Algorithms for Conic Fitting Through Given Proper and Improper Waypoints in Geometric Algebra for Conics

Abstract

As an addition to proper points of the real plane, we introduce a representation of improper points, i.e. points at infinity, in terms of Geometric Algebra for Conics (GAC) and offer possible use of both types of points. More precisely, we present two algorithms fitting a conic to a dataset with a certain number of points lying on the conic precisely, referred to as the waypoints. Furthermore, we consider inclusion of one or two improper waypoints, which leads to the asymptotic directions of the fitted conic. The number of used waypoints may be up to four and we classify all the cases.

Appendix A. MATLAB Implementation of Conic Fitting Algorithms

Below, we summarise both presented algorithms for conic fitting with given waypoints implemented as a MATLAB function. Let us note that the reduced forms of some vectors and matrices were employed to avoid a few unnecessary computations with zero elements, similarly to the conic fitting algorithms in [9, 10].

Both algorithms receive the same types of inputs and generate the same types of outputs, in particular:

Inputs:

$\textbf{a}$, $\textbf{b}$, $\textbf{c}$:

column vectors of x, y, z homogeneous coordinates of waypoints

${\textbf{p}}{\textbf{x}}$, ${\textbf{p}}{\textbf{y}}$:

column vectors of x, y coordinates of data points

Outputs:

Conic:

fitted conic in the form (2.7)

obj_function:

value of objective function (3.1) for fitted conic

Let us also reiterate that when given affine coordinates of a proper point or the elements of a vector corresponding to an improper point, these can be easily converted into homogeneous coordinates using the mapping (2.1).