Infinite implicit replication: case study for voxelizing and representing cyclical parametric surfaces implicitly

This article proposes a method to infinitely replicate implicit defined objects which is at the same time simple and efficient. The technique is implemented by including replication factors (involving truncation of floating point values corresponding to the coordinates of the point to be evaluated) in the implicit object equation in order to create the clones. These replication factors serve to identify the exact region in the space where each object clone will show up during evaluation. The method is illustrated in the cases of replicating simple objects or replicating cylindrical/spherical coordinate cyclical objects. Cyclical objects are often represented parametrically using cylindrical/spherical coordinates because in this representation angles can be associated with the idea of creating infinite cycles when the associated angles tend to infinity. However, representing these objects implicitly cannot reproduce more than one cycle because the obtained angles are generally limited to the values in the principal branch of the corresponding inverse trigonometric function. Infinite implicit replication solves this problem and introduces new possibilities where Cartesian coordinates could be intermingled with cylindrical/spherical coordinates in the same implicit function. The case study of voxelizing the replicated objects using interval arithmetic is also presented in detail. The efficiency of the infinite replication method comes from the fact that the equation has to be evaluated just once per point even though an infinite number of clones exist.