Euclidean distance fit of conics using differential evolution

In this chapter, we apply the Differential Evolution (DE) algorithm to fit conic curves, ellipse, parabola and hyperbola, to a set of given points. Our proposal minimizes the sum of orthogonal Euclidean distances from the given points to the curve; this is a nonlinear problem that is usually solved by minimizing the square of the Euclidean distances, which allows the usage of the gradient and some numerical methods based on it, such as the Gauss-Newton method. The novelty of the proposed approach is that we can utilize any distance function as the objective function because we are using an Evolutionary Algorithm. For the parabola case, it is proved that the calculation of the orthogonal point to a given point is a simple problem that can be solved using a cubic equation.We also show how to combine DE with a conventional deterministic algorithm to initialize it. We present experiments that show better results than those previously reported. In addition, our solutions have a very low variance, which indicates the robustness of the approach.