混沌分形之谢尔宾斯基（Sierpinski）

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]; };

[b]（5）其他[/b]

谢尔宾斯基三角形是一种很神的东西，我写过一些生成图像的算法，常常不知不觉中就出现了谢尔宾斯基三角形。如细胞生长机



再如：



之前我写过几篇与谢尔宾斯基分形相关的文章

分形之谢尔宾斯基(Sierpinski)三角形

分形之谢尔宾斯基(Sierpinski)地毯

分形之谢尔宾斯基(Sierpinski)四面体