Measuring Conicity from Shape Boundaries

There has been a lot of research on ellipse fitting and measuring ellipticity of a set of points. However, when the shape is primarily hyperbolic or parabolic, there are no existing methods to measure such properties. This paper describes the first known methods of measuring conicity, hyperbolicity and parabolicity of a set of points. After finding the best conic fit, we measure the corresponding ellipticity (using a known method), hyperbolicity or parabolicity value with respect to that best fit. We are interested in measures which rely exclusively on shape boundary points. They should also be calculated very quickly, be invariant to rotation, scaling and translation. The evaluation of fits transforms the point data into polar representation where the radius in this representation is equal to the difference of distances from each point to both foci (for hyperbolas), and the sum of distances from each point to the focus and a line parallel to the directrix line (for parabolas). The linearity of the polar representation will correspond to the quality of the fit for the original data. The conicity measure is tested on a set