Building a Surface from Torus Knots
In the previous question, you parametrized the two curves on the
torus that are the image under T of the
lines t=Pi/2 and t=3 Pi/2. Now we want to
parametrize the surface defined by inserting a line segment from the point
T(s,Pi/2) to the point T(s,3 Pi/2) for every value of
s in [0,2 Pi].
Question #2
Refer to your sketches from the previous question as you answer this question.
Some people may
want to consider the following hint: if you want a linear combination
of two vectors u and v, the Maple command
add(u,v,a,b) will give you the vector au+bv.
- Parametrize the line segment between the points
T(0,Pi/2) and T(0,3*Pi/2).
- Parametrize the line segment between the points
T(s,Pi/2) and T(s,3*Pi/2).
Hint: Notice that T(s,Pi/2) and T(s,3*Pi/2)
describe two different curves on the torus.
- Use the
plot3d command to plot your
parametric surface. What surface is it? What is the boundary of
this surface?
- Include a plot of this surface in your lab writeup with
the outside portion of the surface colored red and the
inside portion of the surface colored green.
Question #3
- Change the previous problem so that you
- parametrize the line segment from the point
T(0,Pi/2) to the point T(Pi/2,3*Pi/2).
- parametrize the surface defined by inserting a
line segment from the point
T(s,Pi/2) to the point T(s+Pi/2,3*Pi/2)
for s in [0,2*Pi].
Hint: These are the same curves as in the previous
question. The only thing changing is the
way you are connecting the curves.
- Now what surface do you have?
What is the boundary of this surface?
- Given the two curves
s-> T(s,Pi/2) and s-> T(s,3*Pi/2),
describe the family of surfaces that you obtain by inserting a
line segment from T(s,Pi/2) to the point T(s+c,3*Pi/2).
(We've already examined the cases c=0 and c=Pi/2.
Now let's see what happens if we start with two curves that
are torus knots of type (1,1).
Make sure that you can identify the two curves parametrized by
s-> T(s,s) and s-> T(s,s+Pi) before you start
the next question.
Question #4
- Use the method of the previous problems to
parametrize the surface generated by
inserting a line segment from the point
T(s,s) on the first curve to the point T(s,s+Pi)
on the second curve.
- What surface do you get? What is the boundary of this surface?
Question #5
Conjecture on the surface that you obtain by starting with two
torus knots of type (1,n) and connecting the curves with line segments
from T(s,ns) to T(s,ns+Pi).
Next: Building a Surface from One Knot
Up: Introduction
Previous: Torus Knots
Frederick J. Wicklin<fjw@geom.umn.edu>
Document Created: Thu Feb 23 1995
Last modified: Sat Feb 25 09:46:47 1995