Lecture Notes for 120 - UCLA Department of Mathematics
Lecture Notes for 120 - UCLA Department of Mathematics
Lecture Notes for 120 - UCLA Department of Mathematics
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3.1. THE FUNDAMENTAL EQUATIONS 59<br />
Pro<strong>of</strong>. By assumption<br />
0 0 and apple>0. This <strong>for</strong>ces<br />
d✓<br />
ds = apple<br />
and<br />
This establishes the <strong>for</strong>mulas<br />
dT<br />
d✓ = N<br />
There is a very elegant way <strong>of</strong> collecting the Serret-Frenet <strong>for</strong>mulas.<br />
Corollary 3.1.3. (Darboux) For a space curve as above define the Darboux<br />
vector<br />
D = ⌧T + appleB<br />
then<br />
Pro<strong>of</strong>. We have<br />
d ⇥ ⇤ ds<br />
T N B =<br />
dt<br />
dt D ⇥ ⇥ T N B ⇤<br />
D ⇥ T = appleN<br />
D ⇥ N = ⌧B appleT<br />
D ⇥ B = ⌧N<br />
so the equation follows directly from the Serret-Frenet <strong>for</strong>mulas.<br />
⇤<br />
⇤<br />
Exercises.<br />
(1) Find the curvature, torsion, normal, and binormal <strong>for</strong> the twisted cubic<br />
q (t) = t, t 2 ,t 3<br />
(2) Consider a regular space curve q (t) with non-vanishing curvature and<br />
torsion. Let k be a fixed vector and denote T, N, B the angles<br />
between T, N, B and k. Showthat<br />
and<br />
d N<br />
dt<br />
apple =<br />
sin<br />
cos<br />
T d T<br />
N dt<br />
sin N = apple cos T ⌧ cos B<br />
d B<br />
dt<br />
⌧ = sin B d B<br />
cos N dt<br />
sin B = ⌧ d T<br />
sin<br />
apple dt<br />
T