Generalized distance functions

In this paper, we obtain a generalized version of the well-known distance function family L-p norm. We prove that the new functions satisfy distance function properties. By using these functions, convex symmetric shapes can be described as loci, the set of points which are in equal distance from a given point. We also show that these symmetric convex shapes can be easily parameterized. We also show these distance functions satisfy a Lipschitz type Condition. We provide a fast ray marching algorithm for rendering shapes described by these distance functions. These distance functions can be used as building blocks for some implicit modeling tools such as soft objects. constructive soft geometry, freps or ray-quadrics.