本文關鍵詞：謝爾賓斯基，，由筆耕文化傳播整理發(fā)布。

本文以使用混沌方法生成若干種謝爾賓斯基相關的分形圖形。   （1）謝爾賓斯基三角形   給三角形的3個頂點，和一個當前點，然后以以下的方式進行迭代處理：   a.隨機選擇三角形的某一個頂點，計算出它與當前點的中點位置；   b.將計算出的中點做為當前點，再重新執(zhí)行操作a   相關代碼如下：   復制代碼 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]; }; 復制代碼 關于基類FractalEquation的定義見：混沌與分形   最終生成的圖形為：       通過這一算法可以生成如下圖像：       （2）謝爾賓斯基矩形   既然能生成三角形的圖形，那么對于矩形會如何呢？嘗試下吧：   復制代碼 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]; }; 復制代碼 圖形如下：       噢，SHIT，毫無規(guī)律可言。   那就變動一下吧：   復制代碼 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]; }; 復制代碼     看上去還有點樣。   （3）謝爾賓斯基五邊形   四邊形是不行的，那再試下五邊：   復制代碼 // 五邊形 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]; }; 復制代碼     有點樣子，那就以此算法為基礎，生成幅圖像看看：       有人稱謝爾賓斯基三角形為謝爾賓斯基墳垛，當我看到這幅圖時，有一種恐怖的感覺。邪惡的五角形，總感覺里面有數(shù)不清的骷髏。   看來二維空間中謝爾賓斯基的單數(shù)可以生成分形圖形，而雙數(shù)則為無序的混沌。   （4）謝爾賓斯基四面體   再由二維擴展到三維看看：   復制代碼 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]; };

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  本文關鍵詞：謝爾賓斯基，由筆耕文化傳播整理發(fā)布。