# Efficient representation and manipulation of quadratic surfaces using Geometric Algebras

## TL;DR

This paper unifies various Geometric Algebra frameworks to comprehensively represent, transform, and intersect quadratic surfaces, enabling complete handling of these surfaces through a single, versatile approach.

## Contribution

It introduces a unified framework that supports all essential operations for quadratic surfaces, combining previous methods into a comprehensive solution.

## Key findings

- Supports construction of quadratic surfaces from control points and implicit coefficients
- Enables transformations and intersections of quadratic surfaces
- Allows extraction of geometric properties easily

## Abstract

Quadratic surfaces gain more and more attention among the Geometric Algebra community and some frameworks were proposed in order to represent, transform, and intersect these quadratic surfaces. As far as the authors know, none of these frameworks support all the operations required to completely handle these surfaces. Some frameworks do not allow the construction of quadratic surfaces from control points when others do not allow to transform these quadratic surfaces. However , if we consider all the frameworks together, then all the required operations over quadratic are covered. This paper presents a unification of these frameworks that enables to represent any quadratic surfaces either using control points or from the coefficients of its implicit form. The proposed approach also allows to transform any quadratic surfaces and to compute their intersection and to easily extract some geometric properties .