Lecture Notes for 120 - UCLA Department of Mathematics

3.1. THE FUNDAMENTAL EQUATIONS 58
B · dN dt
= v ⇥ a
|v ⇥ a| · d ✓ ◆
a (a · T) T
dt |a (a · T) T|
= v ⇥ a
|v ⇥ a| · j
|a (a · T) T|
(v ⇥ a) · j
=
|v ⇥ a| 2 |v|
In the third line all of the missing terms disappear as they are perpendicular to
v ⇥ a. The last line follows from our formulas for the area of the parallelogram
spanned by v and a. Aslightlymoreconvincingproofworksbyfirstnoticingthat
Thus
Next we recall that
v = (v · T) T
a = (a · T) T +(a · N) N
j = (j · T) T +(j · N) N +(j · B) B
det ⇥ v a j ⇤ =(v · T)(a · N)(j · B)
v · T = |v|
a · N = |v| 2 apple
so we have to calculate j · B. Keepinginmindthata · B =0we obtain
and finally combine this with
to obtain the desired identity.
j · B = a · dB
dt
= ⌧ |v| a · N
apple =
= ⌧apple|v| 3
|v ⇥ a|
|v| 3
The curvature and torsion can also be described by the formulas
⌧ =
apple =
area of parallelogram (v, a)
|v| 3
signed volume of the parallepiped (v, a, j)
(area of the parallelogram (v, a)) 2
Corollary 3.1.2. If q (t) is a regular space curve with linearly independent
velocity and acceleration, then T is regular and if ✓ is its arclength parameter, then
and
d✓
ds = dT
ds · N
dT
d✓ = a (a · T) T
|a (a · T) T|
⇤