Worksheet: Graphing Parametric Surfaces

Introduction

As we have done before with worksheets:

Please work through all the code and examples in this worksheet.

Things in this font are Octave commands — don't cut and paste them, type the commands yourself. Trust me, this is the better way to learn these commands.

Please do the following to show your work:

1. Start a diary.
2. Type all the commands.
3. Also start a separate .m file in which you put all the commands which work, in their best forms. (So this will be basically an edited diary, with only the good stuff.)
4. Wherever there is a plot you are to make, save off a copy of it. Make sure the plot has some title and is saved in a file name which is (part of) that title.

Parametric Surfaces in Octave

Example 1: A Cone

  rv=linspace(0,1,30);
  thetav=linspace(0,2*pi,30);

  [r,theta]=meshgrid(rv,thetav);

  x=r.*cos(theta);
  y=r.*sin(theta);
  z=r;

  mesh(x,y,z);

  axis tight
  box on

  xlabel('x-axis')
  ylabel('y-axis')
  zlabel('z-axis')

Example 2: A Sphere

  phiv=linspace(0,pi,30);
  thetav=linspace(0,2*pi,30);
  [phi,theta]=meshgrid(phiv,thetav);

  x=2*sin(phi).*cos(theta);
  y=2*sin(phi).*sin(theta);
  z=2*cos(phi);

  mesh(x,y,z)
  axis equal
  view(135,30)
  box on
  xlabel('x-axis')
  ylabel('y-axis')
  zlabel('z-axis')

Example 3: A Snail Shell

  surf(x,y,z);

  surf(x,y,z,'EdgeColor','none');

  axis equal
  axis off

Exercises

• autumn
• bone
• colorcube
• cool
• copper
• flag
• gray
• hot
• hsv
• jet
• ocean
• pink
• prism
• rainbow
• spring
• summer
• white
• winter

1. The Möbius Strip: Parametric equations $$\begin{eqnarray*}x&=&\frac{v}{2}\sin\frac{u}{2}\\y&=&\left(1+\frac{v}{2}\cos\frac{u}{2}\right)\sin u\\z&=&\left(1+\frac{v}{2}\cos\frac{u}{2}\right)\cos u\end{eqnarray*}$$ with parameters $u$ and $v$ having ranges $0\le u\le2\pi$ and $-1\le v\le1$.
2. The Klein Bottle: Parametric equations $$\begin{eqnarray*}x&=&\left(r+\cos\frac{u}{2}\sin v-\sin\frac{u}{2}\sin2v\right)\cos u\\y&=&\left(r+\cos\frac{u}{2}\sin v-\sin\frac{u}{2}\sin2v\right)\sin u\\z&=&\sin\frac{u}{2}\sin v+\cos\frac{u}{2}\sin2v\end{eqnarray*}$$ with parameters $r=1$, and $u$ and $v$ having ranges $0\le u,v\le2\pi$. (Try colormap('hot').)
3. Enneper's Surface: Parametric equations $$\begin{eqnarray*}x&=&u-u^3/3+uv^2\\y&=&v-v^3/3+u^2v\\z&=&u^2-v^2\end{eqnarray*}$$ with parameters $u$ and $v$ having ranges $-1.5\le u,v\le1.5$.
4. Ellipsoids: Parametric equations $$\begin{eqnarray*}x&=&a\cos u\sin v\\y&=&b\sin u\sin v\\z&=&c\cos v\end{eqnarray*}$$ with parameters $u$ and $v$ having ranges $0\le u\le2\pi$ and $0\le v\le\pi$. Try $a=3$, $b=4$, and $c=5$. See what happens when you vary $a$, $b$, and $c$ — can you explain why? (Use axis equal.)
5. Hyperboloids of One Sheet: Parametric equations $$\begin{eqnarray*}x&=&a\cosh(u)\cos(v)\\y&=&b\cosh(u)\sin(v)\\z&=&c\sinh(u)\end{eqnarray*}$$ with parameters $u$ and $v$ having ranges $-2\le u\le2$ and $0\le v\le2\pi$. Try $a=b=c=1$, then vary them. Again, use axis equal.
6. Hyperboloids of Two Sheets: Parametric equations $$\begin{eqnarray*}x&=&a\sinh(u)\cos(v)\\y&=&b\sinh(u)\sin(v)\\z&=&c\cosh(u)\end{eqnarray*}$$ with parameters $u$ and $v$ having ranges $-2\le u\le2$ and $0\le v\le2\pi$. Try $a=b=c=1$, then $a=b=1$ but $c=-1$. Again, use axis equal.
7. The Lissajous Surface: Parametric equations $$\begin{eqnarray*}x&=&\sin u\\y&=&\sin v\\z&=&\sin((d-au-bv)/c)\end{eqnarray*}$$ with parameters $u$ and $v$ having ranges $-\pi\le u,v\le\pi$. Start with $a=b=c=1$ and $d=0$, then vary these. Again, use axis equal.
8. Whiney Umbrella: Parametric equations $$\begin{eqnarray*}x&=&uv\\y&=&u\\z&=&v^2\end{eqnarray*}$$ with parameters $u$ and $v$ having ranges $-1\le u,v\le1$. Use axis equal.
9. Steiner's Surface: Parametric equations $$\begin{eqnarray*}x&=&\sin(2u)\cos^2(v)\\y&=&\sin(u)\sin(2v)\\z&=&\cos(u)\sin(2v)\end{eqnarray*}$$ with parameters $u$ and $v$ having ranges $0\le u\le\pi$ and $-\pi/2\le v\le\pi/2$. Use axis equal.
10. The Pseudosphere: Parametric equations $$\begin{eqnarray*}x&=&\cos(u)\sin(v)\\y&=&\sin(u)\sin(v)\\z&=&\cos(v)+\ln(\tan(v/2))\end{eqnarray*}$$ with parameters $u$ and $v$ having ranges $0\le u\le2\pi$ and $0<v<\pi$. Be careful in your choices of grid points since as $v$ gets close to 0 or $\pi$, $\ln(\tan(v/2))$ goes to $\pm\infty$.
11. Dini's Surface: Parametric equations $$\begin{eqnarray*}x&=&\cos(u)\sin(v)\\y&=&\sin(u)\sin(v)\\z&=&\cos(v)+\ln(\tan(v/2))+au\end{eqnarray*}$$ with parameters $u$ and $v$ having ranges $0\le u\le2\pi$ and $0<v<\pi$. Again, the $\ln$ term has problems as $v$ gets close to 0 or $\pi$.
12. The Helicoid: Parametric equations $$\begin{eqnarray*}x&=&av\cos(u)\\y&=&av\sin(u)\\z&=&bu\end{eqnarray*}$$ with parameters $u$ and $v$ having ranges $0\le u\le2\pi$ and $-d<v<d$. Try different values of $a$, $b$, and $d$, and describe in words what you see.
13. Catenoid: Parametric equations $$\begin{eqnarray*}x&=&a\cos(u)\cosh(v/a)\\y&=&a\sin(u)\cosh(v/a)\\z&=&v\end{eqnarray*}$$ with parameters $u$ and $v$ having ranges $0\le u\le2\pi$ and $-c<v<c$. Try different values of $a$ and $c$, and describe what you see.
14. The Torus: Parametric equations $$\begin{eqnarray*}x&=&(a+b\cos(v))\cos(u)\\y&=&(a+b\cos(v))\sin(u)\\z&=&b\sin(v)\end{eqnarray*}$$ with parameters $u$ and $v$ having ranges $0\le u\le2\pi$ and $0\le v\le2\pi$. Try varying $a$ and $b$, describe.
15. Alien Space Ship: Let $a=.4$ and $w=\sqrt{1-a^2}$. Use the parametric equations $$\begin{eqnarray*}x&=&-u+2w^2\cosh(au)\sinh(au)/d\\y&=&2w\cosh(au)\left(-w\cos(v)\cos(wv)-\sin(v)\sin(wv)\right)/d\\z&=&2w\cosh(au)\left(-w\sin(v)\cos(wv)+\cos(v)\sin(wv)\right)/d\end{eqnarray*}$$ with parameters $u$ and $v$ having ranges $-13.4\le u\le13.4$ and $-37.2\le v\le37.2$, where $d=a\left((w\cosh(au))^2+(a\sin(wv))^2\right)$.