Math 423 Differential Geometry Fall 2006
Homework 4: The Main Theorem of Space Curves
Due Thursday Sept. 28
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1. Section 3.5 #3
2. Section 3.5 #6
3. Section 3.5 #7
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Bonus Questions for those taking the course for four credits or for those who
want more of a challenge.
4. The Frenet frame is not the only useful way to frame a curve. Let ฮฑ:IโR3
be a unit speed curve and consider a frame (T(t), E1(t), E2(t)) alonf ฮฑwhich is
positively oriented and satisfies
E0
1(t) = โk1(t)T(t) and E0
2(t) = โk2(t)T(t)
for some functions k1(t) and k2(t).
(a) Prove that two curves with the same k1(t) and k2(t) are congruent.
(b) Assuming ฮบ(t)>0, express k1(t) and k2(t) in terms of ฮบ(t) and ฮธ(t) where
ฮธ(t) is defined by the equation
N(t) = cos ฮธ(t)E1(t) + sin ฮธ(t)E2(t).
(c) Express ฮบand ฯin terms of k1and k2.
(d) Show that T0=k1E1+k2E2.
(e) Show that k1(t) and k2(t) are constant iff ฮฑis part of a circle. What is the
relation of the radius of the circle and k1and k2?
(f) Show that ฮฑis a plane curve iff k1(t)/k2(t) is constant.
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