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Jetbrains全系列IDE稳定放心使用

情形一：函数有显式表达式 $z=f(x,y)$
主要使用函数：meshgrid,mesh,fmesh

例1：圆锥面：$z=\sqrt{x^2+y^2}$

clc,clear,close all
x=-5:0.1:5;
[X,Y]=meshgrid(x);
Z=sqrt(X.^2+Y.^2);
mesh(X,Y,Z)

例2：旋转抛物面 $z=2-x^2-y^2$

clc,clear,close all
x=-5:0.1:5;
[X,Y]=meshgrid(x);
Z=2-X.^2-Y.^2;
mesh(X,Y,Z)

例3：抛物柱面 $z=1-x^2$

clc,clear,close all
x=-5:0.1:5;
[X,Y]=meshgrid(x);
Z=1-X.^2;
mesh(X,Y,Z)

例3：平面 $z=1$

clc,clear,close all
x=-5:0.1:5;
[X,Y]=meshgrid(x);
Z=zeros(size(X))+1;
mesh(X,Y,Z)

此外，还可利用 fmesh 函数

例1：$z=e^y\sin x-e^x\cos y+e^x+e^y$

clc,clear,close all
syms x y
f=sin(x)*exp(y)-cos(y)*exp(x)+exp(x)+exp(y);
fmesh(f)

情形二：

情形三：函数表达式不含有 $z$
主要使用函数：meshgrid,isosurface

例1：抛物柱面 $x=2y^2$

clc,clear,close all

x=-5:0.1:5;
y=-5:0.1:5;
z=[-5,5];

[X,Y,Z] = meshgrid(x,y,z);
v = 2*Y.^2-X;
isosurface(X,Y,Z,v,0)
grid on

例2：平面 $y=0$

clc,clear,close all
x=-5:0.1:5;
y=-5:0.1:5;	
z=[-5,5];	
[X,Y,Z] = meshgrid(x,y,z);
v = Y;	
isosurface(X,Y,Z,v,0)

例3：平面 $x+y=0$

clc,clear,close all
x=-5:0.1:5;
y=-5:0.1:5;
z=[-5,5];
[X,Y,Z] = meshgrid(x,y,z);
v = X+Y-1;
isosurface(X,Y,Z,v,0)

情形3：函数有参数表达式
主要使用函数 fplot3

例1：
$$\begin{array}{rl}x& =\sin (t)\\ y& =\cos (t)\\ z& =t\end{array}$$

clc,clear,close all
xt = @(t) sin(t);
yt = @(t) cos(t);
zt = @(t) t;
fplot3(xt,yt,zt)

例2：
$$\begin{array}{rl}x& =e^{-t/10}\sin (5t)\\ y& =e^{-t/10}\cos (5t)\\ z& =t\end{array}$$

clc,clear,close all
xt = @(t) exp(-t/10).*sin(5*t);
yt = @(t) exp(-t/10).*cos(5*t);
zt = @(t) t;
fplot3(xt,yt,zt,[-10 10])

特殊情形1：取定 $x,y$ 后， $z$ 的值不唯一

这种情况往往需要分别求出每一个z，然后多次利用 mesh 函数绘图，比较复杂。（也可能有别的方法，但我不是很懂）

例1： $x^2+y^2+z^2=1$ （此例也可利用参数方程来绘图，此处使用mesh函数仅做示例用，效果并不如fplot3函数好用）

clc,clear,close all
x=-1:0.01:1;
[X,Y]=meshgrid(x);
Z=1-X.^2-Y.^2;
Z(Z<0)=nan; %这一步是为了后面对Z取根号的时候不会得到虚数
Z1=sqrt(Z);
Z2=-sqrt(Z);
mesh(X,Y,Z1)
hold on
mesh(X,Y,Z2)

一个比较复杂的例子

$1⩽x^2⩽y<4$
$z^2⩽x^2+y^2$

clc,clear,close all

x = -2:0.01:2;
y = 1:0.01:4;

[X,Y]=meshgrid(x,y);
index1 = X.^2-Y>0;
X(index1) = nan;
Y(index1) = nan;
index2 = X.^2<1;
X(index2) = nan;
Y(index2) = nan;
Z1 = sqrt(X.^2+Y.^2);
mesh(X,Y,Z1)
hold on 
Z2 = -sqrt(X.^2+Y.^2);
mesh(X,Y,Z2)


hold on
x = -2:0.01:-1;
z = linspace(-5,5,length(x));
[X,Z] = meshgrid(x,z);
Y = ones(length(x))*4;
index = Z.^2-X.^2-Y.^2>0;
X(index) = nan;
Y(index) = nan;
Z(index) = nan;
mesh(X,Y,Z)

x = 1:0.01:2;
z = linspace(-5,5,length(x));
[X,Z] = meshgrid(x,z);
Y = ones(length(x))*4;
index = Z.^2-X.^2-Y.^2>0;
X(index) = nan;
Y(index) = nan;
Z(index) = nan;
mesh(X,Y,Z)



y = 1:0.01:4;
z = linspace(-5,5,length(y));
[Y,Z] = meshgrid(y,z);
X = ones(length(y))*(-1);
index = Z.^2-X.^2-Y.^2>0;
X(index) = nan;
Y(index) = nan;
Z(index) = nan;
mesh(X,Y,Z)

y = 1:0.01:4;
z = linspace(-5,5,length(y));
[Y,Z] = meshgrid(y,z);
X = ones(length(y))*(1);
index = Z.^2-X.^2-Y.^2>0;
X(index) = nan;
Y(index) = nan;
Z(index) = nan;
mesh(X,Y,Z)



x = -2:0.01:-1;
z = linspace(-5,5,length(x));
[X,Z] = meshgrid(x,z);
Y = X.^2;
index = Z.^2-X.^2-Y.^2>0;
X(index) = nan;
Y(index) = nan;
Z(index) = nan;
mesh(X,Y,Z)

x = 1:0.01:2;
z = linspace(-5,5,length(x));
[X,Z] = meshgrid(x,z);
Y = X.^2;
index = Z.^2-X.^2-Y.^2>0;
X(index) = nan;
Y(index) = nan;
Z(index) = nan;
mesh(X,Y,Z)

2022年5月16日18:23:26