godot/core/math/face3.cpp

392 lines
11 KiB
C++
Raw Normal View History

2014-02-10 01:10:30 +00:00
/*************************************************************************/
/* face3.cpp */
/*************************************************************************/
/* This file is part of: */
/* GODOT ENGINE */
/* https://godotengine.org */
2014-02-10 01:10:30 +00:00
/*************************************************************************/
/* Copyright (c) 2007-2021 Juan Linietsky, Ariel Manzur. */
/* Copyright (c) 2014-2021 Godot Engine contributors (cf. AUTHORS.md). */
2014-02-10 01:10:30 +00:00
/* */
/* Permission is hereby granted, free of charge, to any person obtaining */
/* a copy of this software and associated documentation files (the */
/* "Software"), to deal in the Software without restriction, including */
/* without limitation the rights to use, copy, modify, merge, publish, */
/* distribute, sublicense, and/or sell copies of the Software, and to */
/* permit persons to whom the Software is furnished to do so, subject to */
/* the following conditions: */
/* */
/* The above copyright notice and this permission notice shall be */
/* included in all copies or substantial portions of the Software. */
/* */
/* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, */
/* EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF */
/* MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.*/
/* IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY */
/* CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, */
/* TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE */
/* SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. */
/*************************************************************************/
2014-02-10 01:10:30 +00:00
#include "face3.h"
#include "core/math/geometry_3d.h"
2014-02-10 01:10:30 +00:00
int Face3::split_by_plane(const Plane &p_plane, Face3 p_res[3], bool p_is_point_over[3]) const {
ERR_FAIL_COND_V(is_degenerate(), 0);
2014-02-10 01:10:30 +00:00
Vector3 above[4];
int above_count = 0;
2014-02-10 01:10:30 +00:00
Vector3 below[4];
int below_count = 0;
2014-02-10 01:10:30 +00:00
for (int i = 0; i < 3; i++) {
if (p_plane.has_point(vertex[i], CMP_EPSILON)) { // point is in plane
2014-02-10 01:10:30 +00:00
ERR_FAIL_COND_V(above_count >= 4, 0);
above[above_count++] = vertex[i];
ERR_FAIL_COND_V(below_count >= 4, 0);
below[below_count++] = vertex[i];
2014-02-10 01:10:30 +00:00
} else {
if (p_plane.is_point_over(vertex[i])) {
2014-02-10 01:10:30 +00:00
//Point is over
ERR_FAIL_COND_V(above_count >= 4, 0);
above[above_count++] = vertex[i];
2014-02-10 01:10:30 +00:00
} else {
//Point is under
ERR_FAIL_COND_V(below_count >= 4, 0);
below[below_count++] = vertex[i];
2014-02-10 01:10:30 +00:00
}
/* Check for Intersection between this and the next vertex*/
Vector3 inters;
if (!p_plane.intersects_segment(vertex[i], vertex[(i + 1) % 3], &inters)) {
2014-02-10 01:10:30 +00:00
continue;
}
2014-02-10 01:10:30 +00:00
/* Intersection goes to both */
ERR_FAIL_COND_V(above_count >= 4, 0);
above[above_count++] = inters;
ERR_FAIL_COND_V(below_count >= 4, 0);
below[below_count++] = inters;
2014-02-10 01:10:30 +00:00
}
}
int polygons_created = 0;
2014-02-10 01:10:30 +00:00
ERR_FAIL_COND_V(above_count >= 4 && below_count >= 4, 0); //bug in the algo
2014-02-10 01:10:30 +00:00
if (above_count >= 3) {
p_res[polygons_created] = Face3(above[0], above[1], above[2]);
p_is_point_over[polygons_created] = true;
2014-02-10 01:10:30 +00:00
polygons_created++;
if (above_count == 4) {
p_res[polygons_created] = Face3(above[2], above[3], above[0]);
p_is_point_over[polygons_created] = true;
2014-02-10 01:10:30 +00:00
polygons_created++;
}
}
if (below_count >= 3) {
p_res[polygons_created] = Face3(below[0], below[1], below[2]);
p_is_point_over[polygons_created] = false;
2014-02-10 01:10:30 +00:00
polygons_created++;
if (below_count == 4) {
p_res[polygons_created] = Face3(below[2], below[3], below[0]);
p_is_point_over[polygons_created] = false;
2014-02-10 01:10:30 +00:00
polygons_created++;
}
}
return polygons_created;
}
bool Face3::intersects_ray(const Vector3 &p_from, const Vector3 &p_dir, Vector3 *p_intersection) const {
return Geometry3D::ray_intersects_triangle(p_from, p_dir, vertex[0], vertex[1], vertex[2], p_intersection);
2014-02-10 01:10:30 +00:00
}
bool Face3::intersects_segment(const Vector3 &p_from, const Vector3 &p_dir, Vector3 *p_intersection) const {
return Geometry3D::segment_intersects_triangle(p_from, p_dir, vertex[0], vertex[1], vertex[2], p_intersection);
2014-02-10 01:10:30 +00:00
}
bool Face3::is_degenerate() const {
Vector3 normal = vec3_cross(vertex[0] - vertex[1], vertex[0] - vertex[2]);
2014-02-10 01:10:30 +00:00
return (normal.length_squared() < CMP_EPSILON2);
}
Face3::Side Face3::get_side_of(const Face3 &p_face, ClockDirection p_clock_dir) const {
int over = 0, under = 0;
2014-02-10 01:10:30 +00:00
Plane plane = get_plane(p_clock_dir);
2014-02-10 01:10:30 +00:00
for (int i = 0; i < 3; i++) {
const Vector3 &v = p_face.vertex[i];
2014-02-10 01:10:30 +00:00
if (plane.has_point(v)) { //coplanar, don't bother
2014-02-10 01:10:30 +00:00
continue;
}
2014-02-10 01:10:30 +00:00
if (plane.is_point_over(v)) {
2014-02-10 01:10:30 +00:00
over++;
} else {
2014-02-10 01:10:30 +00:00
under++;
}
2014-02-10 01:10:30 +00:00
}
if (over > 0 && under == 0) {
2014-02-10 01:10:30 +00:00
return SIDE_OVER;
} else if (under > 0 && over == 0) {
2014-02-10 01:10:30 +00:00
return SIDE_UNDER;
} else if (under == 0 && over == 0) {
2014-02-10 01:10:30 +00:00
return SIDE_COPLANAR;
} else {
2014-02-10 01:10:30 +00:00
return SIDE_SPANNING;
}
2014-02-10 01:10:30 +00:00
}
Vector3 Face3::get_random_point_inside() const {
real_t a = Math::random(0.0, 1.0);
real_t b = Math::random(0.0, 1.0);
if (a > b) {
SWAP(a, b);
2014-02-10 01:10:30 +00:00
}
return vertex[0] * a + vertex[1] * (b - a) + vertex[2] * (1.0 - b);
2014-02-10 01:10:30 +00:00
}
Plane Face3::get_plane(ClockDirection p_dir) const {
return Plane(vertex[0], vertex[1], vertex[2], p_dir);
2014-02-10 01:10:30 +00:00
}
Vector3 Face3::get_median_point() const {
return (vertex[0] + vertex[1] + vertex[2]) / 3.0;
2014-02-10 01:10:30 +00:00
}
real_t Face3::get_area() const {
return vec3_cross(vertex[0] - vertex[1], vertex[0] - vertex[2]).length() * 0.5;
2014-02-10 01:10:30 +00:00
}
ClockDirection Face3::get_clock_dir() const {
Vector3 normal = vec3_cross(vertex[0] - vertex[1], vertex[0] - vertex[2]);
2014-02-10 01:10:30 +00:00
//printf("normal is %g,%g,%g x %g,%g,%g- wtfu is %g\n",tofloat(normal.x),tofloat(normal.y),tofloat(normal.z),tofloat(vertex[0].x),tofloat(vertex[0].y),tofloat(vertex[0].z),tofloat( normal.dot( vertex[0] ) ) );
return (normal.dot(vertex[0]) >= 0) ? CLOCKWISE : COUNTERCLOCKWISE;
2014-02-10 01:10:30 +00:00
}
2017-11-17 02:09:00 +00:00
bool Face3::intersects_aabb(const AABB &p_aabb) const {
2014-02-10 01:10:30 +00:00
/** TEST PLANE **/
if (!p_aabb.intersects_plane(get_plane())) {
2014-02-10 01:10:30 +00:00
return false;
}
2014-02-10 01:10:30 +00:00
#define TEST_AXIS(m_ax) \
/** TEST FACE AXIS */ \
{ \
real_t aabb_min = p_aabb.position.m_ax; \
real_t aabb_max = p_aabb.position.m_ax + p_aabb.size.m_ax; \
Fix more "may be used initialized" warnings from GCC 7 Fixes the following GCC 7 warnings: ``` core/cowdata.h:269:47: warning: 'alloc_size' may be used uninitialized in this function [-Wmaybe-uninitialized] core/error_macros.h:163:26: warning: 'nearest_point' may be used uninitialized in this function [-Wmaybe-uninitialized] core/image.cpp:1579:5: warning: 'colormap_size' may be used uninitialized in this function [-Wmaybe-uninitialized] core/image.cpp:1582:12: warning: 'size_height' may be used uninitialized in this function [-Wmaybe-uninitialized] core/image.cpp:1590:23: warning: 'size_width' may be used uninitialized in this function [-Wmaybe-uninitialized] core/image.cpp:1599:29: warning: 'pixel_size' may be used uninitialized in this function [-Wmaybe-uninitialized] core/math/face3.cpp:207:15: warning: 'tri_max' may be used uninitialized in this function [-Wmaybe-uninitialized] core/math/face3.cpp:209:15: warning: 'tri_min' may be used uninitialized in this function [-Wmaybe-uninitialized] drivers/gles3/rasterizer_scene_gles3.cpp:665:22: warning: 'best_used_frame' may be used uninitialized in this function [-Wmaybe-uninitialized] drivers/gles3/rasterizer_storage_gles3.cpp:865:27: warning: 'blit_target' may be used uninitialized in this function [-Wmaybe-uninitialized] drivers/gles3/rasterizer_storage_gles3.cpp:980:29: warning: 'blit_target' may be used uninitialized in this function [-Wmaybe-uninitialized] drivers/gles3/shader_gles3.h:122:9: warning: '<anonymous>.ShaderGLES3::Version::frag_id' may be used uninitialized in this function [-Wmaybe-uninitialized] drivers/gles3/shader_gles3.h:122:9: warning: '<anonymous>.ShaderGLES3::Version::id' may be used uninitialized in this function [-Wmaybe-uninitialized] drivers/gles3/shader_gles3.h:122:9: warning: '<anonymous>.ShaderGLES3::Version::vert_id' may be used uninitialized in this function [-Wmaybe-uninitialized] editor/plugins/script_editor_plugin.cpp:1980:31: warning: 'se' may be used uninitialized in this function [-Wmaybe-uninitialized] editor/scene_tree_dock.cpp:840:30: warning: 'new_node' may be used uninitialized in this function [-Wmaybe-uninitialized] editor/spatial_editor_gizmos.cpp:4259:9: warning: 'a1' may be used uninitialized in this function [-Wmaybe-uninitialized] editor/spatial_editor_gizmos.cpp:4259:9: warning: 'lll' may be used uninitialized in this function [-Wmaybe-uninitialized] editor/spatial_editor_gizmos.cpp:4259:9: warning: 'lul' may be used uninitialized in this function [-Wmaybe-uninitialized] editor/spatial_editor_gizmos.cpp:4260:9: warning: 'a2' may be used uninitialized in this function [-Wmaybe-uninitialized] editor/spatial_editor_gizmos.cpp:4261:9: warning: 'a3' may be used uninitialized in this function [-Wmaybe-uninitialized] editor/spatial_editor_gizmos.cpp:4265:3: warning: 'enable_lin' may be used uninitialized in this function [-Wmaybe-uninitialized] editor/spatial_editor_gizmos.cpp:4294:3: warning: 'enable_ang' may be used uninitialized in this function [-Wmaybe-uninitialized] editor/spatial_editor_gizmos.cpp:4311:34: warning: 'll' may be used uninitialized in this function [-Wmaybe-uninitialized] editor/spatial_editor_gizmos.cpp:4311:34: warning: 'ul' may be used uninitialized in this function [-Wmaybe-uninitialized] scene/3d/voxel_light_baker.cpp:1655:47: warning: 'cone_dirs' may be used uninitialized in this function [-Wmaybe-uninitialized] scene/3d/voxel_light_baker.cpp:1656:73: warning: 'cone_weights' may be used uninitialized in this function [-Wmaybe-uninitialized] scene/gui/texture_progress.cpp:181:6: warning: 'cp' may be used uninitialized in this function [-Wmaybe-uninitialized] scene/gui/texture_progress.cpp:181:6: warning: 'cq' may be used uninitialized in this function [-Wmaybe-uninitialized] servers/physics/shape_sw.cpp:1056:19: warning: 'support_max' may be used uninitialized in this function [-Wmaybe-uninitialized] ```
2018-10-04 16:54:20 +00:00
real_t tri_min = vertex[0].m_ax; \
real_t tri_max = vertex[0].m_ax; \
for (int i = 1; i < 3; i++) { \
if (vertex[i].m_ax > tri_max) \
tri_max = vertex[i].m_ax; \
Fix more "may be used initialized" warnings from GCC 7 Fixes the following GCC 7 warnings: ``` core/cowdata.h:269:47: warning: 'alloc_size' may be used uninitialized in this function [-Wmaybe-uninitialized] core/error_macros.h:163:26: warning: 'nearest_point' may be used uninitialized in this function [-Wmaybe-uninitialized] core/image.cpp:1579:5: warning: 'colormap_size' may be used uninitialized in this function [-Wmaybe-uninitialized] core/image.cpp:1582:12: warning: 'size_height' may be used uninitialized in this function [-Wmaybe-uninitialized] core/image.cpp:1590:23: warning: 'size_width' may be used uninitialized in this function [-Wmaybe-uninitialized] core/image.cpp:1599:29: warning: 'pixel_size' may be used uninitialized in this function [-Wmaybe-uninitialized] core/math/face3.cpp:207:15: warning: 'tri_max' may be used uninitialized in this function [-Wmaybe-uninitialized] core/math/face3.cpp:209:15: warning: 'tri_min' may be used uninitialized in this function [-Wmaybe-uninitialized] drivers/gles3/rasterizer_scene_gles3.cpp:665:22: warning: 'best_used_frame' may be used uninitialized in this function [-Wmaybe-uninitialized] drivers/gles3/rasterizer_storage_gles3.cpp:865:27: warning: 'blit_target' may be used uninitialized in this function [-Wmaybe-uninitialized] drivers/gles3/rasterizer_storage_gles3.cpp:980:29: warning: 'blit_target' may be used uninitialized in this function [-Wmaybe-uninitialized] drivers/gles3/shader_gles3.h:122:9: warning: '<anonymous>.ShaderGLES3::Version::frag_id' may be used uninitialized in this function [-Wmaybe-uninitialized] drivers/gles3/shader_gles3.h:122:9: warning: '<anonymous>.ShaderGLES3::Version::id' may be used uninitialized in this function [-Wmaybe-uninitialized] drivers/gles3/shader_gles3.h:122:9: warning: '<anonymous>.ShaderGLES3::Version::vert_id' may be used uninitialized in this function [-Wmaybe-uninitialized] editor/plugins/script_editor_plugin.cpp:1980:31: warning: 'se' may be used uninitialized in this function [-Wmaybe-uninitialized] editor/scene_tree_dock.cpp:840:30: warning: 'new_node' may be used uninitialized in this function [-Wmaybe-uninitialized] editor/spatial_editor_gizmos.cpp:4259:9: warning: 'a1' may be used uninitialized in this function [-Wmaybe-uninitialized] editor/spatial_editor_gizmos.cpp:4259:9: warning: 'lll' may be used uninitialized in this function [-Wmaybe-uninitialized] editor/spatial_editor_gizmos.cpp:4259:9: warning: 'lul' may be used uninitialized in this function [-Wmaybe-uninitialized] editor/spatial_editor_gizmos.cpp:4260:9: warning: 'a2' may be used uninitialized in this function [-Wmaybe-uninitialized] editor/spatial_editor_gizmos.cpp:4261:9: warning: 'a3' may be used uninitialized in this function [-Wmaybe-uninitialized] editor/spatial_editor_gizmos.cpp:4265:3: warning: 'enable_lin' may be used uninitialized in this function [-Wmaybe-uninitialized] editor/spatial_editor_gizmos.cpp:4294:3: warning: 'enable_ang' may be used uninitialized in this function [-Wmaybe-uninitialized] editor/spatial_editor_gizmos.cpp:4311:34: warning: 'll' may be used uninitialized in this function [-Wmaybe-uninitialized] editor/spatial_editor_gizmos.cpp:4311:34: warning: 'ul' may be used uninitialized in this function [-Wmaybe-uninitialized] scene/3d/voxel_light_baker.cpp:1655:47: warning: 'cone_dirs' may be used uninitialized in this function [-Wmaybe-uninitialized] scene/3d/voxel_light_baker.cpp:1656:73: warning: 'cone_weights' may be used uninitialized in this function [-Wmaybe-uninitialized] scene/gui/texture_progress.cpp:181:6: warning: 'cp' may be used uninitialized in this function [-Wmaybe-uninitialized] scene/gui/texture_progress.cpp:181:6: warning: 'cq' may be used uninitialized in this function [-Wmaybe-uninitialized] servers/physics/shape_sw.cpp:1056:19: warning: 'support_max' may be used uninitialized in this function [-Wmaybe-uninitialized] ```
2018-10-04 16:54:20 +00:00
if (vertex[i].m_ax < tri_min) \
tri_min = vertex[i].m_ax; \
} \
\
if (tri_max < aabb_min || aabb_max < tri_min) \
return false; \
2014-02-10 01:10:30 +00:00
}
TEST_AXIS(x);
TEST_AXIS(y);
TEST_AXIS(z);
/** TEST ALL EDGES **/
Vector3 edge_norms[3] = {
vertex[0] - vertex[1],
vertex[1] - vertex[2],
vertex[2] - vertex[0],
2014-02-10 01:10:30 +00:00
};
for (int i = 0; i < 12; i++) {
Vector3 from, to;
p_aabb.get_edge(i, from, to);
Vector3 e1 = from - to;
for (int j = 0; j < 3; j++) {
Vector3 e2 = edge_norms[j];
2014-02-10 01:10:30 +00:00
Vector3 axis = vec3_cross(e1, e2);
2014-02-10 01:10:30 +00:00
if (axis.length_squared() < 0.0001) {
2014-02-10 01:10:30 +00:00
continue; // coplanar
}
2014-02-10 01:10:30 +00:00
axis.normalize();
real_t minA, maxA, minB, maxB;
p_aabb.project_range_in_plane(Plane(axis, 0), minA, maxA);
project_range(axis, Transform3D(), minB, maxB);
2014-02-10 01:10:30 +00:00
if (maxA < minB || maxB < minA) {
2014-02-10 01:10:30 +00:00
return false;
}
2014-02-10 01:10:30 +00:00
}
}
return true;
}
Face3::operator String() const {
return String() + vertex[0] + ", " + vertex[1] + ", " + vertex[2];
2014-02-10 01:10:30 +00:00
}
void Face3::project_range(const Vector3 &p_normal, const Transform3D &p_transform, real_t &r_min, real_t &r_max) const {
for (int i = 0; i < 3; i++) {
Vector3 v = p_transform.xform(vertex[i]);
real_t d = p_normal.dot(v);
2014-02-10 01:10:30 +00:00
if (i == 0 || d > r_max) {
r_max = d;
}
2014-02-10 01:10:30 +00:00
if (i == 0 || d < r_min) {
r_min = d;
}
2014-02-10 01:10:30 +00:00
}
}
void Face3::get_support(const Vector3 &p_normal, const Transform3D &p_transform, Vector3 *p_vertices, int *p_count, int p_max) const {
2017-07-08 15:12:18 +00:00
#define _FACE_IS_VALID_SUPPORT_THRESHOLD 0.98
#define _EDGE_IS_VALID_SUPPORT_THRESHOLD 0.05
2014-02-10 01:10:30 +00:00
if (p_max <= 0) {
2014-02-10 01:10:30 +00:00
return;
}
2014-02-10 01:10:30 +00:00
Vector3 n = p_transform.basis.xform_inv(p_normal);
2014-02-10 01:10:30 +00:00
/** TEST FACE AS SUPPORT **/
2017-07-08 15:12:18 +00:00
if (get_plane().normal.dot(n) > _FACE_IS_VALID_SUPPORT_THRESHOLD) {
*p_count = MIN(3, p_max);
2014-02-10 01:10:30 +00:00
for (int i = 0; i < *p_count; i++) {
p_vertices[i] = p_transform.xform(vertex[i]);
2014-02-10 01:10:30 +00:00
}
return;
}
/** FIND SUPPORT VERTEX **/
int vert_support_idx = -1;
real_t support_max = 0;
2014-02-10 01:10:30 +00:00
for (int i = 0; i < 3; i++) {
real_t d = n.dot(vertex[i]);
2014-02-10 01:10:30 +00:00
if (i == 0 || d > support_max) {
support_max = d;
vert_support_idx = i;
2014-02-10 01:10:30 +00:00
}
}
/** TEST EDGES AS SUPPORT **/
for (int i = 0; i < 3; i++) {
if (i != vert_support_idx && i + 1 != vert_support_idx) {
2014-02-10 01:10:30 +00:00
continue;
}
2014-02-10 01:10:30 +00:00
// check if edge is valid as a support
real_t dot = (vertex[i] - vertex[(i + 1) % 3]).normalized().dot(n);
dot = ABS(dot);
2017-07-08 15:12:18 +00:00
if (dot < _EDGE_IS_VALID_SUPPORT_THRESHOLD) {
*p_count = MIN(2, p_max);
2014-02-10 01:10:30 +00:00
for (int j = 0; j < *p_count; j++) {
p_vertices[j] = p_transform.xform(vertex[(j + i) % 3]);
}
2014-02-10 01:10:30 +00:00
return;
}
}
*p_count = 1;
p_vertices[0] = p_transform.xform(vertex[vert_support_idx]);
2014-02-10 01:10:30 +00:00
}
Vector3 Face3::get_closest_point_to(const Vector3 &p_point) const {
Vector3 edge0 = vertex[1] - vertex[0];
Vector3 edge1 = vertex[2] - vertex[0];
Vector3 v0 = vertex[0] - p_point;
real_t a = edge0.dot(edge0);
real_t b = edge0.dot(edge1);
real_t c = edge1.dot(edge1);
real_t d = edge0.dot(v0);
real_t e = edge1.dot(v0);
real_t det = a * c - b * b;
real_t s = b * e - c * d;
real_t t = b * d - a * e;
if (s + t < det) {
if (s < 0.f) {
if (t < 0.f) {
if (d < 0.f) {
s = CLAMP(-d / a, 0.f, 1.f);
t = 0.f;
} else {
s = 0.f;
t = CLAMP(-e / c, 0.f, 1.f);
}
} else {
s = 0.f;
t = CLAMP(-e / c, 0.f, 1.f);
}
} else if (t < 0.f) {
s = CLAMP(-d / a, 0.f, 1.f);
t = 0.f;
} else {
real_t invDet = 1.f / det;
s *= invDet;
t *= invDet;
}
} else {
if (s < 0.f) {
real_t tmp0 = b + d;
real_t tmp1 = c + e;
if (tmp1 > tmp0) {
real_t numer = tmp1 - tmp0;
real_t denom = a - 2 * b + c;
s = CLAMP(numer / denom, 0.f, 1.f);
t = 1 - s;
} else {
t = CLAMP(-e / c, 0.f, 1.f);
s = 0.f;
}
} else if (t < 0.f) {
if (a + d > b + e) {
real_t numer = c + e - b - d;
real_t denom = a - 2 * b + c;
s = CLAMP(numer / denom, 0.f, 1.f);
t = 1 - s;
} else {
s = CLAMP(-d / a, 0.f, 1.f);
t = 0.f;
}
} else {
real_t numer = c + e - b - d;
real_t denom = a - 2 * b + c;
s = CLAMP(numer / denom, 0.f, 1.f);
t = 1.f - s;
}
}
2014-02-10 01:10:30 +00:00
return vertex[0] + s * edge0 + t * edge1;
}