Frenet
formulas: Optional extra-credit
due: by Monday Jan 31.
Will be graded out of a total of 25 points, and added onto your
homework score for the course.
Please only hand in well-written solutions/explanations.
Assignment - purpose is to understand concepts under Frenet formulas,
idea of moving coordinate frame along a curve and connections to
curvature and torsion of the curve.
Please read the optional reading at the end of section 2.3
In what follows, assume the curve is parametrized by arclength s.
The following exercises:
1. If T'(t) is not the zero vector, N(t) := unit vector in
direction of T' is perpindicular to T (and is called the principal
normal). Let B = T x N. B is called the binormal. Together
T, N, B form a right handed system that move along the path as t
changes. Show:
(a) dB/ds . B = 0 (the . means dot product)
(b) dB/ds. T = 0
(c) dB/ds is a scalar multiple of N
(note
it is also helpful to look at equation (1.26) on page 55 of your text
and section 1.14)
2. Now we know for some scalar , dB/ds = -t N. This t is called the torsion of
the curve. Show that if a path lies in a plane then the torsion
is zero. (hint: what can you say about dB/ds in this case?)
3. Combine the results from 1 and 2 to show the Frenet formulas
hold:
(a) dB/ds = -tN
(b) dT/ds = kN
for some k (referred to as the curvature)
[what exactly is k??]
(c) dN/ds = -kT
+ tB (hint: dN/ds is normal to
N, so must lie in T, B plane...)
4. Describe in your own words what the Frenet frame and the
curvature and torsion tell you about a curve.
5. Find the curvature and torsion of any helix you like.