需求：
對於給定的三角形面片3個頂點，和一條直線的2個點，求三角面和直線的交點，若無交點，輸出-1。
思路：
利用海倫公式，可以得到三角形的面積，然后用3個點的2個向量，進行叉乘，得到面的法向量。ax+by+cz=d可以表示面，求出常數d，聯力面的方程和直線方程，求解交點。
代碼：
class CVector
{
public:
union
{
float vec[3];
struct { float x, y, z; };
};
};
class CrossPoint
{
public:
CrossPoint();
virtual ~CrossPoint();
public:
static bool ValidPoint(CVector &LinePoint, CVector &LineV,
CVector &TrianglePoint1, CVector &TrianglePoint2, CVector &TrianglePoint3, CVector &result);
static float Area(float a, float b, float c);
static float Distance(CVector &p1, CVector &p2);
};
///////////////////////////////
CrossPoint::CrossPoint()
{
}
CrossPoint::~CrossPoint()
{
}
//計算p1到p2的距離的平方
float CrossPoint::Distance(CVector &p1, CVector &p2)
{
float dist;
dist = ((p2.x - p1.x)*(p2.x - p1.x)
+ (p2.y - p1.y)*(p2.y - p1.y)
+ (p2.z - p1.z)*(p2.z - p1.z));
return (float)sqrt(dist);
}
//利用海倫公式求變成為a,b,c的三角形的面積
float CrossPoint::Area(float a, float b, float c)
{
float s = (a + b + c) / 2;
return (float)sqrt(s*(s - a)*(s - b)*(s - c));
}
bool CrossPoint::ValidPoint(CVector &LinePoint1, CVector &LinePoint2, CVector &TrianglePoint1, CVector
&TrianglePoint2, CVector &TrianglePoint3, CVector &result)
{
//三角形所在平面的法向量
CVector TriangleV;
//三角形的邊方向向量
CVector VP12, VP13;
//直線與平面的交點
CVector CrossPoint;
//平面方程常數項
float TriD;
//CVector LineV = LinePoint2 - LinePoint1;
CVector LineV;
LineV.x = 0, LineV.y = 0, LineV.z = 100;
/*-------計算平面的法向量及常數項-------*/
//point1->point2
VP12.x = TrianglePoint2.x - TrianglePoint1.x;
VP12.y = TrianglePoint2.y - TrianglePoint1.y;
VP12.z = TrianglePoint2.z - TrianglePoint1.z;
//point1->point3
VP13.x = TrianglePoint3.x - TrianglePoint1.x;
VP13.y = TrianglePoint3.y - TrianglePoint1.y;
VP13.z = TrianglePoint3.z - TrianglePoint1.z;
//VP12xVP13
TriangleV.x = VP12.y*VP13.z - VP12.z*VP13.y;
TriangleV.y = -(VP12.x*VP13.z - VP12.z*VP13.x);
TriangleV.z = VP12.x*VP13.y - VP12.y*VP13.x;
//計算常數項
TriD = -(TriangleV.x*TrianglePoint1.x
+ TriangleV.y*TrianglePoint1.y
+ TriangleV.z*TrianglePoint1.z);
/*-------求解直線與平面的交點坐標---------*/
/* 思路：
* 首先將直線方程轉換為參數方程形式，然后代入平面方程，求得參數t，
* 將t代入直線的參數方程即可求出交點坐標
*/
float tempU, tempD; //臨時變量
tempU = TriangleV.x*LinePoint1.x + TriangleV.y*LinePoint1.y
+ TriangleV.z*LinePoint1.z + TriD;
tempD = TriangleV.x*LineV.x + TriangleV.y*LineV.y + TriangleV.z*LineV.z;
//直線與平面平行或在平面上
if (tempD == 0.0)
{
//printf("The line is parallel with the plane.\n");
return false;
}
//計算參數t
float t = -tempU / tempD;
//計算交點坐標
CrossPoint.x = LineV.x*t + LinePoint1.x;
CrossPoint.y = LineV.y*t + LinePoint1.y;
CrossPoint.z = LineV.z*t + LinePoint1.z;
/*----------判斷交點是否在三角形內部---*/
//計算三角形三條邊的長度
float d12 = Distance(TrianglePoint1, TrianglePoint2);
float d13 = Distance(TrianglePoint1, TrianglePoint3);
float d23 = Distance(TrianglePoint2, TrianglePoint3);
//計算交點到三個頂點的長度
float c1 = Distance(CrossPoint, TrianglePoint1);
float c2 = Distance(CrossPoint, TrianglePoint2);
float c3 = Distance(CrossPoint, TrianglePoint3);
//求三角形及子三角形的面積
float areaD = Area(d12, d13, d23); //三角形面積
float area1 = Area(c1, c2, d12); //子三角形1
float area2 = Area(c1, c3, d13); //子三角形2
float area3 = Area(c2, c3, d23); //子三角形3
//根據面積判斷點是否在三角形內部
if (fabs(area1 + area2 + area3 - areaD) > 0.001)
{
return false;
}
result = CrossPoint;
return true;
}