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 p and q, the Maple command
add(p,q,a,b) will give you the vector ap+bq.
- Parametrize the line segment between the points
p=T(0,Pi/2) and q=T(0,3*Pi/2). Hint: p
and q are in R^3. First write out the coordinates
of p and q, then parametrize the line
segment between them.
- For each value of s,
parametrize the line segment between the points
p(s)=T(s,Pi/2) and q(s)=T(s,3*Pi/2).
- For each fixed value of s, a line segment
connects p(s) and q(s). As we allow
s to vary, these segments sweep out a
surface. Use the
plot3d command to plot this
parametric surface. Describe it. We call the curves
forming the outside edges of a surface its boundary. 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).
- For Bonus Points: Create a physical model from wire and string that
illustrates this family of surfaces!
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
Robert E. Thurman<thurman@geom.umn.edu>
Original lab created by Frederick J. Wicklin
Document Created: Thu Feb 23 1995
Last modified: Mon Mar 10 16:53:50 1997