Multivariate Calculus - Bruce E. Shapiro

LECTURE 5. VELOCITY, ACCELERATION, AND CURVATURE 43
Definition 5.4 The Principal unit normal vector N is
N =
dT/ ds
∥
∥dT / ds ∥ = 1 dT
κ ds
(5.17)
N is not the only unit normal vector to the curve at P; in fact, on could define
an infinite number of unit normal vectors to a curve at any given point. To do so,
merely find the plane perpendicular to the tangent vector. An vector in this plane
is normal to the curve. One such vector that is often used is the following, which is
perpendicular to both T and N.
Definition 5.5 The binormal vector is B = T × N .
Definition 5.6 The triple of normal vectors {T, N, B} is called the trihedral at
P.
Definition 5.7 The plane formed by T and N is called the osculating plane at
P.
Since T = v / ‖v‖ we can write
v = T ‖v‖ = T ds
dt
(see equation (5.10).) The acceleration vector (equation (5.11)) is
a = dv
dt = d (
T ds )
= T d2 s
dt dt dt + dT ds
dt dt
( )
= T d2 s dT
dt + ds ds
ds dt dt
= T d2 s
dt + dT ( ) ds 2
ds dt
From equation (5.17),
(5.18)
( )
a = T d2 s ds 2
dt + κN (5.19)
dt
Equation (5.19) breaks the acceleration into two perpendicular components, one
that is tangent to the curve:
a ‖ = d2 s
dt
and one that is perpendicular to the curve:
a ⊥ = κ
( ) ds 2
dt
so that
a = a ‖ T + a ⊥ N
Math 250, Fall 2006 Revised December 6, 2006.