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莫比乌斯环的可视化

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莫比乌斯环的参数方程： \[ \vec{f}:(t,\varphi)\mapsto \begin{bmatrix} \displaystyle\Big(1+\frac{t}{2}\cos\frac{\varphi}{2}\Big)\cos\varphi \\[8pt] \displaystyle\Big(1+\frac{t}{2}\cos\frac{\varphi}{2}\Big)\sin\varphi \\[8pt] \displaystyle\frac{t}{2}\sin\frac{\varphi}{2} \end{bmatrix} \]

其中，\(t\in[-1,1]\) ，\(\varphi\in[0,2\pi]\) 。

可视化效果如下：

代码如下：

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 绘制莫比乌斯环
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% INPUT
% num1   :  - 标量, 线数
% num2   :  - 标量, 环数
% k      :  - 标量, 设置环的颜色, 0=单色环, 1=彩色环
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% OUTPUT    - NONE
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 调用示例1 - 无输入, 随机初始化参数
% fig_mobius;
% 调用示例2 - 50条线, 15个彩色环
% fig_mobius(50,15,1);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function fig_mobius(num1,num2,k)

if nargin==0
    num1=randi([40,60]);num2=randi([10,20]);k=randi([0,1]);
end

% 取值范围: t=[-1,1], phi=[0,2*pi]
mobius=@(t,phi) [(1+t/2*cos(phi/2)).*cos(phi);
                 (1+t/2*cos(phi/2)).*sin(phi);
                    t/2*sin(phi/2)];

phin=linspace(0,2*pi,num1);
tn=linspace(-1,1,num2);

figure(1);hold on;
xlim([-1,1]*1.5);ylim([-1,1]*1.5);zlim([-1,1]*0.5);
view([52,67]);

switch k
    case 0
        for i=1:num2
            temp=mobius(tn(i),phin);
            plot3(temp(1,:),temp(2,:),temp(3,:),'color','blue');
            pause(2/num2);
        end
    case 1
        color_table=parula;% parula, jet, spring, summer
        [numc1,~]=size(color_table);
        xt=linspace(0,1,numc1);
        numc2=ceil(num2/2);
        xq=linspace(0,1,numc2);
        tempc=interp1(xt,color_table,xq);
        for i=1:numc2
            temp1=mobius(tn(i),phin);
            temp2=mobius(tn(num2-i+1),phin);
            plot3(temp1(1,:),temp1(2,:),temp1(3,:),'color',tempc(i,:));
            plot3(temp2(1,:),temp2(2,:),temp2(3,:),'color',tempc(i,:));
            pause(2/numc2);
        end
    otherwise
        error('INPUT k must be 0 or 1');
end

for i=1:num1
    temp=mobius(tn,phin(i));
    plot3(temp(1,:),temp(2,:),temp(3,:),'color',[0.65,0.65,0.65]);
    pause(1.5/num1);
end

end