由离散样本点生成Delaunay三角网有多种算法，每个算法的执行效率都不一样，这里介绍一种最简单，最低效，但是算法逻辑最清晰的一种。

Delaunay三角网必须满足的一个条件是任何一个三角形的外接圆都不能包含其他任何一个样本点，因此，本算法通过枚举所有可能的三角形，再经过其外接圆不包含任何其他样本点的判断，如果满足，则记录该三角形，直到所有三角形枚举完毕。

效果图：

本算法用C++实现，核心代码如下：

const double EP = 0.00000001;

// 点结构

typedef struct PT{

double x;

double y;

}PT;

//线结构

typedef struct SEGMENT

{

PT* ptStart;

PT* ptEnd;

}SEGMENT;

//三角形类

class TRIANGLE

{

public:

TRIANGLE(PT* pt1, PT* pt2, PT* pt3)

{

//确保三个点x坐标升序排序

PT* temp;

if(pt1->x > pt2->x){temp = pt1; pt1 = pt2; pt2 = temp;}

if(pt3->x < pt1->x  && pt3->x )

{

temp = pt3; pt3 = pt1; pt1 = temp;

if(pt3->x < pt2->x){temp=pt3;pt3=pt2;pt2=temp;}

}

if(pt3->x < pt2->x){temp=pt3;pt3=pt2;pt2=temp;}

_ptFirst = pt1;

_ptSecond = pt2;

_ptThird = pt3;

InitData();

}

PT* GetFirstPt(){return _ptFirst;}

PT* GetSecondPt(){return _ptSecond;}

PT* GetThirdPt(){return _ptThird;}

SEGMENT GetSegment1(){SEGMENT seg = {_ptFirst,_ptSecond};return seg;}

SEGMENT GetSegment2(){SEGMENT seg = {_ptFirst,_ptThird};return seg;}

SEGMENT GetSegment3(){SEGMENT seg = {_ptSecond,_ptThird};return seg;}

BOOL IsTriangleSame(TRIANGLE& tri)

{

if(tri.GetFirstPt() == _ptFirst &&

tri.GetSecondPt() == _ptSecond &&

tri.GetThirdPt() == _ptThird)

return TRUE;

return FALSE;

}

BOOL IsPtInCircle(PT* pt)

{

double offsetx = pt->x - _ptCenter.x;

double offsety = pt->y - _ptCenter.y;

if(sqrt(offsetx*offsetx + offsety*offsety) <= _Radius)

return TRUE;

return FALSE;

}

protected:

// 获取三角形外接圆中心点及半径

void InitData()

{

double x0 = _ptFirst->x;

double y0 = _ptFirst->y;

double x1 = _ptSecond->x;

double y1 = _ptSecond->y;

double x2 = _ptThird->x;

double y2 = _ptThird->y;

double y10 = y1 - y0;

double y21 = y2 - y1;

bool b21zero = y21 > -REAL_EPSILON && y21 < REAL_EPSILON;

if (y10 > -REAL_EPSILON && y10 < REAL_EPSILON)

{

if (b21zero)

{

if (x1 > x0)

{

if (x2 > x1) x1 = x2;

}

else

{

if (x2 < x0) x0 = x2;

}

_ptCenter.x = (x0 + x1) * .5F;

_ptCenter.y = y0;

}

else

{

double m1 = - (x2 - x1) / y21;

double mx1 = (x1 + x2) * .5F;

double my1 = (y1 + y2) * .5F;

_ptCenter.x = (x0 + x1) * .5F;

_ptCenter.y = m1 * (_ptCenter.x - mx1) + my1;

}

}

else if (b21zero)

{

double m0 = - (x1 - x0) / y10;

double mx0 = (x0 + x1) * .5F;

double my0 = (y0 + y1) * .5F;

_ptCenter.x = (x1 + x2) * .5F;

_ptCenter.y = m0 * (_ptCenter.x - mx0) + my0;

}

else

{

double m0 = - (x1 - x0) / y10;

double m1 = - (x2 - x1) / y21;

double mx0 = (x0 + x1) * .5F;

double my0 = (y0 + y1) * .5F;

double mx1 = (x1 + x2) * .5F;

double my1 = (y1 + y2) * .5F;

_ptCenter.x = (m0 * mx0 - m1 * mx1 + my1 - my0) / (m0 - m1);

_ptCenter.y = m0 * (_ptCenter.x - mx0) + my0;

}

double dx = x0 - _ptCenter.x;

double dy = y0 - _ptCenter.y;

_Radius2 = dx * dx + dy * dy;    // the radius of the circumcircle, squared

_Radius = (double) sqrt(_Radius2);    // the proper radius

_Radius2 *= 1.000001f;

}

private:

PT* _ptFirst;

PT* _ptSecond;

PT* _ptThird;

PT            _ptCenter;

double        _Radius2;

double        _Radius;

};

class CMyDelaunay

{

public:

// 通过样本点集，生成三角形集，由vecTriangleWork输出

void BuildDelaunayEx(vector& vecPT, vector& vecTriangleWork);

BOOL IsPtsBuildTriangle(PT* pt1, PT* pt2, PT* pt3);

public:

CMyDelaunay();

virtual ~CMyDelaunay();

};

//判断三个点能否组成一个三角形

BOOL CMyDelaunay::IsPtsBuildTriangle(PT* pt1, PT* pt2, PT* pt3)

{

double offset_x1 = pt2->x - pt1->x;

double offset_x2 = pt3->x - pt2->x;

double offset_y1 = pt2->y - pt1->y;

double offset_y2 = pt3->y - pt2->y;

if((fabs(offset_x1) < EP) && (fabs(offset_x2) < EP))    //竖直

{

return FALSE;

}

if(fabs(offset_x1) > EP && fabs(offset_x2) > EP)

{

if(fabs(offset_y1/offset_x1 - offset_y2/offset_x2) < EP)

return FALSE;

}

return TRUE;

}

void CMyDelaunay::BuildDelaunayEx(vector& vecPT, vector& vecTriangleWork)

{

int nSize = vecPT.size();

if(nSize < 3)

return;

for(int i = 0; i < nSize - 2; ++i)

{

for(int j = i + 1;  j < nSize - 1; ++j)

{

for(int k = j + 1; k < nSize; ++k)

{

PT* pt1 = &vecPT[i];

PT* pt2 = &vecPT[j];

PT* pt3 = &vecPT[k];

BOOL bFind = TRUE;

for(int m = 0; m < nSize; ++m)

{

PT* pt = &vecPT[m];

if(pt != pt1 && pt != pt2 && pt != pt3 && IsPtsBuildTriangle(pt1, pt2, pt3))

{

TRIANGLE tri(pt1, pt2, pt3);

if(tri.IsPtInCircle(pt))

{

bFind = FALSE;

break;

}

}

}

if(bFind)

{

vecTriangleWork.push_back(tri);

}

}

}

}

}