# 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.

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*The Columbia Electronic Encyclopedia,* 6th ed. Copyright © 2012, Columbia University Press. All rights reserved.

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