[算法]检测空间三角形相交算法(Devillers & Guigue算法)

#pragma once
//GYDevillersTriangle.h
/*      快速检测空间三角形相交算法的代码实现(Devillers & Guigue算法)
博客原地址:http://blog.csdn.net/fourierfeng/article/details/11969915#

Devillers & Guigue算法(简称Devillers 算法) 通过三角形各顶点构成的行列式正负的几何意义来判断三角形中点、线、面之间的相对位置关系,
从而判断两三角形是否相交。其基本原理如下:给定空间四个点：a(ax, ay, az), b = (bx, by, bz), c = (cx, cy, cz), d = (dx, dy, dz), 定义行列式如下：

[a, b, c, d] 采用右手螺旋法则定义了四个空间点的位置关系。
[a, b, c, d] > 0 表示 d 在 a、b、c 按逆时针顺序所组成的三角形的正法线方向(即上方);
[a, b, c, d] < 0 表示 d 在 △abc的下方; [a, b, c, d] = 0 表示四点共面。

设两个三角形T1和T2，顶点分别为：V10，V11，V12和V20，V21，V22，
三角形所在的平面分别为π1和π2，其法向量分别为N1和N2.算法先判别三角形和另一个三角形所在的平面的相互位置关系, 提前排除不相交的情况。
通过计算[V20, V21, V22, V1i].(i = 0, 1, 2)来判断T1和π2的关系：如果所有的行列式的值都不为零且同号，则T1和T2不相交；否则T1和π2相交。
相交又分为如下几种情况：
a)如果所有的行列式的值为零，则T1和T2共面，转化为共面的线段相交问题。
b)如果其中一个行列式的值为零，而其他两个行列式同号，则只有一个点在平面内，测试顶点是否则T2内部，是则相交，否则不相交；
c)否则T1的顶点位于平面π2两侧(包含T1的一条边在平面π2中的情况)。

再按照类似的方法对 T 2 和 π 1 作进一步的测试。如果通过测试, 则每个三角形必有确定的一点位于另一个三角形所在平面的一侧,
而另外两点位于其另一侧。算法分别循环置换每个三角形的顶点, 以使V10(V20)位于π2(π1)的一侧，另两个点位于其另一侧；
同时对顶点V21，V22(V11, V12)进行交换操作，以确保V10(V20)位于π2(π1)的上方，即正法线方向。
经过以上的预排除和置换操作，V10的邻边V10V11，V10V12和V20的邻边V20V21和V20V22与两平面的交线L相交于固定形式的点上，
分别记为i，j，k，l(i<j, k<l), 如图：(参看原博客)
这些点在L上形成的封闭区间为i1 = [i, j], i2 = [k, l].至此，两个三角形的相交测试问题转换为封闭区间i1，i2的重叠问题。
若重叠则相交，否则不相交。由于交点形式固定，只需满足条件k <= j且i <= l即表明区间重叠，条件还可进一步缩减为判别式
(1)是否成立：
[V10, V11, V20, V21] <= 0 && [V10, V12, V22, V20] <= 0        判别式(1)
*/

typedef float float3[3];

enum TopologicalStructure
{
INTERSECT, NONINTERSECT
};

struct Triangle
{
//float3 Normal_0;
float3 Vertex_1, Vertex_2, Vertex_3;
};

/*******************************************************************************************************/
//Devillers算法主函数
TopologicalStructure judge_triangle_topologicalStructure(Triangle* tri1, Triangle* tri2);

//返回bool值
bool isTriangleTntersect(Triangle* tri1, Triangle* tri2)
{
TopologicalStructure  intersectSt = judge_triangle_topologicalStructure(tri1, tri2);
if (intersectSt == INTERSECT)
return true;
return false;
}

//GYDevillersTriangle.cpp
#include "GYDevillersTriangle.h"
#pragma once

struct point
{
float x, y;
};

//三维点拷贝为二维点
static void copy_point(point& p, float3 f)
{
p.x = f[0];
p.y = f[1];
}

//四点行列式
inline float get_vector4_det(float3 v1, float3 v2, float3 v3, float3 v4)
{
float a[3][3];
for (int i = 0; i != 3; ++i)
{
a[0][i] = v1[i] - v4[i];
a[1][i] = v2[i] - v4[i];
a[2][i] = v3[i] - v4[i];
}

return a[0][0] * a[1][1] * a[2][2]
+ a[0][1] * a[1][2] * a[2][0]
+ a[0][2] * a[1][0] * a[2][1]
- a[0][2] * a[1][1] * a[2][0]
- a[0][1] * a[1][0] * a[2][2]
- a[0][0] * a[1][2] * a[2][1];
}

//利用叉积计算点p相对线段p1p2的方位
inline double direction(point p1, point p2, point p) {
return (p.x - p1.x) * (p2.y - p1.y) - (p2.x - p1.x) * (p.y - p1.y);
}

//确定与线段p1p2共线的点p是否在线段p1p2上
inline int on_segment(point p1, point p2, point p) {
double max = p1.x > p2.x ? p1.x : p2.x;
double min = p1.x < p2.x ? p1.x : p2.x;
double max1 = p1.y > p2.y ? p1.y : p2.y;
double min1 = p1.y < p2.y ? p1.y : p2.y;
if (p.x >= min && p.x <= max && p.y >= min1 && p.y <= max1)
{
return 1;
}
else
{
return 0;
}
}

//判断线段p1p2与线段p3p4是否相交的主函数
inline int segments_intersert(point p1, point p2, point p3, point p4) {
double d1, d2, d3, d4;
d1 = direction(p3, p4, p1);
d2 = direction(p3, p4, p2);
d3 = direction(p1, p2, p3);
d4 = direction(p1, p2, p4);
if (d1 * d2 < 0 && d3 * d4 < 0)
{
return 1;
}
else if (d1 == 0 && on_segment(p3, p4, p1) == 1)
{
return 1;
}
else if (d2 == 0 && on_segment(p3, p4, p2) == 1)
{
return 1;
}
else if (d3 == 0 && on_segment(p1, p2, p3) == 1)
{
return 1;
}
else if (d4 == 0 && on_segment(p1, p2, p4) == 1)
{
return 1;
}
return 0;
}

//判断同一平面的直线和三角形是否相交
inline bool line_triangle_intersert_inSamePlane(Triangle* tri, float3 f1, float3 f2)
{
point p1, p2, p3, p4;

copy_point(p1, f1);

copy_point(p2, f2);

copy_point(p3, tri->Vertex_1);

copy_point(p4, tri->Vertex_2);

if (segments_intersert(p1, p2, p3, p4))
{
return true;
}

copy_point(p3, tri->Vertex_2);

copy_point(p4, tri->Vertex_3);

if (segments_intersert(p1, p2, p3, p4))
{
return true;
}

copy_point(p3, tri->Vertex_1);

copy_point(p4, tri->Vertex_3);

if (segments_intersert(p1, p2, p3, p4))
{
return true;
}

return false;
}

inline void get_central_point(float3 centralPoint, Triangle* tri)
{
centralPoint[0] = (tri->Vertex_1[0] + tri->Vertex_2[0] + tri->Vertex_3[0]) / 3;

centralPoint[1] = (tri->Vertex_1[1] + tri->Vertex_2[1] + tri->Vertex_3[1]) / 3;

centralPoint[2] = (tri->Vertex_1[2] + tri->Vertex_2[2] + tri->Vertex_3[2]) / 3;
}

//向量之差
inline void get_vector_diff(float3& aimV, const float3 a, const float3 b)
{
aimV[0] = b[0] - a[0];

aimV[1] = b[1] - a[1];

aimV[2] = b[2] - a[2];
}

//向量内积
inline float Dot(const float3& v1, const float3& v2)
{
return v1[0] * v2[0] + v1[1] * v2[1] + v1[2] * v2[2];
}

//重心法判断点是否在三角形内部
inline bool is_point_within_triangle(Triangle* tri, float3 point)
{
float3 v0;
get_vector_diff(v0, tri->Vertex_1, tri->Vertex_3);
float3 v1;
get_vector_diff(v1, tri->Vertex_1, tri->Vertex_2);
float3 v2;
get_vector_diff(v2, tri->Vertex_1, point);
float dot00 = Dot(v0, v0);
float dot01 = Dot(v0, v1);
float dot02 = Dot(v0, v2);
float dot11 = Dot(v1, v1);
float dot12 = Dot(v1, v2);
float inverDeno = 1 / (dot00* dot11 - dot01* dot01);
float u = (dot11* dot02 - dot01* dot12) * inverDeno;
if (u < 0 || u > 1) // if u out of range, return directly
{
return false;
}
float v = (dot00* dot12 - dot01* dot02) * inverDeno;
if (v < 0 || v > 1) // if v out of range, return directly
{
return false;
}
return u + v <= 1;
}

//判断同一平面内的三角形是否相交
inline bool triangle_intersert_inSamePlane(Triangle* tri1, Triangle* tri2)
{
if (line_triangle_intersert_inSamePlane(tri2, tri1->Vertex_1, tri1->Vertex_2))
{
return true;
}
else if (line_triangle_intersert_inSamePlane(tri2, tri1->Vertex_2, tri1->Vertex_3))
{
return true;
}
else if (line_triangle_intersert_inSamePlane(tri2, tri1->Vertex_1, tri1->Vertex_3))
{
return true;
}
else
{
float3 centralPoint1, centralPoint2;

get_central_point(centralPoint1, tri1);

get_central_point(centralPoint2, tri2);

if (is_point_within_triangle(tri2, centralPoint1) || is_point_within_triangle(tri1, centralPoint2))
{
return true;
}

return false;
}
}

//Devillers算法主函数
TopologicalStructure judge_triangle_topologicalStructure(Triangle* tri1, Triangle* tri2)
{
//设tri1所在的平面为p1,tri2所在的平面为p2
float p1_tri2_vertex1 = get_vector4_det(tri1->Vertex_1, tri1->Vertex_2, tri1->Vertex_3, tri2->Vertex_1);

float p1_tri2_vertex2 = get_vector4_det(tri1->Vertex_1, tri1->Vertex_2, tri1->Vertex_3, tri2->Vertex_2);

float p1_tri2_vertex3 = get_vector4_det(tri1->Vertex_1, tri1->Vertex_2, tri1->Vertex_3, tri2->Vertex_3);

if (p1_tri2_vertex1 > 0 && p1_tri2_vertex2 > 0 && p1_tri2_vertex3 > 0)
{
return NONINTERSECT;
}

if (p1_tri2_vertex1 < 0 && p1_tri2_vertex2 < 0 && p1_tri2_vertex3 < 0)
{
return NONINTERSECT;
}

if (p1_tri2_vertex1 == 0 && p1_tri2_vertex2 == 0 && p1_tri2_vertex3 == 0)
{
if (triangle_intersert_inSamePlane(tri1, tri2))
{
return INTERSECT;
}
else
{
return NONINTERSECT;
}
}

if (p1_tri2_vertex1 == 0 && p1_tri2_vertex2 * p1_tri2_vertex3 > 0)
{
if (is_point_within_triangle(tri1, tri2->Vertex_1))
{
return INTERSECT;
}
else
{
return NONINTERSECT;
}
}
else if (p1_tri2_vertex2 == 0 && p1_tri2_vertex1 * p1_tri2_vertex3 > 0)
{
if (is_point_within_triangle(tri1, tri2->Vertex_2))
{
return INTERSECT;
}
else
{
return NONINTERSECT;
}
}
else if (p1_tri2_vertex3 == 0 && p1_tri2_vertex1 * p1_tri2_vertex2 > 0)
{
if (is_point_within_triangle(tri1, tri2->Vertex_3))
{
return INTERSECT;
}
else
{
return NONINTERSECT;
}
}

float p2_tri1_vertex1 = get_vector4_det(tri2->Vertex_1, tri2->Vertex_2, tri2->Vertex_3, tri1->Vertex_1);

float p2_tri1_vertex2 = get_vector4_det(tri2->Vertex_1, tri2->Vertex_2, tri2->Vertex_3, tri1->Vertex_2);

float p2_tri1_vertex3 = get_vector4_det(tri2->Vertex_1, tri2->Vertex_2, tri2->Vertex_3, tri1->Vertex_3);

if (p2_tri1_vertex1 > 0 && p2_tri1_vertex2 > 0 && p2_tri1_vertex3 > 0)
{
return NONINTERSECT;
}

if (p2_tri1_vertex1 < 0 && p2_tri1_vertex2 < 0 && p2_tri1_vertex3 < 0)
{
return NONINTERSECT;
}

if (p2_tri1_vertex1 == 0 && p2_tri1_vertex2 * p2_tri1_vertex3 > 0)
{
if (is_point_within_triangle(tri2, tri1->Vertex_1))
{
return INTERSECT;
}
else
{
return NONINTERSECT;
}
}

if (p2_tri1_vertex2 == 0 && p2_tri1_vertex1 * p2_tri1_vertex3 > 0)
{
if (is_point_within_triangle(tri2, tri1->Vertex_2))
{
return INTERSECT;
}
else
{
return NONINTERSECT;
}
}

if (p2_tri1_vertex3 == 0 && p2_tri1_vertex1 * p2_tri1_vertex2 > 0)
{
if (is_point_within_triangle(tri2, tri1->Vertex_3))
{
return INTERSECT;
}
else
{
return NONINTERSECT;
}
}

float* tri1_a = tri1->Vertex_1, *tri1_b = tri1->Vertex_2, *tri1_c = tri1->Vertex_3
, *tri2_a = tri2->Vertex_1, *tri2_b = tri2->Vertex_2, *tri2_c = tri2->Vertex_3;

float* m;

float im;

if (p2_tri1_vertex2 * p2_tri1_vertex3 >= 0 && p2_tri1_vertex1 != 0)
{
if (p2_tri1_vertex1 < 0)
{
m = tri2_b;
tri2_b = tri2_c;
tri2_c = m;

im = p1_tri2_vertex2;
p1_tri2_vertex2 = p1_tri2_vertex3;
p1_tri2_vertex3 = im;
}
}
else if (p2_tri1_vertex1 * p2_tri1_vertex3 >= 0 && p2_tri1_vertex2 != 0)
{
m = tri1_a;
tri1_a = tri1_b;
tri1_b = tri1_c;
tri1_c = m;

if (p2_tri1_vertex2 < 0)
{
m = tri2_b;
tri2_b = tri2_c;
tri2_c = m;

im = p1_tri2_vertex2;
p1_tri2_vertex2 = p1_tri2_vertex3;
p1_tri2_vertex3 = im;
}
}
else if (p2_tri1_vertex1 * p2_tri1_vertex2 >= 0 && p2_tri1_vertex3 != 0)
{
m = tri1_a;

tri1_a = tri1_c;

tri1_c = tri1_b;

tri1_b = m;

if (p2_tri1_vertex3 < 0)
{
m = tri2_b;
tri2_b = tri2_c;
tri2_c = m;

im = p1_tri2_vertex2;
p1_tri2_vertex2 = p1_tri2_vertex3;
p1_tri2_vertex3 = im;
}
}

if (p1_tri2_vertex2 * p1_tri2_vertex3 >= 0 && p1_tri2_vertex1 != 0)
{
if (p1_tri2_vertex1 < 0)
{
m = tri1_b;
tri1_b = tri1_c;
tri1_c = m;
}
}
else if (p1_tri2_vertex1 * p1_tri2_vertex3 >= 0 && p1_tri2_vertex2 != 0)
{
m = tri2_a;

tri2_a = tri2_b;

tri2_b = tri2_c;

tri2_c = m;

if (p1_tri2_vertex2 < 0)
{
m = tri1_b;
tri1_b = tri1_c;
tri1_c = m;
}
}
else if (p1_tri2_vertex1 * p1_tri2_vertex2 >= 0 && p1_tri2_vertex3 != 0)
{
m = tri2_a;

tri2_a = tri2_c;

tri2_c = tri2_b;

tri2_b = m;

if (p1_tri2_vertex3 < 0)
{
m = tri1_b;
tri1_b = tri1_c;
tri1_c = m;
}
}

if (get_vector4_det(tri1_a, tri1_b, tri2_a, tri2_b) <= 0 && get_vector4_det(tri1_a, tri1_c, tri2_c, tri2_a) <= 0)
{
return INTERSECT;
}
else
{
return NONINTERSECT;
}
}