欧拉公式的三维演示

下面的MATLAB 程序以三维形式演示欧拉公式

Exp(t) = cos(t) + j sin(t)

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% 本程序演示欧拉公式

% Jan.25th,2012

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h_fig1 = figure;

set(h_fig1, 'unit', 'normalized', 'position', [0.1, 0.1, 0.9, 0.9]);

set(h_fig1, 'defaultuicontrolunits', 'normalized');

h_text1 = uicontrol(h_fig1, 'Style', 'text', 'Position', [0.71, 0.73, 0.25, 0.05],... % 创建文本框

'String', '▲是cos 曲线的起点', 'ForegroundColor', 'r', 'FontName', '黑体',...

'FontSize', 12, 'FontWeight', 'Bold', 'BackgroundColor', [1, 1, 1]);

h_text2 = uicontrol(h_fig1, 'Style', 'text', 'Position', [0.71, 0.78, 0.25, 0.05],... % 创建文本框

'String', 'Δ是sin 和exp 曲线的起点', 'ForegroundColor', 'r', 'FontName', '黑体',...

'FontSize', 12, 'FontWeight', 'Bold', 'BackgroundColor', [1, 1, 1]);

h_pushbutton1 = uicontrol(h_fig1, 'Style', 'PushButton', 'Position', [0.82, 0.12, 0.07, 0.06],...

'string', '退出', 'BackgroundColor', [0.8 0.9 0.8], 'ForegroundColor', 'r', 'FontSize', 14, 'FontWeight', 'Bold',...

'callback', 'delete(h_fig1),')

h_axes0 = axes('Box', 'on', 'Position', [0.15, 0.18, 0.56, 0.68], 'FontSize', 8)

set(gcf,'color','w');

w = 0.1*pi

t = 0:40; % 在前进方向绕了2 圈

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a = -ones(1,length(t));

plot3(cos(w*t),t,sin(w*t),'b', 'LineWidth', 2);

grid on; hold on;

hc = plot3(cos(w*t),t,a,'k--'); hold on;

set(hc, 'Color', 'r', 'LineWidth', 2);

a=-a;

hs = plot3(a,t,sin(w*t),'r-.'); hold on;

set(hs, 'Color', 'k', 'LineWidth', 2);

text(0.7,0.3,0.6, ' <-- CCW', 'FontSize', 14, 'FontWeight', 'Bold'); text(1,0,-1, ' ▲Cos', 'Color', 'r', 'FontSize', 14, 'FontWeight', 'Bold'); text(1,0,0, ' Δ Sin', 'FontSize', 14, 'FontWeight', 'Bold');

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xlabel('x', 'FontSize', 14, 'FontWeight', 'Bold');

ylabel('t', 'FontSize', 14, 'FontWeight', 'Bold');

zlabel('y', 'FontSize', 14, 'FontWeight', 'Bold');

title('演示欧拉公式y = exp(jwt) = cos(wt) + jsin(wt)', 'Color', 'b', …

'FontSize', 18, 'FontWeight', 'Bold');

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line([-1,-1],[39.9,39.9],[-1,1],'LineWidth',3, 'Color', 'r');

line([1,1],[39.9,39.9],[-1,1],'LineWidth',3, 'Color', 'r');

line([-1,-1],[0,0],[-1,1],'LineWidth',3, 'Color', 'r');

line([1,1],[0,0],[-1,1],'LineWidth',3, 'Color', 'r');

line([-1,-1],[0,40],[-1,-1],'LineWidth',3, 'Color', 'k');

line([-1,1],[0,0],[-1,-1],'LineWidth',3, 'Color', 'b')

line([-1,1],[40,40],[1,1],'LineWidth',3, 'Color', 'b')

line([-1,1],[40,40],[-1,-1],'LineWidth',3, 'Color', 'b')

line([-1,1],[0,0],[1,1],'LineWidth',3, 'Color', 'b')

line([-1,1],[0,0],[0,0],'LineWidth',2, 'Color', 'k');

line([0,0],[0,0],[-1,1],'LineWidth',2, 'Color', 'k');

line([0,0],[40,40],[-1,1],'LineWidth',2, 'Color', 'k');

line([0,0],[0,40],[0,0],'LineWidth',2, 'Color', 'k');

line([-1,1],[40,40],[0,0],'LineWidth',2, 'Color', 'k');

line([0,0],[0,40],[0,0],'LineWidth',2, 'Color', 'k');

text(0,0,0.12,'O', 'FontSize', 14, 'FontWeight', 'Bold', 'Color', 'r')

text(0,40,0.12,'O', 'FontSize', 14, 'FontWeight', 'Bold', 'Color', 'b')

程序运行结果如下所示。

按下工具条的Tool | Rotate 3D 选项，用鼠标带动上述的图形，可以任意改变视角。