Calculating curvature

Curvature is conceptually simple but usually difficult to calculate. For a level set curve f(x, y) = c, such as in the previous couple posts, the equation for curvature is Even when f has a fairly simple expression, the expression for κ can be complicated. If we define then the level set of f(x, y) = c is an equilateral triangle when c = −4. The level sets are smoothed triangles for −4 < c < 0. The curvature of these level sets at any point is given by Simplification But there is one instance in which curvature is easy to calculate. For the graph of a function g(x) = y, the curvature is approximately the absolute value of the second derivative of g, provided the first derivative is small. At a local maximum or local minimum of g(x) the approximation is exact because the first derivative is zero. Max and min curvature of smoothed triangles This means that in the example above, we can calculate the maximum and minimum curvature of the level sets. The maximum curvature occurs in the…