Math 3298 - Hampton - Lab 11 - Due 12/4 (last lab).

Lab reports should conform to the guidelines explained on the handout.  Collaboration during the lab is strongly encouraged, although the write-up of the lab report must be done individually.

Tori and torus knot surfaces

A torus with radii 1 and 2 can be parameterized as follows:

T[u_,v_]:={(2+Cos[v])Cos[u],(2+Cos[v])Sin[u],Sin[v]}

We can plot parameterized surfaces with ParametricPlot3D:

ParametricPlot3D[T[u,v],{u,0,2*Pi},{v,0,2*Pi}]

We can use this parameterization to form a space curve called a torus knot by setting u = m t, v = n t for integers m and n and using t as a parameter.  For example, here are the (2,3) and(11,17) torus knots:

ParametricPlot3D[T[2 t,3 t],{t,0,2*Pi},PlotPoints->1000]

ParametricPlot3D[T[11 t,17 t],{t,0,2*Pi},PlotPoints->1000]

We can create a surface from a torus knot by connecting each point on the curve to the point opposite it with a line segment; here is an example for the (2,3) knot:

ParametricPlot3D[T[2t,3t]*s+T[2t,3t+Pi]*(1-s),{s,0,1},{t,0,Pi}]

Exercise

1. 	a. Make a plot of the surface formed by taking a (1,2) torus knot and connecting each point to the point opposite it with a line segment.  
	b. Use mathematica to compute the surface area of the surface in part 1.  Note that there is a built-in cross-product command, Cross.
2.  Repeat the above exercises for the (2,1) torus knot. More care is required here.