Visualization of Complex Function Mapping Homotopy

版本 1.0.0 (3.3 KB) 作者: Chun
A simple function artistically visualizes the complex function transformation with homotopy.
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更新时间 2025/11/20

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function complex_viz_combined()
% --- Configuration ---
config.density_2d = 100; % Grid density for 2D
config.density_3d = 80; % Grid density for 3D
config.anim_steps = 150; % Frames per animation
config.bg_color = [0.02 0.02 0.05]; % Dark background
config.clamp_limit = 12; % Visual extent limit
config.z_limit = 8; % Max height for 3D plots
config.line_width = 1.0;
% --- Setup Canvas ---
f = figure('Name', 'The Ultimate Math Visualization', ...
'NumberTitle', 'off', ...
'Color', config.bg_color, ...
'Position', [100, 100, 1280, 720], ...
'MenuBar', 'none', 'ToolBar', 'none');
ax = axes('Parent', f, 'Color', config.bg_color, ...
'XColor', [0.4 0.4 0.5], 'YColor', [0.4 0.4 0.5], 'ZColor', [0.4 0.4 0.5], ...
'LineWidth', 1, 'NextPlot', 'replacechildren');
% --- Scene Definitions ---
scenes = struct();
idx = 1;
% 1. Squaring (z^2)
scenes(idx).mode = '2D'; scenes(idx).name = 'Squaring (2D): z^2';
scenes(idx).func = @(z) z.^2;
scenes(idx).xlim = [-3, 3]; scenes(idx).ylim = [-3, 3];
scenes(idx).view = [-9, 9];
idx = idx + 1;
scenes(idx).mode = '3D'; scenes(idx).name = 'The Bowl (3D): z^2';
scenes(idx).func = @(z) z.^2;
scenes(idx).xlim = [-2, 2]; scenes(idx).ylim = [-2, 2];
scenes(idx).view = [-4, 4];
idx = idx + 1;
% 2. Exponential (e^z)
scenes(idx).mode = '2D'; scenes(idx).name = 'Exponential (2D): e^z';
scenes(idx).func = @(z) exp(z);
scenes(idx).xlim = [-2.5, 2.5]; scenes(idx).ylim = [-2*pi, 2*pi];
scenes(idx).view = [-10, 10];
idx = idx + 1;
scenes(idx).mode = '3D'; scenes(idx).name = 'The Spiral Slope (3D): e^z';
scenes(idx).func = @(z) exp(z);
scenes(idx).xlim = [-2, 2]; scenes(idx).ylim = [-3*pi, 3*pi];
scenes(idx).view = [-8, 8];
idx = idx + 1;
% 3. Sine (sin(z))
scenes(idx).mode = '2D'; scenes(idx).name = 'Sine Wave (2D): sin(z)';
scenes(idx).func = @(z) sin(z);
scenes(idx).xlim = [-2*pi, 2*pi]; scenes(idx).ylim = [-2.5, 2.5];
scenes(idx).view = [-6, 6];
idx = idx + 1;
scenes(idx).mode = '3D'; scenes(idx).name = 'The Modular Wave (3D): sin(z)';
scenes(idx).func = @(z) sin(z);
scenes(idx).xlim = [-2*pi, 2*pi]; scenes(idx).ylim = [-2, 2];
scenes(idx).view = [-4, 4];
idx = idx + 1;
% 4. Inversion (1/z)
scenes(idx).mode = '2D'; scenes(idx).name = 'Inversion (2D): 1/z';
scenes(idx).func = @(z) 1./z;
scenes(idx).xlim = [-3, 3]; scenes(idx).ylim = [-3, 3];
scenes(idx).view = [-6, 6];
idx = idx + 1;
scenes(idx).mode = '3D'; scenes(idx).name = 'Gravity Well (3D): 1/z';
scenes(idx).func = @(z) 1./z;
scenes(idx).xlim = [-2.5, 2.5]; scenes(idx).ylim = [-2.5, 2.5];
scenes(idx).view = [-4, 4];
idx = idx + 1;
% 5. Riemann Zeta
scenes(idx).mode = '2D'; scenes(idx).name = 'Riemann Zeta (2D): Chaos';
scenes(idx).func = @safe_zeta;
scenes(idx).xlim = [-12, 12]; scenes(idx).ylim = [-12, 12];
scenes(idx).view = [-10, 10];
idx = idx + 1;
scenes(idx).mode = '3D'; scenes(idx).name = 'Riemann Landscape (3D)';
scenes(idx).func = @safe_zeta;
scenes(idx).xlim = [-12, 12]; scenes(idx).ylim = [-12, 12];
scenes(idx).view = [-10, 10];
idx = idx + 1;
% --- Main Loop ---
for s = 1:length(scenes)
fprintf('Rendering %d/%d: %s...\n', s, length(scenes), scenes(s).name);
try
if strcmp(scenes(s).mode, '2D')
render_2d(f, ax, scenes(s), config);
else
render_3d(f, ax, scenes(s), config);
end
catch ME
fprintf('Error: %s\n', ME.message);
end
pause(1); % Pause between scenes
end
disp('Visualization Complete.');
end
% ==============================================================================
% 2D RENDER ENGINE
% ==============================================================================
function render_2d(f, ax, scene, config)
cla(ax); view(ax, 2); axis(ax, 'equal'); grid(ax, 'on'); hold(ax, 'on');
ax.GridAlpha = 0.15; ax.GridColor = 'w';
title(ax, scene.name, 'Color', [0.8 0.9 1], 'FontSize', 16, 'FontName', 'FixedWidth');
% Generate Grid
nx = config.density_2d; ny = config.density_2d;
x = linspace(scene.xlim(1), scene.xlim(2), nx);
y = linspace(scene.ylim(1), scene.ylim(2), ny);
[X, Y] = meshgrid(x, y);
Z_orig = X + 1i*Y;
try Z_target = scene.func(Z_orig); catch, return; end
% Apply NaN Clamping for clean cutouts
mask = abs(Z_target) > config.clamp_limit;
Z_target(mask) = NaN;
% Initialize Lines
h_lines = gobjects(1, ny);
v_lines = gobjects(1, nx);
cmap_h = [linspace(0,0,ny)', linspace(1,0.4,ny)', linspace(1,1,ny)'];
cmap_v = [linspace(1,1,nx)', linspace(0,0.5,nx)', linspace(1,0,nx)'];
for k = 1:ny
h_lines(k) = plot(ax, real(Z_orig(k,:)), imag(Z_orig(k,:)), ...
'Color', [cmap_h(k,:) 0.6], 'LineWidth', config.line_width);
end
for k = 1:nx
v_lines(k) = plot(ax, real(Z_orig(:,k)), imag(Z_orig(:,k)), ...
'Color', [cmap_v(k,:) 0.6], 'LineWidth', config.line_width);
end
% Set View
xlim(ax, scene.xlim*1.2); ylim(ax, scene.ylim*1.2);
% Animation Loop
t_vals = linspace(0, 1, config.anim_steps);
t_smooth = 1 ./ (1 + exp(-10 * (t_vals - 0.5)));
t_smooth = (t_smooth - min(t_smooth)) / (max(t_smooth) - min(t_smooth));
for t = t_smooth
Z_curr = (1-t)*Z_orig + t*Z_target;
for k = 1:ny, set(h_lines(k), 'XData', real(Z_curr(k,:)), 'YData', imag(Z_curr(k,:))); end
for k = 1:nx, set(v_lines(k), 'XData', real(Z_curr(:,k)), 'YData', imag(Z_curr(:,k))); end
% Dynamic Camera Zoom
curr_view = (1-t)*(scene.xlim*1.2) + t*[scene.view(1) scene.view(2)];
xlim(ax, curr_view); ylim(ax, curr_view);
drawnow;
end
end
% ==============================================================================
% 3D RENDER ENGINE
% ==============================================================================
function render_3d(f, ax, scene, config)
cla(ax); grid(ax, 'on'); hold(ax, 'on');
ax.GridAlpha = 0.15; ax.GridColor = 'w';
xlabel(ax,'Re'); ylabel(ax,'Im'); zlabel(ax,'Magnitude');
title(ax, scene.name, 'Color', [1 0.8 0.5], 'FontSize', 16, 'FontName', 'FixedWidth');
nx = config.density_3d; ny = config.density_3d;
x = linspace(scene.xlim(1), scene.xlim(2), nx);
y = linspace(scene.ylim(1), scene.ylim(2), ny);
[X, Y] = meshgrid(x, y);
Z_orig = X + 1i*Y;
try Z_target = scene.func(Z_orig); catch, return; end
% 3D Coordinates
X_t = real(Z_target);
Y_t = imag(Z_target);
H_t = abs(Z_target);
% Clamping: NaN for horizontal overflow, Cap for vertical height
radii = abs(Z_target);
mask_xy = radii > config.clamp_limit;
X_t(mask_xy) = NaN;
Y_t(mask_xy) = NaN;
H_t(mask_xy) = NaN;
H_t(H_t > config.z_limit) = config.z_limit;
X_0 = real(Z_orig); Y_0 = imag(Z_orig); H_0 = zeros(size(Z_orig));
% Initialize Lines
h_lines = gobjects(1, ny);
v_lines = gobjects(1, nx);
cmap = spring(ny + nx);
for k = 1:ny
h_lines(k) = plot3(ax, X_0(k,:), Y_0(k,:), H_0(k,:), ...
'Color', [cmap(k,:) 0.7], 'LineWidth', config.line_width);
end
for k = 1:nx
v_lines(k) = plot3(ax, X_0(:,k), Y_0(:,k), H_0(:,k), ...
'Color', [cmap(k+ny,:) 0.7], 'LineWidth', config.line_width);
end
% Fixed View
axis(ax, 'equal');
xlim(ax, scene.view); ylim(ax, scene.view); zlim(ax, [0, config.z_limit]);
view(ax, 45, 35);
% Animation Loop
t_vals = linspace(0, 1, config.anim_steps);
t_smooth = 1 ./ (1 + exp(-10 * (t_vals - 0.5)));
t_smooth = (t_smooth - min(t_smooth)) / (max(t_smooth) - min(t_smooth));
for t = t_smooth
X_curr = (1-t)*X_0 + t*X_t;
Y_curr = (1-t)*Y_0 + t*Y_t;
H_curr = (1-t)*H_0 + t*H_t;
for k = 1:ny, set(h_lines(k), 'XData', X_curr(k,:), 'YData', Y_curr(k,:), 'ZData', H_curr(k,:)); end
for k = 1:nx, set(v_lines(k), 'XData', X_curr(:,k), 'YData', Y_curr(:,k), 'ZData', H_curr(:,k)); end
drawnow;
end
end
% --- Helper Functions ---
function z_out = safe_zeta(z_in)
try
z_out = zeta(z_in);
catch
% Fallback approximation if Symbolic Toolbox is missing
z_out = 0.5 ./ (z_in.^0.5) + sin(z_in*5)*0.1;
end
% Hard clamp for pole at s=1
z_out(abs(z_in - 1) < 0.15) = NaN;
end

引用格式

Chun (2025). Visualization of Complex Function Mapping Homotopy (https://ww2.mathworks.cn/matlabcentral/fileexchange/182649-visualization-of-complex-function-mapping-homotopy), MATLAB Central File Exchange. 检索时间: .

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1.0.0