Example for Curvature First, we'll finish up curvature by doing an example:
Find K at (0,0) for y=x^2
Principal Normal
A Space Curve has an infinite number of normals. By convention, we take
the normal to be the particular normal which is in the plane of the tangent
and which points toward the concave side of the curve. Pictorally this
means:
Example:
r(t) = ti + t^2j + t^3k
=> v x a = 6t^2i - 6tj + 2k
K(0,0,0) [t=0] = 2
The TNB Frame
Imagine a coordinate system defined at a particular point of a curve by the
Tangent (T), the Pricipal Normal (N), and a third Vector
perpendicular to the first two (B).
TwistTwist is represented by the greek letter . Twist (also called torsion) is a scalar.
= 0 for a plane curve.
There is another, alternate definition for .
Example:
Differentiating VectorsSay you have v(t) = 5t^2Ti.e. The velocity is some scalar function times the tangent vector. (It always is) a = dv/dt = 10t*T + 5t^2 * (dT/dt) dT/dt is the direction of the normal. Remember, T is a function of t, and therefore the product rule must be used in differentiating.
Generalization:
Created by Lawrence C. Weintraub on 2/28/95 Edited by Milos Borojevic on 3/4/95 Send comments using feedback page |