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];
};
（5）其他
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
