#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; } }