%% A Surface of Revolution

syms x t
%%
% I will revolve the curve 
%%
% 
% $$y=(x^2\,\mbox{--}\,x^4)/4$$
% 
% about the x-axis.  Here is a plot of the curve.

ezplot((4*x^2-x^4)/4,[-2,2])

%%
% Here is a plot of the surface of revolution.  The *ezsurf* command
% assumes the variables are ordered alphabetically (at least in simple
% examples) when figuring out the domain of each variable, so I have to
% list the t interval before the x interval.

ezsurf(x,(4*x^2-x^4)*cos(t)/4,(4*x^2-x^4)*sin(t)/4,[0,2*pi,-2,2])
title('Surface of revolution')

%%
% Now I want to plot the surface along with its tangent plane at the point
% where $x=0.8$ and $t = \pi/2$.  I begin by finding the equation of the tangent
% plane.

clear all
syms x y z t
f = [x,(4*x^2-x^4)*cos(t),(4*x^2-x^4)*sin(t)];
fx = diff(f,x);
ft = diff(f,t);
P = subs(f,[x,t],[.8,pi/2]);
fxp = subs(fx,[x,t],[.8,pi/2]);
ftp = subs(ft,[x,t],[.8,pi/2]);
normal =cross(fxp,ftp);
planeeqn = normal *transpose([x,y,z]-P)

%%
% The equation of the tangent plane is planeeqn=0.  I'll solve that for z.
planefun = solve(planeeqn,z)

%%
% So the tangent plane is the graph of planefun.
% To plot the surface and the tangent plane, I'll have to use *surf* or 
% *mesh* for at least one of those, rather than an *ez* command.  I'll 
% use *surf* for the surface. Since I'll have to enter the parametrized
% equation in notation MATLAB understands for vectors and matrices, I'll 
% first give the *vectorize* command (which tells me where to put the 
% periods). 

vectorize(f)

%%
% Since x,t are symbolic variables, I'll use u,v as the variables for the
% surface. Here is the graph of the surface, along with the tangent plane.
% I didn't give the *axis equal* command, so things aren't to scale.  If
% you run the Mfile, you'll be able
% to rotate the plot to get a good view. 

[u,v]=meshgrid(-1:.05:1,0:.02*pi:2*pi);
surf(u,(4.*u.^2-u.^4).*cos(v),(4.*u.^2-u.^4).*sin(v)), hold on
ezmesh(planefun,[-1.1,1.1,-2,2]), hold off
title('The surface and its tangent plane at P')