Page 226 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 18: Theory of Space Curves

5. If ................... for all points of a non-inflectional curve x, then the curve is contained in a Notes
plane.

18.13 Review Questions

1. One of the most important space curves is the circular helix x(t) = (a cos t, a sin t, bt), where
a  0 and b are constants. Find the length of this curve over the interval [0, 2].

2. Find a constant c such that the helix x(t) = (a cos(ct), a sin(ct), bt) is parametrized by
arclength, so that |x’(t)| = 1 for all t.
3. The astroid is the curve defined by x(t) = (a cos t, a sin t, 0), on the domain [0, 2]. Find the
3
3
points at which x(t) does not define an immersion, i.e., the points for which x’(t) = 0.
4. The trefoil curve is defined by x(t) = ((a + b cos(3t)) cos(2t), (a + b cos(3t)) sin(2t), b sin(3t)),
where a and b are constants with a > b > 0 and 0  t  2. Sketch this curve, and give an
argument to show why it is knotted, i.e. why it cannot be deformed into a circle without
intersecting itself in the process.
5. Let x be a curve with x’(t )  0. Show that the tangent line at x(t ) can be written as
0
0
y(u) = x(t ) + ux’(t ) where u is a parameter that can take on all real values.
0
0
6. The plane through a point x(t ) perpendicular to the tangent line is called the normal plane
0
at the point. Show that a point y is on the normal plane at x(t ) if and only if
0
x’(t ) . y = x’(t ) . x(t )
0
0
0
7. Show that the curvature k of a circular helix
x(t) = (r cos(t), r sin(t), pt)
|r|
is equal to the constant value  = . Are there any other curves with constant
r + p 2
2
curvature? Give a plausible argument for your answer.
8. Assuming that the level surfaces of two functions f(x , x , x ) = 0 and g(x , x , x ) = 0 meet in
2
3
1
3
1
2
a curve, find an expression for the tangent vector to the curve at a point in terms of the
gradient vectors of f and g (where we assume that these two gradient vectors are linearly
2
independent at any intersection point.) Show that the two level surfaces x – x = 0 and
1
2
0
2
x x – x  consists of a line and a “twisted cubic” x (t) = t, x (t) = t , x (t) = t . What is the
3
2
2
2
3
3 1
1
line?
9. What is the geometric meaning of the function f(t) = x(t) . m used in the proof of Fenchel’s
theorem?
10. Let m be a unit vector and let x be a space curve. Show that the projection of this curve into
the plane perpendicular to m is given by
y(t) = x(t) – (x(t) . m)m.
Under what conditions will there be a t with y’(t ) = 0?
0
0
11. W. Scherrer proved that this property characterized a sphere, i.e. if the total twist of every
curve on a closed surface is zero, then the surface is a sphere.
12. T. Banchoff and J. White proved that the total twist of a closed curve is invariant under
inversion with respect to a sphere with center not lying on the curve.
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