6760, Math 21a, Fall 2009

Exhibits page Math 21a 2009, Multivariable Calculus

Ruled surfaces

Course head: Oliver Knill

Office: SciCtr 434

Email: knill@math.harvard.edu

Problem A: (creative assignment). Submit online your own ruled
surface as well as your own bumpy sphere. The examples should be different
from the examples given. Try to build some examples which look good. We
will make an exhibit with the best submissions. |

Gallery of ruled surfaces 2009 |
Gallery of bumpy spheres 2009 |
Asteroid project 2007 | Marble book project 2006 |

Ruled surface Bumpy sphere |
Your ruled surface and a bumpy sphere can be entered
here in this poll.
Please use Mathematica notation, where functions are capitalized and square brackets are used in arguments.
f[t_]:={Cos[t],Sin[t],0}; g[t_]:= {Cos[t+2],Sin[t+2],2};We will exhibit the best submissions. Be creative! What is a ruled surface? The vector valued functions f[t] and g[t] form two closed curves in space. The ruled surface is obtained by connecting corresponding points on these curves with lines. You can use either a flash applet from this source or use Mathematica. Here is example code for a ruled surface in Mathematica f[t_]:={Cos[t],Sin[t],0}; g[t_]:= {Cos[t+2],Sin[t+2],2}; ParametricPlot3D[s f[t] +(1-s) g[t],{t,0,2Pi},{s,0,1}]And here is an example of a bumpy sphere in Mathematica
rho[t_,s_]:=(4+Sin[s] Sin[2 s + t]); ParametricPlot3D[rho[t,s] {Sin[s] Cos[t],Sin[s] Sin[t],Cos[s]},{s,0,Pi},{t,0,2Pi}]which is equivalent to rho[t_,s_]:=(4+Sin[s] Sin[2 s + t]); SphericalPlot3D[rho[t,s],{s, 0, Pi},{t,0,2Pi}]In a bumpy sphere, the radius rho depends on the Euler angles theta (abbreviated t) and phi (abbreviated s). |

Questions and comments to knill@math.harvard.edu

Math21b | Math 21a | Fall 2009 |
Department of Mathematics |
Faculty of Art and Sciences |
Harvard University