如何创建4D复杂曲面图? [英] How can I create a 4D complex surface plot?

问题描述

我有以下要转换为Python 3的Matlab代码.

I have the following Matlab code that I would like to be converted to a Python 3 one.

r = (0:1:15)';                           % create a matrix of complex inputs
theta = pi*(-2:0.05:2);
z = r*exp(1i*theta);
%w = z.^(1/2)  ;                          % calculate the complex outputs
w = sqrt(r)*exp(1i*theta/2);

figure('Name','Graphique complexe','units','normalized','outerposition',[ 0.08 0.1 0.8 0.55]);
subplot(121)

surf(real(z),imag(z),real(w),imag(w))    % visualize the complex function using surf
xlabel('Real(z)')
ylabel('Imag(z)')
zlabel('Real(u)')
cb = colorbar;
colormap jet;                            % gradient from blue to red
cb.Label.String = 'Imag(v)';

subplot(122)
surf(real(z),imag(z),imag(w),real(w))    % visualize the complex function using surf
xlabel('Real(z)')
ylabel('Imag(z)')
zlabel('Imag(v)')
cb = colorbar;
colormap jet;                            % gradient from blue to red
cb.Label.String = 'Real(u)';

结果和原始讨论可以在此处找到.在 SO中也有讨论页.但是，我无法运行和重现这些代码.我接下来可以尝试什么?

The results and original discussions can be found here. There's also a discussion available on this SO page. However, I failed to run and reproduce those codes. What can I try next?

推荐答案

如果您花时间学习matplotlib(尤其是3d轴)的工作原理，这将非常简单:

This is perfectly straightforward if you spend the time learning how matplotlib (and 3d axes in particular) work:

import numpy as np  
import matplotlib.pyplot as plt  
import matplotlib.cm as cm 
from mpl_toolkits.mplot3d import Axes3D 
 
# compute data to plot 
r, theta = np.mgrid[1:16, -2*np.pi:2*np.pi:50j] 
z = r * np.exp(1j*theta)  
w = np.sqrt(r) * np.exp(1j*theta/2)  
 
# plot data  
fig = plt.figure()  
for plot_index in [1, 2]: 
    if plot_index == 1: 
        z_data, c_data = w.real, w.imag 
        z_comp, c_comp = 'Re', 'Im' 
    else: 
        z_data, c_data = w.imag, w.real 
        z_comp, c_comp = 'Im', 'Re' 
    c_data = (c_data - c_data.min()) / c_data.ptp() 
    colors = cm.viridis(c_data) 
 
    ax = fig.add_subplot(f'12{plot_index}', projection='3d') 
    surf = ax.plot_surface(z.real, z.imag, z_data, facecolors=colors,
                           clim=[z_data.min(), z_data.max()])
    ax.set_xlabel('$Re z$')  
    ax.set_ylabel('$Im z$')   
    ax.set_zlabel(f'${z_comp} w$')  
    cb = plt.colorbar(surf, ax=ax)  
    cb.set_label(f'${c_comp} w$')  
 
plt.show()

结果:

一些应注意的地方:

• Viridis色彩图很好，喷射不好.
• 通常，由于matplotlib具有2d渲染器，复杂的3D几何(互锁)可能会出现渲染问题.幸运的是，在这种情况下，即使您以交互方式围绕图形旋转，数据集也紧密耦合在一起，以至于似乎不会发生这种情况. (但是，如果您将两个相交的表面绘制在一起，则可能是不同.)
• 可能要启用标签的乳胶渲染，以使结果更加清晰.
• 如果根据数据的z分量使用默认的着色选项，则阴影看起来要好得多.

• Viridis colormap is good, jet is bad.
• In general there could be rendering issues with complex (interlocking) 3d geometries, because matplotlib has a 2d renderer. Fortunately, in this case the dataset is tightly coupled enough that this doesn't seem to happen, even if you rotate around the figure interactively. (But if you were to plot two intersecting surfaces together, things would probably be different.)
• One might want to enable latex rendering of labels to make the result extra crispy.
• The shading looks a lot better if you use the default option of colouring according to the z component of the data.

如果我们还想移植我的MATLAB答案的第二部分，则必须

If we also want to port the second part of my MATLAB answer you will have to use a trick to stitch together the two branches of the function (which, as I said, is necessary to render interlocking surfaces properly). For the specific example in the above code this will not give you perfect results, since both branches themselves contain discontinuities in the imaginary part, so regardless of our efforts in rendering the two surfaces nicely, the result will look a bit bad:

import numpy as np 
import matplotlib.pyplot as plt 
import matplotlib.cm as cm 
from mpl_toolkits.mplot3d import Axes3D 
 
# compute data to plot 
r0 = 15 
re, im = np.mgrid[-r0:r0:31j, -r0:r0:31j] 
z = re + 1j*im 
r, theta = abs(z), np.angle(z) 
w1 = np.sqrt(r) * np.exp(1j*theta/2)  # first branch 
w2 = np.sqrt(r) * np.exp(1j*(theta + 2*np.pi)/2)  # second branch 
 
# plot data 
fig = plt.figure() 
for plot_index in [1, 2]: 
    # construct transparent bridge 
    re_bridge = np.vstack([re[-1, :], re[0, :]]) 
    im_bridge = np.vstack([im[-1, :], im[0, :]]) 
    c_bridge = np.full(re_bridge.shape + (4,), [1, 1, 1, 0])  # 0% opacity
 
    re_surf = np.vstack([re, re_bridge, re]) 
    im_surf = np.vstack([im, im_bridge, im]) 
    w12 = np.array([w1, w2]) 
    if plot_index == 1: 
        z_comp, c_comp = 'Re', 'Im' 
        z12, c12 = w12.real, w12.imag 
    else: 
        z_comp, c_comp = 'Im', 'Re' 
        z12, c12 = w12.imag, w12.real 
         
    color_arrays = cm.viridis((c12 - c12.min()) / c12.ptp()) 
    z1,z2 = z12 
    c1,c2 = color_arrays 
     
    z_bridge = np.vstack([z1[-1, :], z2[0, :]]) 
    z_surf = np.vstack([z1, z_bridge, z2]) 
    c_surf = np.vstack([c1, c_bridge, c2]) 
     
    ax = fig.add_subplot(f'12{plot_index}', projection='3d') 
    surf = ax.plot_surface(re_surf, im_surf, z_surf, facecolors=c_surf, 
                           clim=[c12.min(), c12.max()], 
                           rstride=1, cstride=1) 
    ax.set_xlabel('$Re z$') 
    ax.set_ylabel('$Im z$') 
    ax.set_zlabel(f'${z_comp} w$') 
    cb = plt.colorbar(surf, ax=ax) 
    cb.set_label(f'${c_comp} w$') 
  
plt.show()

正确的图形上的丑陋跳转可能需要大量工作才能解决，但这并不容易:这是两个表面数据集中出现在负实参上的实际不连续性.由于您的实际问题可能是更像这样，因此您可能不需要面对这个问题问题，您可以使用上述缝合(桥接)技巧来组合您的曲面.

The ugly jump in the right figure might be fixed with a lot of work, but it won't be easy: it's an actual discontinuity in both surface datasets occuring at negative real arguments. Since your actual problem is probably more like this, you will probably not need to face this issue, and you can use the above stitching (bridging) trick to combine your surfaces.

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