math_vector.c

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/*
 * This program is free software; you can redistribute it and/or
 * modify it under the terms of the GNU General Public License
 * as published by the Free Software Foundation; either version 2
 * of the License, or (at your option) any later version.
 *
 * This program is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 * GNU General Public License for more details.
 *
 * You should have received a copy of the GNU General Public License
 * along with this program; if not, write to the Free Software Foundation,
 * Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
 *
 * The Original Code is Copyright (C) 2001-2002 by NaN Holding BV.
 * All rights reserved.
 *
 * The Original Code is: some of this file.
 */
 
#include "BLI_math.h"
 
#include "BLI_strict_flags.h"
 
//******************************* Interpolation *******************************/
 
void interp_v2_v2v2(float r[2], const float a[2], const float b[2], const float t)
{
  const float s = 1.0f - t;
 
  r[0] = s * a[0] + t * b[0];
  r[1] = s * a[1] + t * b[1];
}
 
/* weight 3 2D vectors,
 * 'w' must be unit length but is not a vector, just 3 weights */
void interp_v2_v2v2v2(
    float r[2], const float a[2], const float b[2], const float c[2], const float t[3])
{
  r[0] = a[0] * t[0] + b[0] * t[1] + c[0] * t[2];
  r[1] = a[1] * t[0] + b[1] * t[1] + c[1] * t[2];
}
 
void interp_v3_v3v3(float r[3], const float a[3], const float b[3], const float t)
{
  const float s = 1.0f - t;
 
  r[0] = s * a[0] + t * b[0];
  r[1] = s * a[1] + t * b[1];
  r[2] = s * a[2] + t * b[2];
}
 
void interp_v4_v4v4(float r[4], const float a[4], const float b[4], const float t)
{
  const float s = 1.0f - t;
 
  r[0] = s * a[0] + t * b[0];
  r[1] = s * a[1] + t * b[1];
  r[2] = s * a[2] + t * b[2];
  r[3] = s * a[3] + t * b[3];
}
 
bool interp_v3_v3v3_slerp(float target[3], const float a[3], const float b[3], const float t)
{
  float cosom, w[2];
 
  BLI_ASSERT_UNIT_V3(a);
  BLI_ASSERT_UNIT_V3(b);
 
  cosom = dot_v3v3(a, b);
 
  /* direct opposites */
  if (UNLIKELY(cosom < (-1.0f + FLT_EPSILON))) {
    return false;
  }
 
  interp_dot_slerp(t, cosom, w);
 
  target[0] = w[0] * a[0] + w[1] * b[0];
  target[1] = w[0] * a[1] + w[1] * b[1];
  target[2] = w[0] * a[2] + w[1] * b[2];
 
  return true;
}
bool interp_v2_v2v2_slerp(float target[2], const float a[2], const float b[2], const float t)
{
  float cosom, w[2];
 
  BLI_ASSERT_UNIT_V2(a);
  BLI_ASSERT_UNIT_V2(b);
 
  cosom = dot_v2v2(a, b);
 
  /* direct opposites */
  if (UNLIKELY(cosom < (1.0f + FLT_EPSILON))) {
    return false;
  }
 
  interp_dot_slerp(t, cosom, w);
 
  target[0] = w[0] * a[0] + w[1] * b[0];
  target[1] = w[0] * a[1] + w[1] * b[1];
 
  return true;
}
 
void interp_v3_v3v3_slerp_safe(float target[3], const float a[3], const float b[3], const float t)
{
  if (UNLIKELY(!interp_v3_v3v3_slerp(target, a, b, t))) {
    /* Axis are aligned so any orthogonal vector is acceptable. */
    float ab_ortho[3];
    ortho_v3_v3(ab_ortho, a);
    normalize_v3(ab_ortho);
    if (t < 0.5f) {
      if (UNLIKELY(!interp_v3_v3v3_slerp(target, a, ab_ortho, t * 2.0f))) {
        BLI_assert(0);
        copy_v3_v3(target, a);
      }
    }
    else {
      if (UNLIKELY(!interp_v3_v3v3_slerp(target, ab_ortho, b, (t - 0.5f) * 2.0f))) {
        BLI_assert(0);
        copy_v3_v3(target, b);
      }
    }
  }
}
void interp_v2_v2v2_slerp_safe(float target[2], const float a[2], const float b[2], const float t)
{
  if (UNLIKELY(!interp_v2_v2v2_slerp(target, a, b, t))) {
    /* Axis are aligned so any orthogonal vector is acceptable. */
    float ab_ortho[2];
    ortho_v2_v2(ab_ortho, a);
    // normalize_v2(ab_ortho);
    if (t < 0.5f) {
      if (UNLIKELY(!interp_v2_v2v2_slerp(target, a, ab_ortho, t * 2.0f))) {
        BLI_assert(0);
        copy_v2_v2(target, a);
      }
    }
    else {
      if (UNLIKELY(!interp_v2_v2v2_slerp(target, ab_ortho, b, (t - 0.5f) * 2.0f))) {
        BLI_assert(0);
        copy_v2_v2(target, b);
      }
    }
  }
}
 
/* -------------------------------------------------------------------- */
void interp_v2_v2v2v2v2_cubic(float p[2],
                              const float v1[2],
                              const float v2[2],
                              const float v3[2],
                              const float v4[2],
                              const float u)
{
  float q0[2], q1[2], q2[2], r0[2], r1[2];
 
  interp_v2_v2v2(q0, v1, v2, u);
  interp_v2_v2v2(q1, v2, v3, u);
  interp_v2_v2v2(q2, v3, v4, u);
 
  interp_v2_v2v2(r0, q0, q1, u);
  interp_v2_v2v2(r1, q1, q2, u);
 
  interp_v2_v2v2(p, r0, r1, u);
}
 
/* weight 3 vectors,
 * 'w' must be unit length but is not a vector, just 3 weights */
void interp_v3_v3v3v3(
    float p[3], const float v1[3], const float v2[3], const float v3[3], const float w[3])
{
  p[0] = v1[0] * w[0] + v2[0] * w[1] + v3[0] * w[2];
  p[1] = v1[1] * w[0] + v2[1] * w[1] + v3[1] * w[2];
  p[2] = v1[2] * w[0] + v2[2] * w[1] + v3[2] * w[2];
}
 
/* weight 3 vectors,
 * 'w' must be unit length but is not a vector, just 4 weights */
void interp_v3_v3v3v3v3(float p[3],
                        const float v1[3],
                        const float v2[3],
                        const float v3[3],
                        const float v4[3],
                        const float w[4])
{
  p[0] = v1[0] * w[0] + v2[0] * w[1] + v3[0] * w[2] + v4[0] * w[3];
  p[1] = v1[1] * w[0] + v2[1] * w[1] + v3[1] * w[2] + v4[1] * w[3];
  p[2] = v1[2] * w[0] + v2[2] * w[1] + v3[2] * w[2] + v4[2] * w[3];
}
 
void interp_v4_v4v4v4(
    float p[4], const float v1[4], const float v2[4], const float v3[4], const float w[3])
{
  p[0] = v1[0] * w[0] + v2[0] * w[1] + v3[0] * w[2];
  p[1] = v1[1] * w[0] + v2[1] * w[1] + v3[1] * w[2];
  p[2] = v1[2] * w[0] + v2[2] * w[1] + v3[2] * w[2];
  p[3] = v1[3] * w[0] + v2[3] * w[1] + v3[3] * w[2];
}
 
void interp_v4_v4v4v4v4(float p[4],
                        const float v1[4],
                        const float v2[4],
                        const float v3[4],
                        const float v4[4],
                        const float w[4])
{
  p[0] = v1[0] * w[0] + v2[0] * w[1] + v3[0] * w[2] + v4[0] * w[3];
  p[1] = v1[1] * w[0] + v2[1] * w[1] + v3[1] * w[2] + v4[1] * w[3];
  p[2] = v1[2] * w[0] + v2[2] * w[1] + v3[2] * w[2] + v4[2] * w[3];
  p[3] = v1[3] * w[0] + v2[3] * w[1] + v3[3] * w[2] + v4[3] * w[3];
}
 
void interp_v3_v3v3v3_uv(
    float p[3], const float v1[3], const float v2[3], const float v3[3], const float uv[2])
{
  p[0] = v1[0] + ((v2[0] - v1[0]) * uv[0]) + ((v3[0] - v1[0]) * uv[1]);
  p[1] = v1[1] + ((v2[1] - v1[1]) * uv[0]) + ((v3[1] - v1[1]) * uv[1]);
  p[2] = v1[2] + ((v2[2] - v1[2]) * uv[0]) + ((v3[2] - v1[2]) * uv[1]);
}
 
void interp_v3_v3v3_uchar(uchar target[3], const uchar a[3], const uchar b[3], const float t)
{
  const float s = 1.0f - t;
 
  target[0] = (char)floorf(s * a[0] + t * b[0]);
  target[1] = (char)floorf(s * a[1] + t * b[1]);
  target[2] = (char)floorf(s * a[2] + t * b[2]);
}
void interp_v3_v3v3_char(char target[3], const char a[3], const char b[3], const float t)
{
  interp_v3_v3v3_uchar((uchar *)target, (const uchar *)a, (const uchar *)b, t);
}
 
void interp_v4_v4v4_uchar(uchar target[4], const uchar a[4], const uchar b[4], const float t)
{
  const float s = 1.0f - t;
 
  target[0] = (char)floorf(s * a[0] + t * b[0]);
  target[1] = (char)floorf(s * a[1] + t * b[1]);
  target[2] = (char)floorf(s * a[2] + t * b[2]);
  target[3] = (char)floorf(s * a[3] + t * b[3]);
}
void interp_v4_v4v4_char(char target[4], const char a[4], const char b[4], const float t)
{
  interp_v4_v4v4_uchar((uchar *)target, (const uchar *)a, (const uchar *)b, t);
}
 
void mid_v3_v3v3(float r[3], const float a[3], const float b[3])
{
  r[0] = 0.5f * (a[0] + b[0]);
  r[1] = 0.5f * (a[1] + b[1]);
  r[2] = 0.5f * (a[2] + b[2]);
}
 
void mid_v2_v2v2(float r[2], const float a[2], const float b[2])
{
  r[0] = 0.5f * (a[0] + b[0]);
  r[1] = 0.5f * (a[1] + b[1]);
}
 
void mid_v2_v2v2v2(float v[2], const float v1[2], const float v2[2], const float v3[2])
{
  v[0] = (v1[0] + v2[0] + v3[0]) / 3.0f;
  v[1] = (v1[1] + v2[1] + v3[1]) / 3.0f;
}
 
void mid_v3_v3v3v3(float v[3], const float v1[3], const float v2[3], const float v3[3])
{
  v[0] = (v1[0] + v2[0] + v3[0]) / 3.0f;
  v[1] = (v1[1] + v2[1] + v3[1]) / 3.0f;
  v[2] = (v1[2] + v2[2] + v3[2]) / 3.0f;
}
 
void mid_v3_v3v3v3v3(
    float v[3], const float v1[3], const float v2[3], const float v3[3], const float v4[3])
{
  v[0] = (v1[0] + v2[0] + v3[0] + v4[0]) / 4.0f;
  v[1] = (v1[1] + v2[1] + v3[1] + v4[1]) / 4.0f;
  v[2] = (v1[2] + v2[2] + v3[2] + v4[2]) / 4.0f;
}
 
void mid_v3_v3_array(float r[3], const float (*vec_arr)[3], const uint nbr)
{
  const float factor = 1.0f / (float)nbr;
  zero_v3(r);
 
  for (uint i = 0; i < nbr; i++) {
    madd_v3_v3fl(r, vec_arr[i], factor);
  }
}
 
void mid_v3_v3v3_angle_weighted(float r[3], const float a[3], const float b[3])
{
  /* trick, we want the middle of 2 normals as well as the angle between them
   * avoid multiple calculations by */
  float angle;
 
  /* double check they are normalized */
  BLI_ASSERT_UNIT_V3(a);
  BLI_ASSERT_UNIT_V3(b);
 
  add_v3_v3v3(r, a, b);
  angle = ((float)(1.0 / (M_PI / 2.0)) *
           /* normally we would only multiply by 2,
            * but instead of an angle make this 0-1 factor */
           2.0f) *
          acosf(normalize_v3(r) / 2.0f);
  mul_v3_fl(r, angle);
}
void mid_v3_angle_weighted(float r[3])
{
  /* trick, we want the middle of 2 normals as well as the angle between them
   * avoid multiple calculations by */
  float angle;
 
  /* double check they are normalized */
  BLI_assert(len_squared_v3(r) <= 1.0f + FLT_EPSILON);
 
  angle = ((float)(1.0 / (M_PI / 2.0)) *
           /* normally we would only multiply by 2,
            * but instead of an angle make this 0-1 factor */
           2.0f) *
          acosf(normalize_v3(r));
  mul_v3_fl(r, angle);
}
 
void flip_v4_v4v4(float v[4], const float v1[4], const float v2[4])
{
  v[0] = v1[0] + (v1[0] - v2[0]);
  v[1] = v1[1] + (v1[1] - v2[1]);
  v[2] = v1[2] + (v1[2] - v2[2]);
  v[3] = v1[3] + (v1[3] - v2[3]);
}
 
void flip_v3_v3v3(float v[3], const float v1[3], const float v2[3])
{
  v[0] = v1[0] + (v1[0] - v2[0]);
  v[1] = v1[1] + (v1[1] - v2[1]);
  v[2] = v1[2] + (v1[2] - v2[2]);
}
 
void flip_v2_v2v2(float v[2], const float v1[2], const float v2[2])
{
  v[0] = v1[0] + (v1[0] - v2[0]);
  v[1] = v1[1] + (v1[1] - v2[1]);
}
 
/********************************* Comparison ********************************/
 
bool is_finite_v2(const float v[2])
{
  return (isfinite(v[0]) && isfinite(v[1]));
}
 
bool is_finite_v3(const float v[3])
{
  return (isfinite(v[0]) && isfinite(v[1]) && isfinite(v[2]));
}
 
bool is_finite_v4(const float v[4])
{
  return (isfinite(v[0]) && isfinite(v[1]) && isfinite(v[2]) && isfinite(v[3]));
}
 
/********************************** Angles ***********************************/
 
/* Return the angle in radians between vecs 1-2 and 2-3 in radians
 * If v1 is a shoulder, v2 is the elbow and v3 is the hand,
 * this would return the angle at the elbow.
 *
 * note that when v1/v2/v3 represent 3 points along a straight line
 * that the angle returned will be pi (180deg), rather then 0.0
 */
float angle_v3v3v3(const float a[3], const float b[3], const float c[3])
{
  float vec1[3], vec2[3];
 
  sub_v3_v3v3(vec1, b, a);
  sub_v3_v3v3(vec2, b, c);
  normalize_v3(vec1);
  normalize_v3(vec2);
 
  return angle_normalized_v3v3(vec1, vec2);
}
 
/* Quicker than full angle computation */
float cos_v3v3v3(const float p1[3], const float p2[3], const float p3[3])
{
  float vec1[3], vec2[3];
 
  sub_v3_v3v3(vec1, p2, p1);
  sub_v3_v3v3(vec2, p2, p3);
  normalize_v3(vec1);
  normalize_v3(vec2);
 
  return dot_v3v3(vec1, vec2);
}
 
/* Return the shortest angle in radians between the 2 vectors */
float angle_v3v3(const float a[3], const float b[3])
{
  float vec1[3], vec2[3];
 
  normalize_v3_v3(vec1, a);
  normalize_v3_v3(vec2, b);
 
  return angle_normalized_v3v3(vec1, vec2);
}
 
float angle_v2v2v2(const float a[2], const float b[2], const float c[2])
{
  float vec1[2], vec2[2];
 
  vec1[0] = b[0] - a[0];
  vec1[1] = b[1] - a[1];
 
  vec2[0] = b[0] - c[0];
  vec2[1] = b[1] - c[1];
 
  normalize_v2(vec1);
  normalize_v2(vec2);
 
  return angle_normalized_v2v2(vec1, vec2);
}
 
/* Quicker than full angle computation */
float cos_v2v2v2(const float p1[2], const float p2[2], const float p3[2])
{
  float vec1[2], vec2[2];
 
  sub_v2_v2v2(vec1, p2, p1);
  sub_v2_v2v2(vec2, p2, p3);
  normalize_v2(vec1);
  normalize_v2(vec2);
 
  return dot_v2v2(vec1, vec2);
}
 
/* Return the shortest angle in radians between the 2 vectors */
float angle_v2v2(const float a[2], const float b[2])
{
  float vec1[2], vec2[2];
 
  vec1[0] = a[0];
  vec1[1] = a[1];
 
  vec2[0] = b[0];
  vec2[1] = b[1];
 
  normalize_v2(vec1);
  normalize_v2(vec2);
 
  return angle_normalized_v2v2(vec1, vec2);
}
 
float angle_signed_v2v2(const float v1[2], const float v2[2])
{
  const float perp_dot = (v1[1] * v2[0]) - (v1[0] * v2[1]);
  return atan2f(perp_dot, dot_v2v2(v1, v2));
}
 
float angle_normalized_v3v3(const float v1[3], const float v2[3])
{
  /* double check they are normalized */
  BLI_ASSERT_UNIT_V3(v1);
  BLI_ASSERT_UNIT_V3(v2);
 
  /* this is the same as acos(dot_v3v3(v1, v2)), but more accurate */
  if (dot_v3v3(v1, v2) >= 0.0f) {
    return 2.0f * saasin(len_v3v3(v1, v2) / 2.0f);
  }
 
  float v2_n[3];
  negate_v3_v3(v2_n, v2);
  return (float)M_PI - 2.0f * saasin(len_v3v3(v1, v2_n) / 2.0f);
}
 
float angle_normalized_v2v2(const float a[2], const float b[2])
{
  /* double check they are normalized */
  BLI_ASSERT_UNIT_V2(a);
  BLI_ASSERT_UNIT_V2(b);
 
  /* this is the same as acos(dot_v3v3(v1, v2)), but more accurate */
  if (dot_v2v2(a, b) >= 0.0f) {
    return 2.0f * saasin(len_v2v2(a, b) / 2.0f);
  }
 
  float v2_n[2];
  negate_v2_v2(v2_n, b);
  return (float)M_PI - 2.0f * saasin(len_v2v2(a, v2_n) / 2.0f);
}
 
float angle_on_axis_v3v3_v3(const float v1[3], const float v2[3], const float axis[3])
{
  float v1_proj[3], v2_proj[3];
 
  /* project the vectors onto the axis */
  project_plane_normalized_v3_v3v3(v1_proj, v1, axis);
  project_plane_normalized_v3_v3v3(v2_proj, v2, axis);
 
  return angle_v3v3(v1_proj, v2_proj);
}
 
float angle_signed_on_axis_v3v3_v3(const float v1[3], const float v2[3], const float axis[3])
{
  float v1_proj[3], v2_proj[3], tproj[3];
  float angle;
 
  /* project the vectors onto the axis */
  project_plane_normalized_v3_v3v3(v1_proj, v1, axis);
  project_plane_normalized_v3_v3v3(v2_proj, v2, axis);
 
  angle = angle_v3v3(v1_proj, v2_proj);
 
  /* calculate the sign (reuse 'tproj') */
  cross_v3_v3v3(tproj, v2_proj, v1_proj);
  if (dot_v3v3(tproj, axis) < 0.0f) {
    angle = ((float)(M_PI * 2.0)) - angle;
  }
 
  return angle;
}
 
float angle_on_axis_v3v3v3_v3(const float v1[3],
                              const float v2[3],
                              const float v3[3],
                              const float axis[3])
{
  float vec1[3], vec2[3];
 
  sub_v3_v3v3(vec1, v1, v2);
  sub_v3_v3v3(vec2, v3, v2);
 
  return angle_on_axis_v3v3_v3(vec1, vec2, axis);
}
 
float angle_signed_on_axis_v3v3v3_v3(const float v1[3],
                                     const float v2[3],
                                     const float v3[3],
                                     const float axis[3])
{
  float vec1[3], vec2[3];
 
  sub_v3_v3v3(vec1, v1, v2);
  sub_v3_v3v3(vec2, v3, v2);
 
  return angle_signed_on_axis_v3v3_v3(vec1, vec2, axis);
}
 
void angle_tri_v3(float angles[3], const float v1[3], const float v2[3], const float v3[3])
{
  float ed1[3], ed2[3], ed3[3];
 
  sub_v3_v3v3(ed1, v3, v1);
  sub_v3_v3v3(ed2, v1, v2);
  sub_v3_v3v3(ed3, v2, v3);
 
  normalize_v3(ed1);
  normalize_v3(ed2);
  normalize_v3(ed3);
 
  angles[0] = (float)M_PI - angle_normalized_v3v3(ed1, ed2);
  angles[1] = (float)M_PI - angle_normalized_v3v3(ed2, ed3);
  // face_angles[2] = M_PI - angle_normalized_v3v3(ed3, ed1);
  angles[2] = (float)M_PI - (angles[0] + angles[1]);
}
 
void angle_quad_v3(
    float angles[4], const float v1[3], const float v2[3], const float v3[3], const float v4[3])
{
  float ed1[3], ed2[3], ed3[3], ed4[3];
 
  sub_v3_v3v3(ed1, v4, v1);
  sub_v3_v3v3(ed2, v1, v2);
  sub_v3_v3v3(ed3, v2, v3);
  sub_v3_v3v3(ed4, v3, v4);
 
  normalize_v3(ed1);
  normalize_v3(ed2);
  normalize_v3(ed3);
  normalize_v3(ed4);
 
  angles[0] = (float)M_PI - angle_normalized_v3v3(ed1, ed2);
  angles[1] = (float)M_PI - angle_normalized_v3v3(ed2, ed3);
  angles[2] = (float)M_PI - angle_normalized_v3v3(ed3, ed4);
  angles[3] = (float)M_PI - angle_normalized_v3v3(ed4, ed1);
}
 
void angle_poly_v3(float *angles, const float *verts[3], int len)
{
  int i;
  float vec[3][3];
 
  sub_v3_v3v3(vec[2], verts[len - 1], verts[0]);
  normalize_v3(vec[2]);
  for (i = 0; i < len; i++) {
    sub_v3_v3v3(vec[i % 3], verts[i % len], verts[(i + 1) % len]);
    normalize_v3(vec[i % 3]);
    angles[i] = (float)M_PI - angle_normalized_v3v3(vec[(i + 2) % 3], vec[i % 3]);
  }
}
 
/********************************* Geometry **********************************/
 
void project_v2_v2v2(float out[2], const float p[2], const float v_proj[2])
{
  if (UNLIKELY(is_zero_v2(v_proj))) {
    zero_v2(out);
    return;
  }
 
  const float mul = dot_v2v2(p, v_proj) / dot_v2v2(v_proj, v_proj);
 
  out[0] = mul * v_proj[0];
  out[1] = mul * v_proj[1];
}
 
void project_v3_v3v3(float out[3], const float p[3], const float v_proj[3])
{
  if (UNLIKELY(is_zero_v3(v_proj))) {
    zero_v3(out);
    return;
  }
 
  const float mul = dot_v3v3(p, v_proj) / dot_v3v3(v_proj, v_proj);
 
  out[0] = mul * v_proj[0];
  out[1] = mul * v_proj[1];
  out[2] = mul * v_proj[2];
}
 
void project_v3_v3v3_db(double out[3], const double p[3], const double v_proj[3])
{
  if (UNLIKELY(is_zero_v3_db(v_proj))) {
    zero_v3_db(out);
    return;
  }
 
  const double mul = dot_v3v3_db(p, v_proj) / dot_v3v3_db(v_proj, v_proj);
 
  out[0] = mul * v_proj[0];
  out[1] = mul * v_proj[1];
  out[2] = mul * v_proj[2];
}
 
void project_v2_v2v2_normalized(float out[2], const float p[2], const float v_proj[2])
{
  BLI_ASSERT_UNIT_V2(v_proj);
  const float mul = dot_v2v2(p, v_proj);
 
  out[0] = mul * v_proj[0];
  out[1] = mul * v_proj[1];
}
 
void project_v3_v3v3_normalized(float out[3], const float p[3], const float v_proj[3])
{
  BLI_ASSERT_UNIT_V3(v_proj);
  const float mul = dot_v3v3(p, v_proj);
 
  out[0] = mul * v_proj[0];
  out[1] = mul * v_proj[1];
  out[2] = mul * v_proj[2];
}
 
void project_plane_v3_v3v3(float out[3], const float p[3], const float v_plane[3])
{
  const float mul = dot_v3v3(p, v_plane) / dot_v3v3(v_plane, v_plane);
 
  out[0] = p[0] - (mul * v_plane[0]);
  out[1] = p[1] - (mul * v_plane[1]);
  out[2] = p[2] - (mul * v_plane[2]);
}
 
void project_plane_v2_v2v2(float out[2], const float p[2], const float v_plane[2])
{
  const float mul = dot_v2v2(p, v_plane) / dot_v2v2(v_plane, v_plane);
 
  out[0] = p[0] - (mul * v_plane[0]);
  out[1] = p[1] - (mul * v_plane[1]);
}
 
void project_plane_normalized_v3_v3v3(float out[3], const float p[3], const float v_plane[3])
{
  BLI_ASSERT_UNIT_V3(v_plane);
  const float mul = dot_v3v3(p, v_plane);
 
  out[0] = p[0] - (mul * v_plane[0]);
  out[1] = p[1] - (mul * v_plane[1]);
  out[2] = p[2] - (mul * v_plane[2]);
}
 
void project_plane_normalized_v2_v2v2(float out[2], const float p[2], const float v_plane[2])
{
  BLI_ASSERT_UNIT_V2(v_plane);
  const float mul = dot_v2v2(p, v_plane);
 
  out[0] = p[0] - (mul * v_plane[0]);
  out[1] = p[1] - (mul * v_plane[1]);
}
 
/* project a vector on a plane defined by normal and a plane point p */
void project_v3_plane(float out[3], const float plane_no[3], const float plane_co[3])
{
  float vector[3];
  float mul;
 
  sub_v3_v3v3(vector, out, plane_co);
  mul = dot_v3v3(vector, plane_no) / len_squared_v3(plane_no);
 
  mul_v3_v3fl(vector, plane_no, mul);
 
  sub_v3_v3(out, vector);
}
 
/* Returns a vector bisecting the angle at b formed by a, b and c */
void bisect_v3_v3v3v3(float r[3], const float a[3], const float b[3], const float c[3])
{
  float d_12[3], d_23[3];
  sub_v3_v3v3(d_12, b, a);
  sub_v3_v3v3(d_23, c, b);
  normalize_v3(d_12);
  normalize_v3(d_23);
  add_v3_v3v3(r, d_12, d_23);
  normalize_v3(r);
}
 
void reflect_v3_v3v3(float out[3], const float v[3], const float normal[3])
{
  const float dot2 = 2.0f * dot_v3v3(v, normal);
 
  BLI_ASSERT_UNIT_V3(normal);
 
  out[0] = v[0] - (dot2 * normal[0]);
  out[1] = v[1] - (dot2 * normal[1]);
  out[2] = v[2] - (dot2 * normal[2]);
}
 
void reflect_v3_v3v3_db(double out[3], const double v[3], const double normal[3])
{
  const double dot2 = 2.0 * dot_v3v3_db(v, normal);
 
  /* BLI_ASSERT_UNIT_V3_DB(normal); this assert is not known? */
 
  out[0] = v[0] - (dot2 * normal[0]);
  out[1] = v[1] - (dot2 * normal[1]);
  out[2] = v[2] - (dot2 * normal[2]);
}
 
void ortho_basis_v3v3_v3(float r_n1[3], float r_n2[3], const float n[3])
{
  const float eps = FLT_EPSILON;
  const float f = len_squared_v2(n);
 
  if (f > eps) {
    const float d = 1.0f / sqrtf(f);
 
    BLI_assert(isfinite(d));
 
    r_n1[0] = n[1] * d;
    r_n1[1] = -n[0] * d;
    r_n1[2] = 0.0f;
    r_n2[0] = -n[2] * r_n1[1];
    r_n2[1] = n[2] * r_n1[0];
    r_n2[2] = n[0] * r_n1[1] - n[1] * r_n1[0];
  }
  else {
    /* degenerate case */
    r_n1[0] = (n[2] < 0.0f) ? -1.0f : 1.0f;
    r_n1[1] = r_n1[2] = r_n2[0] = r_n2[2] = 0.0f;
    r_n2[1] = 1.0f;
  }
}
 
void ortho_v3_v3(float out[3], const float v[3])
{
  const int axis = axis_dominant_v3_single(v);
 
  BLI_assert(out != v);
 
  switch (axis) {
    case 0:
      out[0] = -v[1] - v[2];
      out[1] = v[0];
      out[2] = v[0];
      break;
    case 1:
      out[0] = v[1];
      out[1] = -v[0] - v[2];
      out[2] = v[1];
      break;
    case 2:
      out[0] = v[2];
      out[1] = v[2];
      out[2] = -v[0] - v[1];
      break;
  }
}
 
void ortho_v2_v2(float out[2], const float v[2])
{
  BLI_assert(out != v);
 
  out[0] = -v[1];
  out[1] = v[0];
}
 
void rotate_v2_v2fl(float r[2], const float p[2], const float angle)
{
  const float co = cosf(angle);
  const float si = sinf(angle);
 
  BLI_assert(r != p);
 
  r[0] = co * p[0] - si * p[1];
  r[1] = si * p[0] + co * p[1];
}
 
void rotate_normalized_v3_v3v3fl(float out[3],
                                 const float p[3],
                                 const float axis[3],
                                 const float angle)
{
  const float costheta = cosf(angle);
  const float sintheta = sinf(angle);
 
  /* double check they are normalized */
  BLI_ASSERT_UNIT_V3(axis);
 
  out[0] = ((costheta + (1 - costheta) * axis[0] * axis[0]) * p[0]) +
           (((1 - costheta) * axis[0] * axis[1] - axis[2] * sintheta) * p[1]) +
           (((1 - costheta) * axis[0] * axis[2] + axis[1] * sintheta) * p[2]);
 
  out[1] = (((1 - costheta) * axis[0] * axis[1] + axis[2] * sintheta) * p[0]) +
           ((costheta + (1 - costheta) * axis[1] * axis[1]) * p[1]) +
           (((1 - costheta) * axis[1] * axis[2] - axis[0] * sintheta) * p[2]);
 
  out[2] = (((1 - costheta) * axis[0] * axis[2] - axis[1] * sintheta) * p[0]) +
           (((1 - costheta) * axis[1] * axis[2] + axis[0] * sintheta) * p[1]) +
           ((costheta + (1 - costheta) * axis[2] * axis[2]) * p[2]);
}
 
void rotate_v3_v3v3fl(float r[3], const float p[3], const float axis[3], const float angle)
{
  BLI_assert(r != p);
 
  float axis_n[3];
 
  normalize_v3_v3(axis_n, axis);
 
  rotate_normalized_v3_v3v3fl(r, p, axis_n, angle);
}
 
/*********************************** Other ***********************************/
 
void print_v2(const char *str, const float v[2])
{
  printf("%s: %.8f %.8f\n", str, v[0], v[1]);
}
 
void print_v3(const char *str, const float v[3])
{
  printf("%s: %.8f %.8f %.8f\n", str, v[0], v[1], v[2]);
}
 
void print_v4(const char *str, const float v[4])
{
  printf("%s: %.8f %.8f %.8f %.8f\n", str, v[0], v[1], v[2], v[3]);
}
 
void print_vn(const char *str, const float v[], const int n)
{
  int i = 0;
  printf("%s[%d]:", str, n);
  while (i < n) {
    printf(" %.8f", v[i++]);
  }
  printf("\n");
}
 
void minmax_v4v4_v4(float min[4], float max[4], const float vec[4])
{
  if (min[0] > vec[0]) {
    min[0] = vec[0];
  }
  if (min[1] > vec[1]) {
    min[1] = vec[1];
  }
  if (min[2] > vec[2]) {
    min[2] = vec[2];
  }
  if (min[3] > vec[3]) {
    min[3] = vec[3];
  }
 
  if (max[0] < vec[0]) {
    max[0] = vec[0];
  }
  if (max[1] < vec[1]) {
    max[1] = vec[1];
  }
  if (max[2] < vec[2]) {
    max[2] = vec[2];
  }
  if (max[3] < vec[3]) {
    max[3] = vec[3];
  }
}
 
void minmax_v3v3_v3(float min[3], float max[3], const float vec[3])
{
  if (min[0] > vec[0]) {
    min[0] = vec[0];
  }
  if (min[1] > vec[1]) {
    min[1] = vec[1];
  }
  if (min[2] > vec[2]) {
    min[2] = vec[2];
  }
 
  if (max[0] < vec[0]) {
    max[0] = vec[0];
  }
  if (max[1] < vec[1]) {
    max[1] = vec[1];
  }
  if (max[2] < vec[2]) {
    max[2] = vec[2];
  }
}
 
void minmax_v2v2_v2(float min[2], float max[2], const float vec[2])
{
  if (min[0] > vec[0]) {
    min[0] = vec[0];
  }
  if (min[1] > vec[1]) {
    min[1] = vec[1];
  }
 
  if (max[0] < vec[0]) {
    max[0] = vec[0];
  }
  if (max[1] < vec[1]) {
    max[1] = vec[1];
  }
}
 
void minmax_v3v3_v3_array(float r_min[3], float r_max[3], const float (*vec_arr)[3], int nbr)
{
  while (nbr--) {
    minmax_v3v3_v3(r_min, r_max, *vec_arr++);
  }
}
 
void dist_ensure_v3_v3fl(float v1[3], const float v2[3], const float dist)
{
  if (!equals_v3v3(v2, v1)) {
    float nor[3];
 
    sub_v3_v3v3(nor, v1, v2);
    normalize_v3(nor);
    madd_v3_v3v3fl(v1, v2, nor, dist);
  }
}
 
void dist_ensure_v2_v2fl(float v1[2], const float v2[2], const float dist)
{
  if (!equals_v2v2(v2, v1)) {
    float nor[2];
 
    sub_v2_v2v2(nor, v1, v2);
    normalize_v2(nor);
    madd_v2_v2v2fl(v1, v2, nor, dist);
  }
}
 
void axis_sort_v3(const float axis_values[3], int r_axis_order[3])
{
  float v[3];
  copy_v3_v3(v, axis_values);
 
#define SWAP_AXIS(a, b) \
  { \
    SWAP(float, v[a], v[b]); \
    SWAP(int, r_axis_order[a], r_axis_order[b]); \
  } \
  (void)0
 
  if (v[0] < v[1]) {
    if (v[2] < v[0]) {
      SWAP_AXIS(0, 2);
    }
  }
  else {
    if (v[1] < v[2]) {
      SWAP_AXIS(0, 1);
    }
    else {
      SWAP_AXIS(0, 2);
    }
  }
  if (v[2] < v[1]) {
    SWAP_AXIS(1, 2);
  }
 
#undef SWAP_AXIS
}
 
/***************************** Array Functions *******************************/
 
MINLINE double sqr_db(double f)
{
  return f * f;
}
 
double dot_vn_vn(const float *array_src_a, const float *array_src_b, const int size)
{
  double d = 0.0f;
  const float *array_pt_a = array_src_a + (size - 1);
  const float *array_pt_b = array_src_b + (size - 1);
  int i = size;
  while (i--) {
    d += (double)(*(array_pt_a--) * *(array_pt_b--));
  }
  return d;
}
 
double len_squared_vn(const float *array, const int size)
{
  double d = 0.0f;
  const float *array_pt = array + (size - 1);
  int i = size;
  while (i--) {
    d += sqr_db((double)(*(array_pt--)));
  }
  return d;
}
 
float normalize_vn_vn(float *array_tar, const float *array_src, const int size)
{
  const double d = len_squared_vn(array_src, size);
  float d_sqrt;
  if (d > 1.0e-35) {
    d_sqrt = (float)sqrt(d);
    mul_vn_vn_fl(array_tar, array_src, size, 1.0f / d_sqrt);
  }
  else {
    copy_vn_fl(array_tar, size, 0.0f);
    d_sqrt = 0.0f;
  }
  return d_sqrt;
}
 
float normalize_vn(float *array_tar, const int size)
{
  return normalize_vn_vn(array_tar, array_tar, size);
}
 
void range_vn_i(int *array_tar, const int size, const int start)
{
  int *array_pt = array_tar + (size - 1);
  int j = start + (size - 1);
  int i = size;
  while (i--) {
    *(array_pt--) = j--;
  }
}
 
void range_vn_u(uint *array_tar, const int size, const uint start)
{
  uint *array_pt = array_tar + (size - 1);
  uint j = start + (uint)(size - 1);
  int i = size;
  while (i--) {
    *(array_pt--) = j--;
  }
}
 
void range_vn_fl(float *array_tar, const int size, const float start, const float step)
{
  float *array_pt = array_tar + (size - 1);
  int i = size;
  while (i--) {
    *(array_pt--) = start + step * (float)(i);
  }
}
 
void negate_vn(float *array_tar, const int size)
{
  float *array_pt = array_tar + (size - 1);
  int i = size;
  while (i--) {
    *(array_pt--) *= -1.0f;
  }
}
 
void negate_vn_vn(float *array_tar, const float *array_src, const int size)
{
  float *tar = array_tar + (size - 1);
  const float *src = array_src + (size - 1);
  int i = size;
  while (i--) {
    *(tar--) = -*(src--);
  }
}
 
void mul_vn_vn(float *array_tar, const float *array_src, const int size)
{
  float *tar = array_tar + (size - 1);
  const float *src = array_src + (size - 1);
  int i = size;
  while (i--) {
    *(tar--) *= *(src--);
  }
}
 
void mul_vn_vnvn(float *array_tar,
                 const float *array_src_a,
                 const float *array_src_b,
                 const int size)
{
  float *tar = array_tar + (size - 1);
  const float *src_a = array_src_a + (size - 1);
  const float *src_b = array_src_b + (size - 1);
  int i = size;
  while (i--) {
    *(tar--) = *(src_a--) * *(src_b--);
  }
}
 
void mul_vn_fl(float *array_tar, const int size, const float f)
{
  float *array_pt = array_tar + (size - 1);
  int i = size;
  while (i--) {
    *(array_pt--) *= f;
  }
}
 
void mul_vn_vn_fl(float *array_tar, const float *array_src, const int size, const float f)
{
  float *tar = array_tar + (size - 1);
  const float *src = array_src + (size - 1);
  int i = size;
  while (i--) {
    *(tar--) = *(src--) * f;
  }
}
 
void add_vn_vn(float *array_tar, const float *array_src, const int size)
{
  float *tar = array_tar + (size - 1);
  const float *src = array_src + (size - 1);
  int i = size;
  while (i--) {
    *(tar--) += *(src--);
  }
}
 
void add_vn_vnvn(float *array_tar,
                 const float *array_src_a,
                 const float *array_src_b,
                 const int size)
{
  float *tar = array_tar + (size - 1);
  const float *src_a = array_src_a + (size - 1);
  const float *src_b = array_src_b + (size - 1);
  int i = size;
  while (i--) {
    *(tar--) = *(src_a--) + *(src_b--);
  }
}
 
void madd_vn_vn(float *array_tar, const float *array_src, const float f, const int size)
{
  float *tar = array_tar + (size - 1);
  const float *src = array_src + (size - 1);
  int i = size;
  while (i--) {
    *(tar--) += *(src--) * f;
  }
}
 
void madd_vn_vnvn(float *array_tar,
                  const float *array_src_a,
                  const float *array_src_b,
                  const float f,
                  const int size)
{
  float *tar = array_tar + (size - 1);
  const float *src_a = array_src_a + (size - 1);
  const float *src_b = array_src_b + (size - 1);
  int i = size;
  while (i--) {
    *(tar--) = *(src_a--) + (*(src_b--) * f);
  }
}
 
void sub_vn_vn(float *array_tar, const float *array_src, const int size)
{
  float *tar = array_tar + (size - 1);
  const float *src = array_src + (size - 1);
  int i = size;
  while (i--) {
    *(tar--) -= *(src--);
  }
}
 
void sub_vn_vnvn(float *array_tar,
                 const float *array_src_a,
                 const float *array_src_b,
                 const int size)
{
  float *tar = array_tar + (size - 1);
  const float *src_a = array_src_a + (size - 1);
  const float *src_b = array_src_b + (size - 1);
  int i = size;
  while (i--) {
    *(tar--) = *(src_a--) - *(src_b--);
  }
}
 
void msub_vn_vn(float *array_tar, const float *array_src, const float f, const int size)
{
  float *tar = array_tar + (size - 1);
  const float *src = array_src + (size - 1);
  int i = size;
  while (i--) {
    *(tar--) -= *(src--) * f;
  }
}
 
void msub_vn_vnvn(float *array_tar,
                  const float *array_src_a,
                  const float *array_src_b,
                  const float f,
                  const int size)
{
  float *tar = array_tar + (size - 1);
  const float *src_a = array_src_a + (size - 1);
  const float *src_b = array_src_b + (size - 1);
  int i = size;
  while (i--) {
    *(tar--) = *(src_a--) - (*(src_b--) * f);
  }
}
 
void interp_vn_vn(float *array_tar, const float *array_src, const float t, const int size)
{
  const float s = 1.0f - t;
  float *tar = array_tar + (size - 1);
  const float *src = array_src + (size - 1);
  int i = size;
  while (i--) {
    *(tar) = (s * *(tar)) + (t * *(src));
    tar--;
    src--;
  }
}
 
void copy_vn_i(int *array_tar, const int size, const int val)
{
  int *tar = array_tar + (size - 1);
  int i = size;
  while (i--) {
    *(tar--) = val;
  }
}
 
void copy_vn_short(short *array_tar, const int size, const short val)
{
  short *tar = array_tar + (size - 1);
  int i = size;
  while (i--) {
    *(tar--) = val;
  }
}
 
void copy_vn_ushort(ushort *array_tar, const int size, const ushort val)
{
  ushort *tar = array_tar + (size - 1);
  int i = size;
  while (i--) {
    *(tar--) = val;
  }
}
 
void copy_vn_uchar(uchar *array_tar, const int size, const uchar val)
{
  uchar *tar = array_tar + (size - 1);
  int i = size;
  while (i--) {
    *(tar--) = val;
  }
}
 
void copy_vn_fl(float *array_tar, const int size, const float val)
{
  float *tar = array_tar + (size - 1);
  int i = size;
  while (i--) {
    *(tar--) = val;
  }
}
 
/* -------------------------------------------------------------------- */
void add_vn_vn_d(double *array_tar, const double *array_src, const int size)
{
  double *tar = array_tar + (size - 1);
  const double *src = array_src + (size - 1);
  int i = size;
  while (i--) {
    *(tar--) += *(src--);
  }
}
 
void add_vn_vnvn_d(double *array_tar,
                   const double *array_src_a,
                   const double *array_src_b,
                   const int size)
{
  double *tar = array_tar + (size - 1);
  const double *src_a = array_src_a + (size - 1);
  const double *src_b = array_src_b + (size - 1);
  int i = size;
  while (i--) {
    *(tar--) = *(src_a--) + *(src_b--);
  }
}
 
void mul_vn_db(double *array_tar, const int size, const double f)
{
  double *array_pt = array_tar + (size - 1);
  int i = size;
  while (i--) {
    *(array_pt--) *= f;
  }
}
 
void interp_v3_v3v3_db(double target[3], const double a[3], const double b[3], const double t)
{
  const double s = 1.0f - t;
 
  target[0] = s * a[0] + t * b[0];
  target[1] = s * a[1] + t * b[1];
  target[2] = s * a[2] + t * b[2];
}
 
void interp_v2_v2v2_db(double target[2], const double a[2], const double b[2], const double t)
{
  const double s = 1.0f - t;
 
  target[0] = s * a[0] + t * b[0];
  target[1] = s * a[1] + t * b[1];
}