Transverse Approach to Geometric Algebra Models for Manipulating Quadratic Surfaces

Quadratic surfaces gain more and more attention in the geometric algebra community and some frameworks to represent, transform, and intersect these quadratic surfaces have been proposed. To the best of our knowledge, however, no framework has yet proposed that supports all the operations required to completely handle these surfaces. Some existing frameworks do not allow the construction of quadratic surfaces from control points while some do not allow to transform these quadratic surfaces. Although a framework does not exist that covers all the required operations, if we consider all already proposed frameworks together, then all the operations over quadratic surfaces are covered there. This paper presents an approach that transversely uses different frameworks for covering all the operations on quadratic surfaces. We employ a framework to represent any quadratic surfaces either using control points or the coefficients of its implicit form and then map the representation into another framework so that we can transform them and compute their intersection. Our approach also allows us to easily extract some geometric properties.