Abstract: The paper presents an original Python-based application that outlines the advantages of combining some elementary notions of mathematics with the study of a programming language. The application support refers to some of the first lessons of analytic geometry, meaning conics and quadrics and their reduction to a standard form, as well as some related notions. The chosen programming language is Python, not only for its closer to an everyday language syntax – and therefore, enhanced readability – but also for its highly reusable code, which is of utmost importance for a mathematician that is accustomed to exploit already known and used problems to solve new ones. The purpose of this paper is, on one hand, to support the idea that one of the most appropriate means to initiate one into programming is throughout mathematics, and reciprocal, one of the most facile and handy ways to assimilate some basic knowledge in the study of mathematics is to apply them in a personal project. On the other hand, besides being a mean of learning both programming and analytic geometry, the application subject to this paper is itself a useful tool for it can be seen as an independent original Python package for analytic geometry.
Abstract: Super-quadrics can represent a set of implicit surfaces,
which can be used furthermore as primitive surfaces to construct a
complex object via Boolean set operations in implicit surface
modeling. In fact, super-quadrics were developed to create a
parametric surface by performing spherical product on two parametric
curves and some of the resulting parametric surfaces were also
represented as implicit surfaces. However, because not every
parametric curve can be redefined implicitly, this causes only implicit
super-elliptic and super-hyperbolic curves are applied to perform
spherical product and so only implicit super-ellipsoids and
hyperboloids are developed in super-quadrics. To create implicit
surfaces with more diverse shapes than super-quadrics, this paper
proposes an implicit representation of spherical product, which
performs spherical product on two implicit curves like super-quadrics
do. By means of the implicit representation, many new implicit curves
such as polygonal, star-shaped and rose-shaped curves can be used to
develop new implicit surfaces with a greater variety of shapes than
super-quadrics, such as polyhedrons, hyper-ellipsoids, superhyperboloids
and hyper-toroids containing star-shaped and roseshaped
major and minor circles. Besides, the newly developed implicit
surfaces can also be used to define new primitive implicit surfaces for
constructing a more complex implicit surface in implicit surface
modeling.