Implicitization is a fundamental change of representation of geometric objects
from a parametric or point cloud representation to an implicit form, namely as
the zero set of one (or more) polynomial equation. This thesis examines three
questions related to expressing the implicit equation of a curve or a surface.

First, we consider a sparse interpolation method for implicitization: When the
basis of the kernel of the interpolation matrix is in reduced row echelon form,
the implicit equation can be readily obtained, without demanding computations
such as multivariate polynomial GCD or factoring. As a second contribution, a
numeric method that computes a multiple of the implicit equation based on the
power method is tested and evaluated.

The third contribution of this thesis is to provide a method for computing a
matrix representation of a rational planar or space curve, or a rational
surface, when we are only given a sufficiently large sample of points (point
cloud) on the object in such a way that the value of the parameter is known per
point. Our method extends the approach of algebraic syzygies for
implicitization to the case where the parameterization is not given but only
assumed.

```
@masterthesis{Gavriil2016,
title = {Implicitization, Interpolation, and Syzygies},
author = {Gavriil, Konstantinos},
year = {2016},
school = {National and Kapodistrian University of Athens}
}
```