# Implicitization, Interpolation, and Syzygies

MSc thesis, National and Kapodistrian University of Athens, 2016.

# Abstract

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.

# BibTex

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