There is a natural way to compute the curvature for a curve whose position is given as a function of time, . The following is the vector proof which shows how the velocity and acceleration vectors can be used to compute the curvature at any time.
Is there a natural way to describe the curvature of a function without using the parametric form? For example, suppose a curve is the consequence of the intersection of three volumes defined using an extended Cartesian system with four coordinates: . So, , and we wish to know the curvature at . 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.
My initial thoughts are that we can still define the differential arc length in this approach ; however, at the point , one’s movement is restricted by the requirement that . So the differential arc length and the unit tangent vector will reflect this in the solution.
Help me out. Add a solution route in a comment below if you have a solution. I have a compact form for the tangent vector, but I haven’t found a method to get the curvature yet.