MATLAB _ 由隐式函数(直接)画曲线曲面
一. 将隐式函数化作参数形式，如球面，x2 +y2 +z2 =R2
- ezsurf
sphere.m
function sphere(R)
x=[num2str(R),'*sin(s)*cos(t)'];
y=[num2str(R),'*sin(s)*sin(t)'];
z=[num2str(R),'*cos(s)'];
ezsurf(x,y,z,[0,pi,0,2*pi]);
axis square;
shading interp;
colormap(spring);
view(10,40)
在命令行中执行命令sphere(r)就可以得到半径为r的球形了
- 转化成参数方程后，用mesh或surf画
% x,y,z are equally sized matrices with coordinates.
s=0:0.05:10;
t=-0.05:0.05:2*pi;%取-0.05 使得圆锥体没有缝隙
[s,t]=meshgrid(s,t);
x=(s+10).*cos(t);
y=(s+10).*sin(t);
z=s;
mesh(x,y,z);
saveobjmesh('temp.obj',x,y,z);
二. 利用isosurface, 可将隐式函数直接画出来
1. isosurface与patch连用，可将曲面显示出来
Example: 在同一个图内，画出球面，圆柱面
x=-10:0.1:10;y=x;z=x;
[x,y,z]=meshgrid(x,y,z);
f1=x.^2+y.^2+z.^2-4;
f2=x.^2+y.^2-1;
f3=x.^2+z.^2-1;
f4=z.^2+y.^2-1;
p1=patch(isosurface(x,y,z,f1,0));
set(p1, 'FaceColor', 'b', 'EdgeColor', 'none');
p2=patch(isosurface(x,y,z,f2,0));
set(p2, 'FaceColor', 'r', 'EdgeColor', 'none');
p3=patch(isosurface(x,y,z,f3,0));
set(p3, 'FaceColor', 'y', 'EdgeColor', 'none');
p4=patch(isosurface(x,y,z,f4,0));
set(p4, 'FaceColor', 'h', 'EdgeColor', 'none');
daspect([1 1 1])
view(3); axis tight
camlight;
lighting phong
2. 将isosurface输出.obj文件
首先，先写vertface2obj.m
function vertface2obj(v,f,name)
% VERTFACE2OBJ Save a set of vertice coordinates and faces as a Wavefront/Alias Obj file
% VERTFACE2OBJ(v,f,fname)
% v is a Nx3 matrix of vertex coordinates.
% f is a Mx3 matrix of vertex indices.
% fname is the filename to save the obj file.
fid = fopen(name,'w');
for i=1:size(v,1)
fprintf(fid,'v %f %f %f/n',v(i,1),v(i,2),v(i,3));
end
fprintf(fid,'g foo/n');
for i=1:size(f,1);
fprintf(fid,'f %d %d %d/n',f(i,1),f(i,2),f(i,3));
end
fprintf(fid,'g/n');
fclose(fid);
Example:
a=2*pi; s=2*pi/60; [x,y,z]=meshgrid(-a:s:a,-a:s:a,-a:s:a); cx = cos(2*x); cy = cos(2*y); cz = cos(2*z); u = 10.0*(cos(x).*sin(y) + cos(y).*sin(z) + cos(z).*sin(x))- 0.5*(cx.*cy + cy.*cz + cz.*cx) - 14.0; [f,v]=isosurface(u,0); vertface2obj(v,f,'gyroid.obj')
Reference: http://www.nada.kth.se/~asa/Ray/matlabobj.html
三. 由ezimplot3，可将隐士函数直接画出曲面
function h = ezimplot3(varargin) % EZIMPLOT3 Easy to use 3D implicit plotter. % EZIMPLOT3(FUN) plots the function FUN(X,Y,Z) = 0 (vectorized or not) % over the default domain: % -2*PI < X < 2*PI, -2*PI < Y < 2*PI, -2*PI < Z < 2*PI. % FUN can be a string, an anonymous function handle, a .M-file handle, an % inline function or a symbolic function (see examples below) % % EZIMPLOT3(FUN,DOMAIN)plots FUN over the specified DOMAIN instead of the % default domain. DOMAIN can be vector [XMIN,XMAX,YMIN,YMAX,ZMIN,ZMAX] or % vector [A,B] (to plot over A < X < B, A < Y < B, A < Z < B). % % EZIMPLOT3(..,N) plots FUN using an N-by-N grid. The default value for % N is 60. % EZIMPLOT3(..,'color') plots FUN with color 'color'. The default value % for 'color' is 'red'. 'color' must be a valid Matlab color identifier. % % EZIMPLOT3(axes_handle,..) plots into the axes with handle axes_handle % instead of into current axes (gca). % % H = EZIMPLOT3(...) returns the handle to the patch object this function % creates. % % Example: % Plot x^3+exp(y)-cosh(z)=4, between -5 and 5 for x,y and z % % via a string: % f = 'x^3+exp(y)-cosh(z)-4' % ezimplot3(f,[-5 5]) % % via a anonymous function handle: % f = @(x,y,z) x^3+exp(y)-cosh(z)-4 % ezimplot3(f,[-5 5]) % % via a function .m file: %------------------------------% % function out = myfun(x,y,z) % out = x^3+exp(y)-cosh(z)-4; %------------------------------% % ezimplot3(@myfun,[-5 5]) or ezimplot('myfun',[-5 5]) % % via a inline function: % f = inline('x^3+exp(y)-cosh(z)-4') % ezimplot3(f,[-5 5]) % % via a symbolic expression: % syms x y z % f = x^3+exp(y)-cosh(z)-4 % ezimplot3(f,[-5 5]) % % Note: this function do not use the "ezgraph3" standard, like ezsurf, % ezmesh, etc, does. Because of this, ezimplot3 only tries to imitate that % interface. A future work must be to modify "ezgraph3" to include a % routine for implicit surfaces based on this file % % Inspired by works of: Artur Jutan UWO 02-02-98 [email protected] % Made by: Gustavo Morales UC 04-12-09 [email protected] % %%% Checking & Parsing input arguments: if ishandle(varargin{1}) cax = varargin{1}; % User selected axes handle for graphics axes(cax); args{:} = varargin{2:end}; %ensuring args be a cell array else args = varargin; end [fun domain n color] = argcheck(args{:}); %%% Generating the volumetric domain data: xm = linspace(domain(1),domain(2),n); ym = linspace(domain(3),domain(4),n); zm = linspace(domain(5),domain(6),n); [x,y,z] = meshgrid(xm,ym,zm); %%% Formatting "fun" [f_handle f_text] = fix_fun(fun); % f_handle is the anonymous f-handle for "fun" % f_text is "fun" ready to be a title %%% Evaluating "f_handle" in domain: try fvalues = f_handle(x,y,z); % fvalues: volume data catch ME error('Ezimplot3:Functions', 'FUN must have no more than 3 arguments'); end %%% Making the 3D graph of the 0-level surface of the 4D function "fun": h = patch(isosurface(x,y,z,fvalues,0)); % "patch" handles the structure... % sent by "isosurface" isonormals(x,y,z,fvalues,h)% Recalculating the isosurface normals based... % on the volume data set(h,'FaceColor',color,'EdgeColor','none'); %%% Aditional graphic details: xlabel('x');ylabel('y');zlabel('z');% naming the axis alpha(0.7) % adjusting for some transparency grid on; view([1,1,1]); axis equal; camlight; lighting gouraud %%% Showing title: title([f_text,' = 0']); % %--------------------------------------------Sub-functions HERE--- function [f dom n color] = argcheck(varargin) %ARGCHECK(arg) parses "args" to the variables "f"(function),"dom"(domain) %,"n"(grid size) and "c"(color)and TRIES to check its validity switch nargin case 0 error('Ezimplot3:Arguments',... 'At least "fun" argument must be given'); case 1 f = varargin{1}; dom = [-2*pi, 2*pi]; % default domain: -2*pi < xi < 2*pi n = 60; % default grid size color = 'red'; % default graph color case 2 f = varargin{1}; if isa(varargin{2},'double') && length(varargin{2})>1 dom = varargin{2}; n = 60; color = 'red'; elseif isa(varargin{2},'double') && length(varargin{2})==1 n = varargin{2}; dom = [-2*pi, 2*pi]; color = 'red'; elseif isa(varargin{2},'char') dom = [-2*pi, 2*pi]; n = 60; color = varargin{2}; end case 3 % If more than 2 arguments are given, it's f = varargin{1}; % assumed they are in the correct order dom = varargin{2}; n = varargin{3}; color = 'red'; % default color case 4 % If more than 2 arguments are given, it's f = varargin{1}; % assumed they are in the correct order dom = varargin{2}; n = varargin{3}; color = varargin{4}; otherwise warning('Ezimplot3:Arguments', ... 'Attempt will be made only with the 4 first arguments'); f = varargin{1}; dom = varargin{2}; n = varargin{3}; color = varargin{4}; end if length(dom) == 2 dom = repmat(dom,1,3); %domain repeated in all variables elseif length(dom) ~= 6 error('Ezimplot3:Arguments',... 'Input argument "domain" must be a row vector of size 2 or size 6'); end % %-------------------------------------------- function [f_hand f_text] = fix_fun(fun) % FIX_FUN(fun) Converts "fun" into an anonymous function of 3 variables (x,y,z) % with handle "f_hand" and a string "f_text" to use it as title types = {'char','sym','function_handle','inline'}; % cell array of 'types' type = ''; %Identifing FUN object class for i=1:size(types,2) if isa(fun,types{i}) type = types{i}; break; end end switch type case 'char' % Formatting FUN if it is char type. There's 2 possibilities: % A string with the name of the .m file if exist([fun,'.m'],'file') syms x y z; if nargin(str2func(fun)) == 3 f_sym = eval([fun,'(x,y,z)']); % evaluating FUN at the sym point (x,y,z) else error('Ezimplot3:Arguments',... '%s must be a function of 3 arguments or unknown function',fun); end f_text = strrep(char(f_sym),' ',''); % converting to char and eliminating spaces f_hand = eval(['@(x,y,z)',vectorize(f_text),';']); % converting string to anonymous f_handle else % A string with the function's expression f_hand = eval(['@(x,y,z)',vectorize(fun),';']); % converting string to anonymous f_handle f_text = strrep(fun,'.',''); f_text = strrep(f_text,' ',''); % removing vectorization & spaces end case 'sym' % Formatting FUN if it is a symbolic object f_hand = eval(['@(x,y,z)',vectorize(fun),';']); % converting string to anonymous f_handle f_text = strrep(char(fun),' ',''); % removing spaces case {'function_handle', 'inline'} % Formatting FUN if it is a function_handle or an inline object syms x y z; if nargin(fun) == 3 && numel(symvar(char(fun))) == 3 % Determining if # variables == 3 f_sym = fun(x,y,z); % evaluating FUN at the sym point (x,y,z) else error('Ezimplot3:Arguments',... '%s must be function of 3 arguments or unknown function',char(fun)); end f_text = strrep(char(f_sym),' ',''); % converting into string to removing spaces f_hand = eval(['@(x,y,z)',vectorize(f_text),';']); % converting string to anonymous f_handle otherwise error('First argument "fun" must be of type character, simbolic, function handle or inline'); end
Example:
Plot x^3+exp(y)-cosh(z)=4, between -5 and 5 for x,y and z
% via a string:
f = 'x^3+exp(y)-cosh(z)-4'
ezimplot3(f,[-5 5])
四. 由implot，可将直接画出隐式曲线
implot.m
function implot(fun,rangexy,ngrid) % Implicit plot function % function implot(fun,rangexy,ngrid) % fun is 'inline' function f(x,y)=0 (Note function written as equal to zero) % rangexy =[xmin,xmax,ymin,ymax] range over which x and y is ploted default(-2*pi,2*pi) % ngrid is the number of grid points used to calculate the plot, % Start with course grid (ngrid =20) and then use finer grid if necessary % default ngrid=50 % % Example % Plot y^3+exp(y)-tanh(x)=0 % % write function f as an 'inline' function of x and y-- right hand side % equal to zero % % f=inline('y^3+exp(y)-tanh(x)','x','y') % implot(f,[-3 3 -2 1]) % A.Jutan UWO 2-2-98 [email protected] if nargin == 1 ;% grid value and ranges not specified calculate default rangexy=[-2*pi,2*pi,-2*pi,2*pi]; ngrid=50; end if nargin == 2; % grid value not specified ngrid=50; end % get 2-D grid for x and y xm=linspace(rangexy(1),rangexy(2),ngrid); ym=linspace(rangexy(3),rangexy(4),ngrid); [x,y]=meshgrid(xm,ym); fvector=vectorize(fun);% vectorize the inline function to handle vectors of x y fvalues=feval(fvector,x,y); %calculate with feval-this works if fvector is an m file too %fvalues=fvector(x,y); % can also calculate directly from the vectorized inline function contour(x,y,fvalues,[0,0],'b-');% plot single contour at f(x,y)=0, blue lines xlabel('x');ylabel('y'); grid
Example:
Example
% Plot y^3+exp(y)-tanh(x)=0
%
% write function f as an 'inline' function of x and y-- right hand side
% equal to zero
%
% f=inline('y^3+exp(y)-tanh(x)','x','y')
% implot(f,[-3 3 -2 1])
五. 由复平面画复杂曲面
%画圆锥曲面
%利用复平面
z=cplxgrid(20);%在复平面上面得到一个21*41（(m+1)*(2m+1)）的网格，x，y分别为
其实部和虚部
x=real(z);
y=imag(z);
fz=sqrt((x.^2+y.^2));%圆锥公式，z^2=x^2+y^2来求fz
cplxmap(z,fz)%作图