Curvature: Definition and Curvature of a Circle is 1/Radius

In this video, I go over the definition of curvature, which describes how quickly a curve changes direction at a given point. It is defined as the magnitude of the derivative of the unit tangent vector in terms of its arc length, so that it is independent of any specific parameterization. I also rewrite this definition to obtain curvature, indicated by the Greek letter Kappa (κ), as a function of the parameter t, as this is often easier to compute. I illustrate this with an example on obtaining the curvature of a circle, which turns out to be equal to 1 divided by its radius. Note that the curvature of a straight line is equal to zero since the unit tangent vector would be zero in that case.

#math #vectors #calculus #arclength #curvature

3 Curvature Circle.png

Timestamps

Curvature: Smooth parameterizations are continuous and the tangent vector is not equal to 0 – 0:00
Recap on Unit Tangent Vector, T – 1:13
Graph of T, which changes quickly when C bends sharply – 2:46
Definition of curvature as the magnitude of the derivative of T in terms of arc length – 5:00
Curvature is easier to compute when written in terms of t – 7:30
Curvature = magnitude of derivative of T divided by derivative of the vector function – 11:35
Example 3: Curvature of a circle is 1/radius – 13:52
- Obtaining parametric equations of a circle – 15:10
- Curvature of a circle with radius a is 1/a: Large circles have small curvature – 24:05
Curvature of a straight line is zero – 25:00