混沌分形之谢尔宾斯基（Sierpinski）

article/2025/8/29 17:38:42

本文以使用混沌方法生成若干种谢尔宾斯基相关的分形图形。

（1）谢尔宾斯基三角形

给三角形的3个顶点，和一个当前点，然后以以下的方式进行迭代处理：

a.随机选择三角形的某一个顶点，计算出它与当前点的中点位置；

b.将计算出的中点做为当前点，再重新执行操作a

相关代码如下：

class SierpinskiTriangle : public FractalEquation
{
public:SierpinskiTriangle(){m_StartX = 0.0f;m_StartY = 0.0f;m_StartZ = 0.0f;m_triangleX[0] = 0.0f;m_triangleY[0] = FRACTAL_RADIUS;m_triangleX[1] = FRACTAL_RADIUS*sinf(PI/3);m_triangleY[1] = -FRACTAL_RADIUS*sinf(PI/6);m_triangleX[2] = -m_triangleX[1];m_triangleY[2] = m_triangleY[1];}void IterateValue(float x, float y, float z, float& outX, float& outY, float& outZ) const{int r = rand()%3;outX = (x + m_triangleX[r])*0.5f;outY = (y + m_triangleY[r])*0.5f;outZ = z;}private:float m_triangleX[3];float m_triangleY[3];
};

关于基类FractalEquation的定义见：混沌与分形

最终生成的图形为：

通过这一算法可以生成如下图像：

（2）谢尔宾斯基矩形

既然能生成三角形的图形，那么对于矩形会如何呢？尝试下吧：

class SierpinskiRectangle : public FractalEquation
{
public:SierpinskiRectangle(){m_StartX = 0.0f;m_StartY = 0.0f;m_StartZ = 0.0f;m_ParamA = 1.0f;m_ParamB = 1.0f;m_rectX[0] = FRACTAL_RADIUS;m_rectY[0] = FRACTAL_RADIUS;m_rectX[1] = FRACTAL_RADIUS;m_rectY[1] = -FRACTAL_RADIUS;m_rectX[2] = -FRACTAL_RADIUS;m_rectY[2] = -FRACTAL_RADIUS;m_rectX[3] = -FRACTAL_RADIUS;m_rectY[3] = FRACTAL_RADIUS;}void IterateValue(float x, float y, float z, float& outX, float& outY, float& outZ) const{int r = rand()%4;outX = (x + m_rectX[r])*0.5f;outY = (y + m_rectY[r])*0.5f;outZ = z;}bool IsValidParamA() const {return true;}bool IsValidParamB() const {return true;}void SetParamA(float v){m_ParamA = v;m_rectX[0] = FRACTAL_RADIUS*m_ParamA;m_rectX[1] = FRACTAL_RADIUS*m_ParamA;m_rectX[2] = -FRACTAL_RADIUS*m_ParamA;m_rectX[3] = -FRACTAL_RADIUS*m_ParamA;}void SetParamB(float v){m_ParamB = v;m_rectY[0] = FRACTAL_RADIUS*m_ParamB;m_rectY[1] = -FRACTAL_RADIUS*m_ParamB;m_rectY[2] = -FRACTAL_RADIUS*m_ParamB;m_rectY[3] = FRACTAL_RADIUS*m_ParamB;}private:float m_rectX[4];float m_rectY[4];
};

图形如下：

噢，SHIT，毫无规律可言。

那就变动一下吧：

class FractalSquare : public FractalEquation
{
public:FractalSquare(){m_StartX = 0.0f;m_StartY = 0.0f;m_StartZ = 0.0f;m_rectX[0] = FRACTAL_RADIUS;m_rectY[0] = FRACTAL_RADIUS;m_rectX[1] = FRACTAL_RADIUS;m_rectY[1] = -FRACTAL_RADIUS;m_rectX[2] = -FRACTAL_RADIUS;m_rectY[2] = -FRACTAL_RADIUS;m_rectX[3] = -FRACTAL_RADIUS;m_rectY[3] = FRACTAL_RADIUS;}void IterateValue(float x, float y, float z, float& outX, float& outY, float& outZ) const{int r = rand()%10;if (r < 4){outX = (x + m_rectX[r])*0.5f;outY = (y + m_rectY[r])*0.5f;}else{outX = x*0.5f;outY = y*0.5f;}outZ = z;}private:float m_rectX[4];float m_rectY[4];
};

看上去还有点样。

（3）谢尔宾斯基五边形

四边形是不行的，那再试下五边：

// 五边形
class SierpinskiPentagon : public FractalEquation
{
public:SierpinskiPentagon(){m_StartX = 0.0f;m_StartY = 0.0f;m_StartZ = 0.0f;for (int i = 0; i < 5; i++){m_pentagonX[i] = sinf(i*PI*2/5);m_pentagonY[i] = cosf(i*PI*2/5);}}void IterateValue(float x, float y, float z, float& outX, float& outY, float& outZ) const{int r = rand()%5;outX = (x + m_pentagonX[r])*0.5f;outY = (y + m_pentagonY[r])*0.5f;outZ = z;}private:float m_pentagonX[5];float m_pentagonY[5];
};

有点样子，那就以此算法为基础，生成幅图像看看：

有人称谢尔宾斯基三角形为谢尔宾斯基坟垛，当我看到这幅图时，有一种恐怖的感觉。邪恶的五角形，总感觉里面有数不清的骷髅。

看来二维空间中谢尔宾斯基的单数可以生成分形图形，而双数则为无序的混沌。

（4）谢尔宾斯基四面体

再由二维扩展到三维看看：

class SierpinskiTetrahedron : public FractalEquation
{
public:SierpinskiTetrahedron(){m_StartX = 0.0f;m_StartY = 0.0f;m_StartZ = 0.0f;m_vTetrahedron[0] = YsVector(0.0f, 0.0f, 0.0f);m_vTetrahedron[1] = YsVector(0.0f, 1.0f, 0.0f);m_vTetrahedron[2] = YsVector(YD_REAL_SQRT_3/2, 0.5f, 0.0f);m_vTetrahedron[3] = YsVector(YD_REAL_SQRT_3/6, 0.5f, YD_REAL_SQRT_3*YD_REAL_SQRT_2/3);YsVector vCenter = m_vTetrahedron[0] + m_vTetrahedron[1] + m_vTetrahedron[2] + m_vTetrahedron[3];vCenter *= 0.25f;m_vTetrahedron[0] -= vCenter;m_vTetrahedron[1] -= vCenter;m_vTetrahedron[2] -= vCenter;m_vTetrahedron[3] -= vCenter;m_vTetrahedron[0] *= FRACTAL_RADIUS;m_vTetrahedron[1] *= FRACTAL_RADIUS;m_vTetrahedron[2] *= FRACTAL_RADIUS;m_vTetrahedron[3] *= FRACTAL_RADIUS;}void IterateValue(float x, float y, float z, float& outX, float& outY, float& outZ) const{int r = rand()%4;outX = (x + m_vTetrahedron[r].x)*0.5f;outY = (y + m_vTetrahedron[r].y)*0.5f;outZ = (z + m_vTetrahedron[r].z)*0.5f;}bool Is3D() const {return true;}private:YsVector m_vTetrahedron[4];
};