# Algebra and Geometry

## Abstract

It is an elementary fact, even known to high schoolers, that there are two different ways of showing “shapes”, the one by *photography* and the other by *drawing*. Mathematically, they correspond to the one by giving *equations* which show algebraic relations among coordinates and the other by using *parameters*, of which coordinates are expressed as functions. Think of a circle of radius one, on one hand expressed by an equation *x* ^{2}+*y* ^{2}=1 and on the other by *x*=cos*t*, *y*=sin*t* with a *parametert*, 0≤*t*≤2*π*. As is well known, if the equation is cubic with no *singularity*, then we have a parametric presentation using elliptic functions.

The correspondence between *equational presentation* and *parametric presentation* becomes complex but interesting in the case of *many variables and presence of singularities*. I will present how the correspondence can be processed in general.