I am very interested in a certain problem and I am wondering what methods currently exist to solve it. Given a curve defined by a function which maps any given arc length, s, from an arbitrarily chosen original point on the curve (so s designates the arc length from the original point to another point on the curve), to a certain tangential angle, theta, what is the area enclosed by this curve if said curve is a Jordan curve?
Edit: More specifically, if a curve, A, is defined by the aforementioned function, and this curve is closed (it returns to its original point after some finite arc length), what is the area within the entire curve. For instance, with this construction, we have a Jordan curve. As the Jordan Curve Theorem states that a Jordan curve divides the plane into two sections, an inner section and an outer section, what is the area of the inner section define by the previous curve?
Edit 2: I understand that you can use Green's Theorem for this solution, but as for many different curves described in the aforementioned way, this would be a difficult, messy, and not very elegant way to solve the problem. Is there a simpler solution?