Serret-Frenet equations in 2

Given a plane curveMathworldPlanetmath, we may associate to each point on the curve an orthonormal basis consisting of the unit normalMathworldPlanetmath tangent vectorMathworldPlanetmath and the unit normal. In general, different points will have different bases associated to them, so we may ask how the basis depends upon the choice of point. The Serret-Frenet equations answer this question by relating the rte of change of the basis vectors to the curvatureMathworldPlanetmathPlanetmath of the curve.

Suppose I is an open intervalDlmfPlanetmath and c:I2 is a twice continuously differentiable curve such that c=1. Let us then define the tangent vector and normal vectorMathworldPlanetmath as

𝐓 = c,
𝐍 = J𝐓,

where J=(0-110) is the rotational matrix that rotates the plane 90 degrees counterclockwise.


Differentiating c,c=1 yields 𝐓,𝐓=0, so 𝐓 is in the orthogonal complementMathworldPlanetmath of 𝐓, which is 1-dimensional. Since J𝐓 is also in the orthogonal complement, it follows that there exists a functionMathworldPlanetmath κ:I such that


Furthermore, κ is uniquely determined by this equation. We define this unique κ to be the curvature of c. Explicitly,


Serret-Frenet equations

By the definition of curvature

𝐓 = κJ𝐓=κ𝐍

and so

𝐍 = J𝐓=κJ𝐍=-κ𝐓

since J2=-I. These are the Serret-Frenet equations

Title Serret-Frenet equations in 2
Canonical name SerretFrenetEquationsInmathbbR2
Date of creation 2013-03-22 15:16:57
Last modified on 2013-03-22 15:16:57
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 8
Author rspuzio (6075)
Entry type Definition
Classification msc 53A04
Related topic SerretFrenetFormulas