Multi Calc, Lec 12B: Curvature (Definition, Meaning, and Formulas)

Given a curve described with an arc length parameterization r(s), we can define the curvature κ(s) as the magnitude of the derivative of the unit tangent vector: ||T'(s)||. For the arc length parameterization, that is the same as the magnitude of the acceleration vector. Small circles have large curvature while large circles have small curvature. In fact, the radius of curvature of an osculating circle is R(s)=1/κ(s). What are the formulas for the curvature in terms of any parameterization r(t)? There are three main types: 1) for the graph of a function y=f(x), 2) for a planar parametric curve (x(t),y(t)) that may not be the graph of a function, 3) for a curve in space. Infinite Powers, How Calculus Reveals the Secrets of the Universe: https://amzn.to/37PBMjb.

Multivariable Calculus, Lecture 12B

#curvature #MultivariableCalculus #DifferentialGeometry

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