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 Matlab 代码，我想将其转换为 Python 3 代码.
r = (0:1:15)';% 创建一个复杂输入矩阵θ = pi*(-2:0.05:2);z = r*exp(1i*theta);%w = z.^(1/2) ;% 计算复杂的输出w = sqrt(r)*exp(1i*theta/2);图('名称'，'Graphique complexe'，'units'，'normalized'，'outerposition'，[0.08 0.1 0.8 0.55])；次要情节(121)surf(real(z),imag(z),real(w),imag(w)) % 使用 surf 可视化复杂函数xlabel('Real(z)')ylabel('图像(z)')zlabel('Real(u)')cb = 颜色条；颜色图喷射；% 从蓝色到红色的渐变cb.Label.String = '图像(v)';子图(122)surf(real(z),imag(z),imag(w),real(w)) % 使用 surf 可视化复杂函数xlabel('Real(z)')ylabel('图像(z)')zlabel('图像(v)')cb = 颜色条；颜色图喷射；% 从蓝色到红色的渐变cb.Label.String = 'Real(u)';可以在
需要注意的几点:
- Viridis 颜色图好，jet 不好.
- 一般来说，复杂(互锁)3d 几何图形可能存在渲染问题，因为 matplotlib 有一个 2d 渲染器.幸运的是，在这种情况下，数据集耦合得足够紧密，即使您以交互方式围绕图形旋转，这似乎也不会发生.(但如果您要将两个相交的曲面绘制在一起，
右图中丑陋的跳跃可能会通过大量工作来修复，但这并不容易:这是两个表面数据集中在负实参数下发生的实际不连续性.由于您的实际问题可能是更像这样，您可能不需要面对这个问题，您可以使用上述缝合(桥接)技巧来组合您的曲面.
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)';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?
解决方案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()The result:
Some things that should be noted:
- 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.
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()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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