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# Shaded & power normalized rendering

The Mandelbrot set rendering can be improved by using a normalized recount associated with a power normalized colormap (gamma=0.3). Rendering can be further enhanced thanks to shading.

The `maxiter` gives the precision of the computation. `maxiter=200` should take a few seconds on most modern laptops.

    import numpy as np

    def mandelbrot_set(xmin, xmax, ymin, ymax, xn, yn, maxiter, horizon=2.0):
        X = np.linspace(xmin, xmax, xn).astype(np.float32)
        Y = np.linspace(ymin, ymax, yn).astype(np.float32)
        C = X + Y[:, None] * 1j
        N = np.zeros_like(C, dtype=int)
        Z = np.zeros_like(C)
        for n in range(maxiter):
            I = np.less(abs(Z), horizon)
            N[I] = n
            Z[I] = Z[I]**2 + C[I]
        N[N == maxiter-1] = 0
        return Z, N

    if __name__ == '__main__':
        import time
        import matplotlib
        from matplotlib import colors
        import matplotlib.pyplot as plt

        xmin, xmax, xn = -2.25, +0.75, 3000/2
        ymin, ymax, yn = -1.25, +1.25, 2500/2
        maxiter = 200
        horizon = 2.0 ** 40
        log_horizon = np.log(np.log(horizon))/np.log(2)
        Z, N = mandelbrot_set(xmin, xmax, ymin, ymax, xn, yn, maxiter, horizon)

        # Normalized recount as explained in:
        # https://linas.org/art-gallery/escape/smooth.html
        # https://www.ibm.com/developerworks/community/blogs/jfp/entry/My_Christmas_Gift

        # This line will generate warnings for null values but it is faster to
        # process them afterwards using the nan_to_num
        with np.errstate(invalid='ignore'):
            M = np.nan_to_num(N + 1 -
                              np.log(np.log(abs(Z)))/np.log(2) +
                              log_horizon)

        dpi = 72
        width = 10
        height = 10*yn/xn
        fig = plt.figure(figsize=(width, height), dpi=dpi)
        ax = fig.add_axes([0.0, 0.0, 1.0, 1.0], frameon=False, aspect=1)

        # Shaded rendering
        light = colors.LightSource(azdeg=315, altdeg=10)
        M = light.shade(M, cmap=plt.cm.hot, vert_exag=1.5,
                        norm=colors.PowerNorm(0.3), blend_mode='hsv')
        plt.imshow(M, extent=[xmin, xmax, ymin, ymax], interpolation="bicubic")
        ax.set_xticks([])
        ax.set_yticks([])

        # Some advertisement for matplotlib
        year = time.strftime("%Y")
        major, minor, micro = matplotlib.__version__.split('.', 2)
        text = ("The Mandelbrot fractal set\n"
                "Rendered with matplotlib %s.%s, %s - http://matplotlib.org"
                % (major, minor, year))
        ax.text(xmin+.025, ymin+.025, text, color="white", fontsize=12, alpha=0.5)

        plt.show()

Total running time of the script: ( 0 minutes 4.798 seconds)

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