differential geometry
The Analysis of Curves If a point r moves along a curve at arc length s from some fixed point, then t = d r/ ds is a unit tangent vector to the curve at r. The normal vector n is perpendicular to the curve at the point and indicates the direction of the rate of change of t, i.e., the tendency of r to bend in the plane containing both r and t, and the binormal vector b is perpendicular to both t and n and indicates the tendency of the curve to twist out of the plane of t and n. These three vectors are related by the three formulas of the French mathematician Jean Frédéric Frenet, which are fundamental to the study of space curves: d t/ ds = κn; d n/ ds =  κt + τb; d b/ ds =  τn, where the constants κ and τ are the curvature and the torsion of the curve, respectively. Of special interest are the curves called evolutes and involutes; the evolute of a curve is another curve whose tangents are the normals to the original curve, and an involute of a curve is a curve whose evolute is the given curve. Sections in this article:
The Columbia Electronic Encyclopedia, 6th ed. Copyright © 2012, Columbia University Press. All rights reserved. More on differential geometry The Analysis of Curves from Fact Monster:
See more Encyclopedia articles on: Mathematics
