In my last post Curvature, I showed how the curvature, along a path is equal to: . I then asked a “challenge” question about how one could compute the curvature without referencing a velocity or acceleration vector. I have slightly rephrased the question below:
Is there a natural way to describe the curvature of a function without using the parametric form? For example, suppose there exists a curve defined by the intersection of three volumes, which are defined using an extended Cartesian system with four coordinates: . So, , and we wish to know the curvature at a point . This is a well-defined question with no reference to a parameter, . It is natural to wonder whether there is a way to find curvature without imposing a new parameter.
Spoiler (my work on the problem)
See if your work agrees with mine:
Part 1: Show that:
Part 2: Show that: , where is defined above.
Part 3: Use Parts 1 and 2 to show that the path which satisfies the system of equations: is a circle of radius two; and the curvature at any point on this path is , as expected since for a circle of radius, .