(alpha++);
      a3 = *(alpha++);
    }
    else {
      p_face_angles(f, &a1, &a2, &a3);
    }

    if (flip_faces) {
      SWAP(float, a2, a3);
      SWAP(PEdge *, e2, e3);
      SWAP(PVert *, v2, v3);
    }

    sina1 = sinf(a1);
    sina2 = sinf(a2);
    sina3 = sinf(a3);

    sinmax = max_fff(sina1, sina2, sina3);

    /* shift vertices to find most stable order */
    if (sina3 != sinmax) {
      SHIFT3(PVert *, v1, v2, v3);
      SHIFT3(float, a1, a2, a3);
      SHIFT3(float, sina1, sina2, sina3);

      if (sina2 == sinmax) {
        SHIFT3(PVert *, v1, v2, v3);
        SHIFT3(float, a1, a2, a3);
        SHIFT3(float, sina1, sina2, sina3);
      }
    }

    /* angle based lscm formulation */
    ratio = (sina3 == 0.0f) ? 1.0f : sina2 / sina3;
    cosine = cosf(a1) * ratio;
    sine = sina1 * ratio;

    EIG_linear_solver_matrix_add(context, row, 2 * v1->u.id, cosine - 1.0f);
    EIG_linear_solver_matrix_add(context, row, 2 * v1->u.id + 1, -sine);
    EIG_linear_solver_matrix_add(context, row, 2 * v2->u.id, -cosine);
    EIG_linear_solver_matrix_add(context, row, 2 * v2->u.id + 1, sine);
    EIG_linear_solver_matrix_add(context, row, 2 * v3->u.id, 1.0);
    row++;

    EIG_linear_solver_matrix_add(context, row, 2 * v1->u.id, sine);
    EIG_linear_solver_matrix_add(context, row, 2 * v1->u.id + 1, cosine - 1.0f);
    EIG_linear_solver_matrix_add(context, row, 2 * v2->u.id, -sine);
    EIG_linear_solver_matrix_add(context, row, 2 * v2->u.id + 1, -cosine);
    EIG_linear_solver_matrix_add(context, row, 2 * v3->u.id + 1, 1.0);
    row++;
  }

  if (EIG_linear_solver_solve(context)) {
    p_chart_lscm_load_solution(chart);
    return true;
  }

  for (v = chart->verts; v; v = v->nextlink) {
    v->uv[0] = 0.0f;
    v->uv[1] = 0.0f;
  }

  return false;
}

static void p_chart_lscm_transform_single_pin(PChart *chart)
{
  PVert *pin = chart->u.lscm.single_pin;

  /* If only one pin, keep UV area the same. */
  const float new_area = p_chart_uv_area(chart);
  if (new_area > 0.0f) {
    const float scale = chart->u.lscm.single_pin_area / new_area;
    if (scale > 0.0f) {
      p_chart_uv_scale(chart, sqrtf(scale));
    }
  }

  /* Translate to keep the pinned vertex in place. */
  float offset[2];
  sub_v2_v2v2(offset, chart->u.lscm.single_pin_uv, pin->uv);
  p_chart_uv_translate(chart, offset);
}

static void p_chart_lscm_end(PChart *chart)
{
  EIG_linear_solver_delete(chart->u.lscm.context);
  chart->u.lscm.context = nullptr;

  MEM_SAFE_FREE(chart->u.lscm.abf_alpha);

  chart->u.lscm.pin1 = nullptr;
  chart->u.lscm.pin2 = nullptr;
  chart->u.lscm.single_pin = nullptr;
  chart->u.lscm.single_pin_area = 0.0f;
}

/* Stretch */

#define P_STRETCH_ITER 20

static void p_stretch_pin_boundary(PChart *chart)
{
  PVert *v;

  for (v = chart->verts; v; v = v->nextlink) {
    if (v->edge->pair == nullptr) {
      v->flag |= PVERT_PIN;
    }
    else {
      v->flag &= ~PVERT_PIN;
    }
  }
}

static float p_face_stretch(PFace *f)
{
  float T, w, tmp[3];
  float Ps[3], Pt[3];
  float a, c, area;
  PEdge *e1 = f->edge, *e2 = e1->next, *e3 = e2->next;
  PVert *v1 = e1->vert, *v2 = e2->vert, *v3 = e3->vert;

  area = p_face_uv_area_signed(f);

  if (area <= 0.0f) {
    /* When a face is flipped, provide a large penalty.
     * Add on a slight gradient to unflip the face, see also: T99781. */
    return 1e8f * (1.0f + p_edge_uv_length(e1) + p_edge_uv_length(e2) + p_edge_uv_length(e3));
  }

  w = 1.0f / (2.0f * area);

  /* compute derivatives */
  copy_v3_v3(Ps, v1->co);
  mul_v3_fl(Ps, (v2->uv[1] - v3->uv[1]));

  copy_v3_v3(tmp, v2->co);
  mul_v3_fl(tmp, (v3->uv[1] - v1->uv[1]));
  add_v3_v3(Ps, tmp);

  copy_v3_v3(tmp, v3->co);
  mul_v3_fl(tmp, (v1->uv[1] - v2->uv[1]));
  add_v3_v3(Ps, tmp);

  mul_v3_fl(Ps, w);

  copy_v3_v3(Pt, v1->co);
  mul_v3_fl(Pt, (v3->uv[0] - v2->uv[0]));

  copy_v3_v3(tmp, v2->co);
  mul_v3_fl(tmp, (v1->uv[0] - v3->uv[0]));
  add_v3_v3(Pt, tmp);

  copy_v3_v3(tmp, v3->co);
  mul_v3_fl(tmp, (v2->uv[0] - v1->uv[0]));
  add_v3_v3(Pt, tmp);

  mul_v3_fl(Pt, w);

  /* Sander Tensor */
  a = dot_v3v3(Ps, Ps);
  c = dot_v3v3(Pt, Pt);

  T = sqrtf(0.5f * (a + c));
  if (f->flag & PFACE_FILLED) {
    T *= 0.2f;
  }

  return T;
}

static float p_stretch_compute_vertex(PVert *v)
{
  PEdge *e = v->edge;
  float sum = 0.0f;

  do {
    sum += p_face_stretch(e->face);
    e = p_wheel_edge_next(e);
  } while (e && e != (v->edge));

  return sum;
}

static void p_chart_stretch_minimize(PChart *chart, RNG *rng)
{
  PVert *v;
  PEdge *e;
  int j, nedges;
  float orig_stretch, low, stretch_low, high, stretch_high, mid, stretch;
  float orig_uv[2], dir[2], random_angle, trusted_radius;

  for (v = chart->verts; v; v = v->nextlink) {
    if ((v->flag & PVERT_PIN) || !(v->flag & PVERT_SELECT)) {
      continue;
    }

    orig_stretch = p_stretch_compute_vertex(v);
    orig_uv[0] = v->uv[0];
    orig_uv[1] = v->uv[1];

    /* move vertex in a random direction */
    trusted_radius = 0.0f;
    nedges = 0;
    e = v->edge;

    do {
      trusted_radius += p_edge_uv_length(e);
      nedges++;

      e = p_wheel_edge_next(e);
    } while (e && e != (v->edge));

    trusted_radius /= 2 * nedges;

    random_angle = BLI_rng_get_float(rng) * 2.0f * (float)M_PI;
    dir[0] = trusted_radius * cosf(random_angle);
    dir[1] = trusted_radius * sinf(random_angle);

    /* calculate old and new stretch */
    low = 0;
    stretch_low = orig_stretch;

    add_v2_v2v2(v->uv, orig_uv, dir);
    high = 1;
    stretch = stretch_high = p_stretch_compute_vertex(v);

    /* binary search for lowest stretch position */
    for (j = 0; j < P_STRETCH_ITER; j++) {
      mid = 0.5f * (low + high);
      v->uv[0] = orig_uv[0] + mid * dir[0];
      v->uv[1] = orig_uv[1] + mid * dir[1];
      stretch = p_stretch_compute_vertex(v);

      if (stretch_low < stretch_high) {
        high = mid;
        stretch_high = stretch;
      }
      else {
        low = mid;
        stretch_low = stretch;
      }
    }

    /* no luck, stretch has increased, reset to old values */
    if (stretch >= orig_stretch) {
      copy_v2_v2(v->uv, orig_uv);
    }
  }
}

/* Minimum area enclosing rectangle for packing */

static int p_compare_geometric_uv(const void *a, const void *b)
{
  const PVert *v1 = *(const PVert *const *)a;
  const PVert *v2 = *(const PVert *const *)b;

  if (v1->uv[0] < v2->uv[0]) {
    return -1;
  }
  if (v1->uv[0] == v2->uv[0]) {
    if (v1->uv[1] < v2->uv[1]) {
      return -1;
    }
    if (v1->uv[1] == v2->uv[1]) {
      return 0;
    }
    return 1;
  }
  return 1;
}

static bool p_chart_convex_hull(PChart *chart, PVert ***r_verts, int *r_nverts, int *r_right)
{
  /* Graham algorithm, taken from:
   * http://aspn.activestate.com/ASPN/Cookbook/Python/Recipe/117225 */

  PEdge *be, *e;
  int npoints = 0, i, ulen, llen;
  PVert **U, **L, **points, **p;

  p_chart_boundaries(chart, &be);

  if (!be) {
    return false;
  }

  e = be;
  do {
    npoints++;
    e = p_boundary_edge_next(e);
  } while (e != be);

  p = points = (PVert **)MEM_mallocN(sizeof(PVert *) * npoints * 2, "PCHullpoints");
  U = (PVert **)MEM_mallocN(sizeof(PVert *) * npoints, "PCHullU");
  L = (PVert **)MEM_mallocN(sizeof(PVert *) * npoints, "PCHullL");

  e = be;
  do {
    *p = e->vert;
    p++;
    e = p_boundary_edge_next(e);
  } while (e != be);

  qsort(points, npoints, sizeof(PVert *), p_compare_geometric_uv);

  ulen = llen = 0;
  for (p = points, i = 0; i < npoints; i++, p++) {
    while ((ulen > 1) && (p_area_signed(U[ulen - 2]->uv, (*p)->uv, U[ulen - 1]->uv) <= 0)) {
      ulen--;
    }
    while ((llen > 1) && (p_area_signed(L[llen - 2]->uv, (*p)->uv, L[llen - 1]->uv) >= 0)) {
      llen--;
    }

    U[ulen] = *p;
    ulen++;
    L[llen] = *p;
    llen++;
  }

  npoints = 0;
  for (p = points, i = 0; i < ulen; i++, p++, npoints++) {
    *p = U[i];
  }

  /* the first and last point in L are left out, since they are also in U */
  for (i = llen - 2; i > 0; i--, p++, npoints++) {
    *p = L[i];
  }

  *r_verts = points;
  *r_nverts = npoints;
  *r_right = ulen - 1;

  MEM_freeN(U);
  MEM_freeN(L);

  return true;
}

static float p_rectangle_area(float *p1, float *dir, float *p2, float *p3, float *p4)
{
  /* given 4 points on the rectangle edges and the direction of on edge,
   * compute the area of the rectangle */

  float orthodir[2], corner1[2], corner2[2], corner3[2];

  orthodir[0] = dir[1];
  orthodir[1] = -dir[0];

  if (!p_intersect_line_2d_dir(p1, dir, p2, orthodir, corner1)) {
    return 1e10;
  }

  if (!p_intersect_line_2d_dir(p1, dir, p4, orthodir, corner2)) {
    return 1e10;
  }

  if (!p_intersect_line_2d_dir(p3, dir, p4, orthodir, corner3)) {
    return 1e10;
  }

  return len_v2v2(corner1, corner2) * len_v2v2(corner2, corner3);
}

static float p_chart_minimum_area_angle(PChart *chart)
{
  /* minimum area enclosing rectangle with rotating calipers, info:
   * http://cgm.cs.mcgill.ca/~orm/maer.html */

  float rotated, minarea, minangle, area, len;
  float *angles, miny, maxy, v[2], a[4], mina;
  int npoints, right, i_min, i_max, i, idx[4], nextidx;
  PVert **points, *p1, *p2, *p3, *p4, *p1n;

  /* compute convex hull */
  if (!p_chart_convex_hull(chart, &points, &npoints, &right)) {
    return 0.0;
  }

  /* find left/top/right/bottom points, and compute angle for each point */
  angles = (float *)MEM_mallocN(sizeof(float) * npoints, "PMinAreaAngles");

  i_min = i_max = 0;
  miny = 1e10;
  maxy = -1e10;

  for (i = 0; i < npoints; i++) {
    p1 = (i == 0) ? points[npoints - 1] : points[i - 1];
    p2 = points[i];
    p3 = (i == npoints - 1) ? points[0] : points[i + 1];

    angles[i] = (float)M_PI - p_vec2_angle(p1->uv, p2->uv, p3->uv);

    if (points[i]->uv[1] < miny) {
      miny = points[i]->uv[1];
      i_min = i;
    }
    if (points[i]->uv[1] > maxy) {
      maxy = points[i]->uv[1];
      i_max = i;
    }
  }

  /* left, top, right, bottom */
  idx[0] = 0;
  idx[1] = i_max;
  idx[2] = right;
  idx[3] = i_min;

  v[0] = points[idx[0]]->uv[0];
  v[1] = points[idx[0]]->uv[1] + 1.0f;
  a[0] = p_vec2_angle(points[(idx[0] + 1) % npoints]->uv, points[idx[0]]->uv, v);

  v[0] = points[idx[1]]->uv[0] + 1.0f;
  v[1] = points[idx[1]]->uv[1];
  a[1] = p_vec2_angle(points[(idx[1] + 1) % npoints]->uv, points[idx[1]]->uv, v);

  v[0] = points[idx[2]]->uv[0];
  v[1] = points[idx[2]]->uv[1] - 1.0f;
  a[2] = p_vec2_angle(points[(idx[2] + 1) % npoints]->uv, points[idx[2]]->uv, v);

  v[0] = points[idx[3]]->uv[0] - 1.0f;
  v[1] = points[idx[3]]->uv[1];
  a[3] = p_vec2_angle(points[(idx[3] + 1) % npoints]->uv, points[idx[3]]->uv, v);

  /* 4 rotating calipers */

  rotated = 0.0;
  minarea = 1e10;
  minangle = 0.0;

  while (rotated <= (float)M_PI_2) { /* INVESTIGATE: how far to rotate? */
    /* rotate with the smallest angle */
    i_min = 0;
    mina = 1e10;

    for (i = 0; i < 4; i++) {
      if (a[i] < mina) {
        mina = a[i];
        i_min = i;
      }
    }

    rotated += mina;
    nextidx = (idx[i_min] + 1) % npoints;

    a[i_min] = angles[nextidx];
    a[(i_min + 1) % 4] = a[(i_min + 1) % 4] - mina;
    a[(i_min + 2) % 4] = a[(i_min + 2) % 4] - mina;
    a[(i_min + 3) % 4] = a[(i_min + 3) % 4] - mina;

    /* compute area */
    p1 = points[idx[i_min]];
    p1n = points[nextidx];
    p2 = points[idx[(i_min + 1) % 4]];
    p3 = points[idx[(i_min + 2) % 4]];
    p4 = points[idx[(i_min + 3) % 4]];

    len = len_v2v2(p1->uv, p1n->uv);

    if (len > 0.0f) {
      len = 1.0f / len;
      v[0] = (p1n->uv[0] - p1->uv[0]) * len;
      v[1] = (p1n->uv[1] - p1->uv[1]) * len;

      area = p_rectangle_area(p1->uv, v, p2->uv, p3->uv, p4->uv);

      /* remember smallest area */
      if (area < minarea) {
        minarea = area;
        minangle = rotated;
      }
    }

    idx[i_min] = nextidx;
  }

  /* try keeping rotation as small as possible */
  if (minangle > (float)M_PI_4) {
    minangle -= (float)M_PI_2;
  }

  MEM_freeN(angles);
  MEM_freeN(points);

  return minangle;
}

static void p_chart_rotate_minimum_area(PChart *chart)
{
  float angle = p_chart_minimum_area_angle(chart);
  float sine = sinf(angle);
  float cosine = cosf(angle);
  PVert *v;

  for (v = chart->verts; v; v = v->nextlink) {
    float oldu = v->uv[0], oldv = v->uv[1];
    v->uv[0] = cosine * oldu - sine * oldv;
    v->uv[1] = sine * oldu + cosine * oldv;
  }
}

static void p_chart_rotate_fit_aabb(PChart *chart)
{
  float(*points)[2] = static_cast<float(*)[2]>(
      MEM_mallocN(sizeof(*points) * chart->nverts, __func__));

  p_chart_uv_to_array(chart, points);

  float angle = BLI_convexhull_aabb_fit_points_2d(points, chart->nverts);

  MEM_freeN(points);

  if (angle != 0.0f) {
    float mat[2][2];
    angle_to_mat2(mat, angle);
    p_chart_uv_transform(chart, mat);
  }
}

/* Exported */

ParamHandle *GEO_uv_parametrizer_construct_begin()
{
  ParamHandle *handle = new ParamHandle();
  handle->construction_chart = (PChart *)MEM_callocN(sizeof(PChart), "PChart");
  handle->state = PHANDLE_STATE_ALLOCATED;
  handle->arena = BLI_memarena_new(MEM_SIZE_OPTIMAL(1 << 16), "param construct arena");
  handle->polyfill_arena = BLI_memarena_new(BLI_MEMARENA_STD_BUFSIZE, "param polyfill arena");
  handle->polyfill_heap = BLI_heap_new_ex(BLI_POLYFILL_ALLOC_NGON_RESERVE);
  handle->aspx = 1.0f;
  handle->aspy = 1.0f;

  handle->hash_verts = phash_new((PHashLink **)&handle->construction_chart->verts, 1);
  handle->hash_edges = phash_new((PHashLink **)&handle->construction_chart->edges, 1);
  handle->hash_faces = phash_new((PHashLink **)&handle->construction_chart->faces, 1);

  return handle;
}

void GEO_uv_parametrizer_aspect_ratio(ParamHandle *phandle, float aspx, float aspy)
{
  phandle->aspx = aspx;
  phandle->aspy = aspy;
}

void GEO_uv_parametrizer_delete(ParamHandle *phandle)
{
  if (!phandle) {
    return;
  }
  param_assert(ELEM(phandle->state, PHANDLE_STATE_ALLOCATED, PHANDLE_STATE_CONSTRUCTED));

  for (int i = 0; i < phandle->ncharts; i++) {
    MEM_SAFE_FREE(phandle->charts[i]);
  }

  MEM_SAFE_FREE(phandle->charts);

  if (phandle->pin_hash) {
    BLI_ghash_free(phandle->pin_hash, nullptr, nullptr);
    phandle->pin_hash = nullptr;
  }

  MEM_SAFE_FREE(phandle->construction_chart);

  phash_delete(phandle->hash_verts);
  phash_delete(phandle->hash_edges);
  phash_delete(phandle->hash_faces);

  BLI_memarena_free(phandle->arena);
  BLI_memarena_free(phandle->polyfill_arena);
  BLI_heap_free(phandle->polyfill_heap, nullptr);

  if (phandle->rng) {
    BLI_rng_free(phandle->rng);
    phandle->rng = nullptr;
  }

  delete phandle;
}

using GeoUVPinIndex = struct GeoUVPinIndex {
  struct GeoUVPinIndex *next;
  float uv[2];
  ParamKey reindex;
};

/* Find a (mostly) unique ParamKey given a BMVert index and UV co-ordinates.
 * For each unique pinned UVs, return a unique ParamKey, starting with
 *  a very large number, and decreasing steadily from there.
 * For non-pinned UVs which share a BMVert with a pinned UV,
 *  return the index corresponding to the closest pinned UV.
 * For everything else, just return the BMVert index.
 * Note that ParamKeys will eventually be hashed, so they don't need to be contiguous.
 */
ParamKey GEO_uv_find_pin_index(ParamHandle *handle, const int bmvertindex, const float uv[2])
{
  if (!handle->pin_hash) {
    return bmvertindex; /* No verts pinned. */
  }

  const GeoUVPinIndex *pinuvlist = (const GeoUVPinIndex *)BLI_ghash_lookup(
      handle->pin_hash, POINTER_FROM_INT(bmvertindex));
  if (!pinuvlist) {
    return bmvertindex; /* Vert not pinned. */
  }

  /* At least one of the UVs associated with bmvertindex is pinned. Find the best one. */
  float bestdistsquared = len_squared_v2v2(pinuvlist->uv, uv);
  ParamKey bestkey = pinuvlist->reindex;
  pinuvlist = pinuvlist->next;
  while (pinuvlist) {
    const float distsquared = len_squared_v2v2(pinuvlist->uv, uv);
    if (bestdistsquared > distsquared) {
      bestdistsquared = distsquared;
      bestkey = pinuvlist->reindex;
    }
    pinuvlist = pinuvlist->next;
  }
  return bestkey;
}

static GeoUVPinIndex *new_geo_uv_pinindex(ParamHandle *handle, const float uv[2])
{
  GeoUVPinIndex *pinuv = (GeoUVPinIndex *)BLI_memarena_alloc(handle->arena, sizeof(*pinuv));
  pinuv->next = nullptr;
  copy_v2_v2(pinuv->uv, uv);
  pinuv->reindex = PARAM_KEY_MAX - (handle->unique_pin_count++);
  return pinuv;
}

void GEO_uv_prepare_pin_index(ParamHandle *handle, const int bmvertindex, const float uv[2])
{
  if (!handle->pin_hash) {
    handle->pin_hash = BLI_ghash_int_new("uv pin reindex");
  }

  GeoUVPinIndex *pinuvlist = (GeoUVPinIndex *)BLI_ghash_lookup(handle->pin_hash,
                                                               POINTER_FROM_INT(bmvertindex));
  if (!pinuvlist) {
    BLI_ghash_insert(
        handle->pin_hash, POINTER_FROM_INT(bmvertindex), new_geo_uv_pinindex(handle, uv));
    return;
  }

  while (true) {
    if (equals_v2v2(pinuvlist->uv, uv)) {
      return;
    }
    if (!pinuvlist->next) {
      pinuvlist->next = new_geo_uv_pinindex(handle, uv);
      return;
    }
    pinuvlist = pinuvlist->next;
  }
}

static void p_add_ngon(ParamHandle *handle,
                       const ParamKey key,
                       const int nverts,
                       const ParamKey *vkeys,
                       const float **co,
                       float **uv, /* Output will eventually be written to `uv`. */
                       const bool *pin,
                       const bool *select)
{
  /* Allocate memory for polyfill. */
  MemArena *arena = handle->polyfill_arena;
  Heap *heap = handle->polyfill_heap;
  uint nfilltri = nverts - 2;
  uint(*tris)[3] = static_cast<uint(*)[3]>(
      BLI_memarena_alloc(arena, sizeof(*tris) * (size_t)nfilltri));
  float(*projverts)[2] = static_cast<float(*)[2]>(
      BLI_memarena_alloc(arena, sizeof(*projverts) * (size_t)nverts));

  /* Calc normal, flipped: to get a positive 2d cross product. */
  float normal[3];
  zero_v3(normal);

  const float *co_curr, *co_prev = co[nverts - 1];
  for (int j = 0; j < nverts; j++) {
    co_curr = co[j];
    add_newell_cross_v3_v3v3(normal, co_prev, co_curr);
    co_prev = co_curr;
  }
  if (UNLIKELY(normalize_v3(normal) == 0.0f)) {
    normal[2] = 1.0f;
  }

  /* Project verts to 2d. */
  float axis_mat[3][3];
  axis_dominant_v3_to_m3_negate(axis_mat, normal);
  for (int j = 0; j < nverts; j++) {
    mul_v2_m3v3(projverts[j], axis_mat, co[j]);
  }

  BLI_polyfill_calc_arena(projverts, nverts, 1, tris, arena);

  /* Beautify helps avoid thin triangles that give numerical problems. */
  BLI_polyfill_beautify(projverts, nverts, tris, arena, heap);

  /* Add triangles. */
  for (uint j = 0; j < nfilltri; j++) {
    uint *tri = tris[j];
    uint v0 = tri[0];
    uint v1 = tri[1];
    uint v2 = tri[2];

    const ParamKey tri_vkeys[3] = {vkeys[v0], vkeys[v1], vkeys[v2]};
    const float *tri_co[3] = {co[v0], co[v1], co[v2]};
    float *tri_uv[3] = {uv[v0], uv[v1], uv[v2]};
    bool tri_pin[3] = {pin[v0], pin[v1], pin[v2]};
    bool tri_select[3] = {select[v0], select[v1], select[v2]};

    GEO_uv_parametrizer_face_add(handle, key, 3, tri_vkeys, tri_co, tri_uv, tri_pin, tri_select);
  }

  BLI_memarena_clear(arena);
}

void GEO_uv_parametrizer_face_add(ParamHandle *phandle,
                                  const ParamKey key,
                                  const int nverts,
                                  const ParamKey *vkeys,
                                  const float **co,
                                  float **uv,
                                  const bool *pin,
                                  const bool *select)
{
  param_assert(phash_lookup(phandle->hash_faces, key) == nullptr);
  param_assert(phandle->state == PHANDLE_STATE_ALLOCATED);
  param_assert(ELEM(nverts, 3, 4));

  if (nverts > 4) {
    /* ngon */
    p_add_ngon(phandle, key, nverts, vkeys, co, uv, pin, select);
  }
  else if (nverts == 4) {
    /* quad */
    if (p_quad_split_direction(phandle, co, vkeys)) {
      p_face_add_construct(phandle, key, vkeys, co, uv, 0, 1, 2, pin, select);
      p_face_add_construct(phandle, key, vkeys, co, uv, 0, 2, 3, pin, select);
    }
    else {
      p_face_add_construct(phandle, key, vkeys, co, uv, 0, 1, 3, pin, select);
      p_face_add_construct(phandle, key, vkeys, co, uv, 1, 2, 3, pin, select);
    }
  }
  else if (!p_face_exists(phandle, vkeys, 0, 1, 2)) {
    /* triangle */
    p_face_add_construct(phandle, key, vkeys, co, uv, 0, 1, 2, pin, select);
  }
}

void GEO_uv_parametrizer_edge_set_seam(ParamHandle *phandle, ParamKey *vkeys)
{
  PEdge *e;

  param_assert(phandle->state == PHANDLE_STATE_ALLOCATED);

  e = p_edge_lookup(phandle, vkeys);
  if (e) {
    e->flag |= PEDGE_SEAM;
  }
}

void GEO_uv_parametrizer_construct_end(ParamHandle *phandle,
                                       bool fill,
                                       bool topology_from_uvs,
                                       int *count_fail)
{
  PChart *chart = phandle->construction_chart;
  int i, j;
  PEdge *outer;

  param_assert(phandle->state == PHANDLE_STATE_ALLOCATED);

  phandle->ncharts = p_connect_pairs(phandle, topology_from_uvs);
  phandle->charts = p_split_charts(phandle, chart, phandle->ncharts);

  MEM_freeN(phandle->construction_chart);
  phandle->construction_chart = nullptr;

  phash_delete(phandle->hash_verts);
  phash_delete(phandle->hash_edges);
  phash_delete(phandle->hash_faces);
  phandle->hash_verts = phandle->hash_edges = phandle->hash_faces = nullptr;

  for (i = j = 0; i < phandle->ncharts; i++) {
    PVert *v;
    chart = phandle->charts[i];

    p_chart_boundaries(chart, &outer);

    if (!topology_from_uvs && chart->nboundaries == 0) {
      MEM_freeN(chart);
      if (count_fail != nullptr) {
        *count_fail += 1;
      }
      continue;
    }

    phandle->charts[j] = chart;
    j++;

    if (fill && (chart->nboundaries > 1)) {
      p_chart_fill_boundaries(phandle, chart, outer);
    }

    for (v = chart->verts; v; v = v->nextlink) {
      p_vert_load_pin_select_uvs(phandle, v);
    }
  }

  phandle->ncharts = j;

  phandle->state = PHANDLE_STATE_CONSTRUCTED;
}

void GEO_uv_parametrizer_lscm_begin(ParamHandle *phandle, bool live, bool abf)
{
  PFace *f;
  int i;

  param_assert(phandle->state == PHANDLE_STATE_CONSTRUCTED);
  phandle->state = PHANDLE_STATE_LSCM;

  for (i = 0; i < phandle->ncharts; i++) {
    for (f = phandle->charts[i]->faces; f; f = f->nextlink) {
      p_face_backup_uvs(f);
    }
    p_chart_lscm_begin(phandle->charts[i], live, abf);
  }
}

void GEO_uv_parametrizer_lscm_solve(ParamHandle *phandle, int *count_changed, int *count_failed)
{
  PChart *chart;
  int i;

  param_assert(phandle->state == PHANDLE_STATE_LSCM);

  for (i = 0; i < phandle->ncharts; i++) {
    chart = phandle->charts[i];

    if (chart->u.lscm.context) {
      const bool result = p_chart_lscm_solve(phandle, chart);

      if (result && !chart->has_pins) {
        p_chart_rotate_minimum_area(chart);
      }
      else if (result && chart->u.lscm.single_pin) {
        p_chart_rotate_fit_aabb(chart);
        p_chart_lscm_transform_single_pin(chart);
      }

      if (!result || !chart->has_pins) {
        p_chart_lscm_end(chart);
      }

      if (result) {
        if (count_changed != nullptr) {
          *count_changed += 1;
        }
      }
      else {
        if (count_failed != nullptr) {
          *count_failed += 1;
        }
      }
    }
  }
}

void GEO_uv_parametrizer_lscm_end(ParamHandle *phandle)
{
  BLI_assert(phandle->state == PHANDLE_STATE_LSCM);

  for (int i = 0; i < phandle->ncharts; i++) {
    p_chart_lscm_end(phandle->charts[i]);
#if 0
    p_chart_complexify(phandle->charts[i]);
#endif
  }

  phandle->state = PHANDLE_STATE_CONSTRUCTED;
}

void GEO_uv_parametrizer_stretch_begin(ParamHandle *phandle)
{
  PChart *chart;
  PVert *v;
  PFace *f;
  int i;

  param_assert(phandle->state == PHANDLE_STATE_CONSTRUCTED);
  phandle->state = PHANDLE_STATE_STRETCH;

  phandle->rng = BLI_rng_new(31415926);
  phandle->blend = 0.0f;

  for (i = 0; i < phandle->ncharts; i++) {
    chart = phandle->charts[i];

    for (v = chart->verts; v; v = v->nextlink) {
      v->flag &= ~PVERT_PIN; /* don't use user-defined pins */
    }

    p_stretch_pin_boundary(chart);

    for (f = chart->faces; f; f = f->nextlink) {
      p_face_backup_uvs(f);
      f->u.area3d = p_face_area(f);
    }
  }
}

void GEO_uv_parametrizer_stretch_blend(ParamHandle *phandle, float blend)
{
  param_assert(phandle->state == PHANDLE_STATE_STRETCH);
  phandle->blend = blend;
}

void GEO_uv_parametrizer_stretch_iter(ParamHandle *phandle)
{
  PChart *chart;
  int i;

  param_assert(phandle->state == PHANDLE_STATE_STRETCH);

  for (i = 0; i < phandle->ncharts; i++) {
    chart = phandle->charts[i];
    p_chart_stretch_minimize(chart, phandle->rng);
  }
}

void GEO_uv_parametrizer_stretch_end(ParamHandle *phandle)
{
  param_assert(phandle->state == PHANDLE_STATE_STRETCH);
  phandle->state = PHANDLE_STATE_CONSTRUCTED;
}

/* don't pack, just rotate (used for better packing) */
static void GEO_uv_parametrizer_pack_rotate(ParamHandle *phandle, bool ignore_pinned)
{
  PChart *chart;
  int i;

  for (i = 0; i < phandle->ncharts; i++) {
    chart = phandle->charts[i];

    if (ignore_pinned && chart->has_pins) {
      continue;
    }

    p_chart_rotate_fit_aabb(chart);
  }
}

void GEO_uv_parametrizer_pack(ParamHandle *handle,
                              float margin,
                              bool do_rotate,
                              bool ignore_pinned)
{
  /* box packing variables */
  BoxPack *boxarray, *box;
  float tot_width, tot_height, scale;

  PChart *chart;
  int i, unpacked = 0;
  float trans[2];
  double area = 0.0;

  if (handle->ncharts == 0) {
    return;
  }

  /* this could be its own function */
  if (do_rotate) {
    GEO_uv_parametrizer_pack_rotate(handle, ignore_pinned);
  }

  if (handle->aspx != handle->aspy) {
    GEO_uv_parametrizer_scale(handle, 1.0f / handle->aspx, 1.0f / handle->aspy);
  }

  /* we may not use all these boxes */
  boxarray = (BoxPack *)MEM_mallocN(handle->ncharts * sizeof(BoxPack), "BoxPack box");

  for (i = 0; i < handle->ncharts; i++) {
    chart = handle->charts[i];

    if (ignore_pinned && chart->has_pins) {
      unpacked++;
      continue;
    }

    box = boxarray + (i - unpacked);

    p_chart_uv_bbox(chart, trans, chart->u.pack.size);

    trans[0] = -trans[0];
    trans[1] = -trans[1];

    p_chart_uv_translate(chart, trans);

    box->w = chart->u.pack.size[0] + trans[0];
    box->h = chart->u.pack.size[1] + trans[1];
    box->index = i; /* Warning this index skips chart->has_pins boxes. */

    if (margin > 0.0f) {
      area += (double)sqrtf(box->w * box->h);
    }
  }

  if (margin > 0.0f) {
    /* multiply the margin by the area to give predictable results not dependent on UV scale,
     * ...Without using the area running pack multiple times also gives a bad feedback loop.
     * multiply by 0.1 so the margin value from the UI can be from
     * 0.0 to 1.0 but not give a massive margin */
    margin = (margin * (float)area) * 0.1f;
    unpacked = 0;
    for (i = 0; i < handle->ncharts; i++) {
      chart = handle->charts[i];

      if (ignore_pinned && chart->has_pins) {
        unpacked++;
        continue;
      }

      box = boxarray + (i - unpacked);
      trans[0] = margin;
      trans[1] = margin;
      p_chart_uv_translate(chart, trans);
      box->w += margin * 2;
      box->h += margin * 2;
    }
  }

  BLI_box_pack_2d(boxarray, handle->ncharts - unpacked, &tot_width, &tot_height);

  if (tot_height > tot_width) {
    scale = tot_height != 0.0f ? (1.0f / tot_height) : 1.0f;
  }
  else {
    scale = tot_width != 0.0f ? (1.0f / tot_width) : 1.0f;
  }

  for (i = 0; i < handle->ncharts - unpacked; i++) {
    box = boxarray + i;
    trans[0] = box->x;
    trans[1] = box->y;

    chart = handle->charts[box->index];
    p_chart_uv_translate(chart, trans);
    p_chart_uv_scale(chart, scale);
  }
  MEM_freeN(boxarray);

  if (handle->aspx != handle->aspy) {
    GEO_uv_parametrizer_scale(handle, handle->aspx, handle->aspy);
  }
}

void GEO_uv_parametrizer_average(ParamHandle *phandle,
                                 bool ignore_pinned,
                                 bool scale_uv,
                                 bool shear)
{
  PChart *chart;
  int i;
  float tot_uvarea = 0.0f, tot_facearea = 0.0f;
  float tot_fac, fac;
  float minv[2], maxv[2], trans[2];

  if (phandle->ncharts == 0) {
    return;
  }

  for (i = 0; i < phandle->ncharts; i++) {
    chart = phandle->charts[i];

    if (ignore_pinned && chart->has_pins) {
      continue;
    }

    p_chart_uv_bbox(chart, minv, maxv);
    mid_v2_v2v2(chart->u.pack.origin, minv, maxv);

    if (scale_uv || shear) {
      /* It's possible that for some "bad" inputs, the following iteration will converge slowly or
       * perhaps even diverge. Rather than infinite loop, we only iterate a maximum of `max_iter`
       * times. (Also useful when making changes to the calculation.) */
      int max_iter = 10;
      for (int j = 0; j < max_iter; j++) {
        /* An island could contain millions of polygons. When summing many small values, we need to
         * use double precision in the accumulator to maintain accuracy. Note that the individual
         * calculations only need to be at single precision.*/
        double scale_cou = 0;
        double scale_cov = 0;
        double scale_cross = 0;
        double weight_sum = 0;
        for (PFace *f = chart->faces; f; f = f->nextlink) {
          float m[2][2], s[2][2];
          PVert *va = f->edge->vert;
          PVert *vb = f->edge->next->vert;
          PVert *vc = f->edge->next->next->vert;
          s[0][0] = va->uv[0] - vc->uv[0];
          s[0][1] = va->uv[1] - vc->uv[1];
          s[1][0] = vb->uv[0] - vc->uv[0];
          s[1][1] = vb->uv[1] - vc->uv[1];
          /* Find the "U" axis and "V" axis in triangle co-ordinates. Normally this would require
           * SVD, but in 2D we can use a cheaper matrix inversion instead.*/
          if (!invert_m2_m2(m, s)) {
            continue;
          }
          float cou[3], cov[3]; /* i.e. Texture "U" and texture "V" in 3D co-ordinates.*/
          for (int k = 0; k < 3; k++) {
            cou[k] = m[0][0] * (va->co[k] - vc->co[k]) + m[0][1] * (vb->co[k] - vc->co[k]);
            cov[k] = m[1][0] * (va->co[k] - vc->co[k]) + m[1][1] * (vb->co[k] - vc->co[k]);
          }
          const float weight = p_face_area(f);
          scale_cou += len_v3(cou) * weight;
          scale_cov += len_v3(cov) * weight;
          if (shear) {
            normalize_v3(cov);
            normalize_v3(cou);

            /* Why is scale_cross called `cross` when we call `dot`? The next line calculates:
             * `scale_cross += length(cross(cross(cou, face_normal), cov))`
             * By construction, both `cou` and `cov` are orthogonal to the face normal.
             * By definition, the normal vector has unit length. */
            scale_cross += dot_v3v3(cou, cov) * weight;
          }
          weight_sum += weight;
        }
        if (scale_cou * scale_cov < 1e-10f) {
          break;
        }
        const float scale_factor_u = scale_uv ? sqrtf(scale_cou / scale_cov) : 1.0f;

        /* Compute correction transform. */
        float t[2][2];
        t[0][0] = scale_factor_u;
        t[1][0] = clamp_f((float)(scale_cross / weight_sum), -0.5f, 0.5f);
        t[0][1] = 0;
        t[1][1] = 1.0f / scale_factor_u;

        /* Apply the correction. */
        p_chart_uv_transform(chart, t);

        /* How far from the identity transform are we? [[1,0],[0,1]] */
        const float err = fabsf(t[0][0] - 1.0f) + fabsf(t[1][0]) + fabsf(t[0][1]) +
                          fabsf(t[1][1] - 1.0f);

        const float tolerance = 1e-6f; /* Trade accuracy for performance. */
        if (err < tolerance) {
          /* Too slow? Use Richardson Extrapolation to accelerate the convergence.*/
          break;
        }
      }
    }

    chart->u.pack.area = 0.0f;    /* 3d area */
    chart->u.pack.rescale = 0.0f; /* UV area, abusing rescale for tmp storage, oh well :/ */

    for (PFace *f = chart->faces; f; f = f->nextlink) {
      chart->u.pack.area += p_face_area(f);
      chart->u.pack.rescale += fabsf(p_face_uv_area_signed(f));
    }

    tot_facearea += chart->u.pack.area;
    tot_uvarea += chart->u.pack.rescale;
  }

  if (tot_facearea == tot_uvarea || tot_facearea == 0.0f || tot_uvarea == 0.0f) {
    /* nothing to do */
    return;
  }

  tot_fac = tot_facearea / tot_uvarea;

  for (i = 0; i < phandle->ncharts; i++) {
    chart = phandle->charts[i];

    if (ignore_pinned && chart->has_pins) {
      continue;
    }

    if (chart->u.pack.area != 0.0f && chart->u.pack.rescale != 0.0f) {
      fac = chart->u.pack.area / chart->u.pack.rescale;

      /* Average scale. */
      p_chart_uv_scale(chart, sqrtf(fac / tot_fac));

      /* Get the current island center. */
      p_chart_uv_bbox(chart, minv, maxv);

      /* Move to original center. */
      mid_v2_v2v2(trans, minv, maxv);
      negate_v2(trans);
      add_v2_v2(trans, chart->u.pack.origin);
      p_chart_uv_translate(chart, trans);
    }
  }
}

void GEO_uv_parametrizer_scale(ParamHandle *phandle, float x, float y)
{
  PChart *chart;
  int i;

  for (i = 0; i < phandle->ncharts; i++) {
    chart = phandle->charts[i];
    p_chart_uv_scale_xy(chart, x, y);
  }
}

void GEO_uv_parametrizer_flush(ParamHandle *phandle)
{
  PChart *chart;
  int i;

  for (i = 0; i < phandle->ncharts; i++) {
    chart = phandle->charts[i];

    if ((phandle->state == PHANDLE_STATE_LSCM) && !chart->u.lscm.context) {
      continue;
    }

    if (phandle->blend == 0.0f) {
      p_flush_uvs(phandle, chart);
    }
    else {
      p_flush_uvs_blend(phandle, chart, phandle->blend);
    }
  }
}

void GEO_uv_parametrizer_flush_restore(ParamHandle *phandle)
{
  PChart *chart;
  PFace *f;
  int i;

  for (i = 0; i < phandle->ncharts; i++) {
    chart = phandle->charts[i];

    for (f = chart->faces; f; f = f->nextlink) {
      p_face_restore_uvs(f);
    }
  }
}