混沌分形之填充集

      通过分形来生成图像，有一个特点是：不想生成什么样的图像就写出相应的算法，而是生成出来的图像像什么，那算法就是什么。总之，当你在写这个算法时或设置相关参数时，你几乎无法猜测出你要生成的图像是什么样子。而生成图像的时间又比较久，无法实时地调整参数。所以我这使用了填充集的方式，先计算少量的顶点，以显示出图像的大致轮廓。确定好参数后再进行图像生成。所谓填充集，就是随机生成顶点位置，当满足要求时顶点保留，否则剔除。这里将填充集的方式来生成Julia集，曼德勃罗集和牛顿迭代集.

（1）Julia集

// 填充Julia集
// http://www.douban.com/note/230496472/
class JuliaSet2 : public FractalEquation
{
public:
    JuliaSet2()
    {
        m_StartX = 0.0f;
        m_StartY = 0.0f;
        m_StartZ = 0.0f;

        m_ParamA = 0.11f;
        m_ParamB = 0.615f;

        m_nIterateCount = 80;
    }

    void IterateValue(float x, float y, float z, float& outX, float& outY, float& outZ) const
    {
        x = outX = yf_rand_real(-1.0f, 1.0f);
        y = outY = yf_rand_real(-1.0f, 1.0f);

        float lengthSqr;
        float temp;
        int count = 0;
        do
        {
            temp = x * x - y * y + m_ParamA;
            y = 2 * x * y + m_ParamB;
            x = temp;

            lengthSqr = x * x + y * y;
            count++;
        }
        while ((lengthSqr < 4.0f) && (count < m_nIterateCount));

        if (lengthSqr > 4.0f)
        {
            outX = 0.0f;
            outY = 0.0f;
        }

        outZ = z;
    }

    bool IsValidParamA() const {return true;}
    bool IsValidParamB() const {return true;}

private:
    int m_nIterateCount;
};

（2）曼德勃罗集

// 曼德勃罗集
// http://www.cnblogs.com/Ninputer/archive/2009/11/24/1609364.html
class MandelbrotSet : public FractalEquation
{
public:
    MandelbrotSet()
    {
        m_StartX = 0.0f;
        m_StartY = 0.0f;
        m_StartZ = 0.0f;

        m_ParamA = -1.5f;
        m_ParamB = 1.0f;
        m_ParamC = -1.0f;
        m_ParamD = 1.0f;

        m_nIterateCount = 100;
    }

    void IterateValue(float x, float y, float z, float& outX, float& outY, float& outZ) const
    {
        float cr = m_ParamA + (m_ParamB - m_ParamA)*((float)rand()/RAND_MAX);
        float ci = m_ParamC + (m_ParamD - m_ParamC)*((float)rand()/RAND_MAX);

        outX = 0.0f;
        outY = 0.0f;

        float lengthSqr;
        float temp;
        int count = 0;
        do
        {
            temp = outX * outX - outY * outY + cr;
            outY = 2 * outX * outY + ci;
            outX = temp;

            lengthSqr = outX * outX + outY * outY;
            count++;
        }
        while ((lengthSqr < 4.0f) && (count < m_nIterateCount));

        if (lengthSqr < 4.0f)
        {
            outX = cr;
            outY = ci;
        }
        else
        {
            outX = 0.0f;
            outY = 0.0f;
        }

        outZ = z;
    }

    bool IsValidParamA() const {return true;}
    bool IsValidParamB() const {return true;}
    bool IsValidParamC() const {return true;}
    bool IsValidParamD() const {return true;}

private:
    int m_nIterateCount;
};

（3）牛顿迭代集

// 牛顿迭代
// http://www.douban.com/note/230496472/
class NewtonIterate : public FractalEquation
{
public:
    NewtonIterate()
    {
        m_StartX = 0.0f;
        m_StartY = 0.0f;
        m_StartZ = 0.0f;

        m_ParamA = 1.0f;

        m_nIterateCount = 64;
    }

    void IterateValue(float x, float y, float z, float& outX, float& outY, float& outZ) const
    {
        x = outX = yf_rand_real(-m_ParamA, m_ParamA);
        y = outY = yf_rand_real(-m_ParamA, m_ParamA);

        float xx, yy, d, tmp;

        for (int i = 0; i < m_nIterateCount; i++)
        {
            xx = x*x;
            yy = y*y;
            d = 3.0f*((xx - yy)*(xx - yy) + 4.0f*xx*yy);
            if (fabsf(d) < EPSILON)
            {
                d = d > 0.0f ? EPSILON : -EPSILON;
            }
            tmp = x;
            x = 0.666667f*x + (xx - yy)/d;
            y = 0.666667f*y - 2.0f*tmp*y/d;
        }

        if (x < 0.0f)
        {
            outX = 0.0f;
            outY = 0.0f;
        }

        outZ = z;
    }

    bool IsValidParamA() const {return true;}

private:
    int m_nIterateCount;
};

（4）

关于基类FractalEquation的定义见：混沌与分形

再发几幅图像：

 

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