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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