Algebra and Geometry
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=cost, y=sint with a parametert, 0≤t≤2π. 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.