Conics, Their Pencils and Intersections in Geometric Algebra

Loučka P., Vašík P.

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.

Chomicki C., Breuils S., Biri V., Nozick V.

Conic sections are extensively encountered in a wide range of disciplines, including optics, physics, and various other fields. Consequently, the geometric algebra community is actively engaged in developing frameworks that enable efficient support and manipulation of conic sections. Conic-conic intersection objects are known and supported by algebras specialized in conic sections representation, but there is yet no elegant formula to extract the intersection points from them. This paper proposes a method for point extraction from an conic intersection through the concept of pencils. It will be based on QC2GA, the 2D version of QCGA (Quadric Conformal Geometric Algebra), that we also prove to be equivalent to GAC (Geometric Algebra for Conics).

Byrtus R., Derevianko A., Vašík P., Hildenbrand D., Steinmetz C.

We describe a possibility for geometric calculation of specific conics’ intersections in Geometric Algebra for Conics (GAC) using its operations that may be expressed as sums of products. The advantage is that no solver for a system of quadratic equations is needed and thus no numerical error is involved. We also describe specific conics connected to intersections of conics in a general mutual position. Then we show how symbolic operations may be calculated directly in GAALOPWeb software, that the basis coefficients may be read off in the appropriate basis and, moreover, the result may be immediately and truly visualized. We compare the functionality with Maple package Clifford.

Breuils S., Nozick V., Fuchs L.

This paper presents both a recursive scheme to perform Geometric Algebra operations over a prefix tree, and Garamon, a C++ library generator implementing these recursive operations. While for low dimension vector spaces, precomputing all the Geometric Algebra products is an efficient strategy, it fails for higher dimensions where the operation should be computed at run time. This paper describes how a prefix tree can be a support for a recursive formulation of Geometric Algebra operations. This recursive approach presents a much better complexity than the usual run time methods. This paper also details how a prefix tree can represent efficiently the dual of a multivector. These results constitute the foundations for Garamon, a C++ library generator synthesizing efficient C++/Python libraries implementing Geometric Algebra in both low and higher dimensions, with any arbitrary metric. Garamon takes advantage of the prefix tree formulation to implement Geometric Algebra operations on high dimensions hardly accessible with state-of-the-art software implementations. Garamon is designed to produce easy to install, easy to use, effective and numerically stable libraries. The design of the libraries is based on a data structure using precomputed functions for low dimensions and a smooth transition to the new recursive products for higher dimensions.

Breuils S., Fuchs L., Hitzer E., Nozick V., Sugimoto A.

We introduce the quadric conformal geometric algebra inside the algebra of $${\mathbb {R}}^{9,6}$$ . In particular, this paper presents how three-dimensional quadratic surfaces can be defined by the outer product of conformal geometric algebra points in higher dimensions, or alternatively by a linear combination of basis vectors with coefficients straight from the implicit quadratic equation. These multivector expressions code all types of quadratic surfaces in arbitrary scale, location, and orientation. Furthermore, we investigate two types of definitions of axis aligned quadric surfaces, from contact points and dually from linear combinations of $${\mathbb {R}}^{9,6}$$ basis vectors.

De Keninck S.

This paper introduces a novel visualization method for elements of arbitrary Geometric Algebras. The algorithm removes the need for a parametric representation, requires no precomputation, and produces high quality images in realtime. It visualizes the outer product null space (OPNS) of 2-dimensional manifolds directly and uses an isosurface approach to display 1- and 0-dimensional manifolds. A multi-platform browser based implementation is publicly available.

Hrdina J., Návrat A., Vašík P.

We present a particular geometric algebra together with such an embedding of two–dimensional Euclidean space that the algebra elements may be in the most efficient way interpreted as arbitrary conic sections. Consequently, in this setting we provide full description of the conic sections analysis, classification and their transformations. Examples that show the functionality and consistency are provided in Maple together with the source code.

Breuils S., Nozick V., Sugimoto A., Hitzer E.

Geometric Algebra can be understood as a set of tools to represent, construct and transform geometric objects. Some Geometric Algebras like the well-studied Conformal Geometric Algebra constructs lines, circles, planes, and spheres from control points just by using the outer product. There exist some Geometric Algebras to handle more complex objects such as quadric surfaces; however in this case, none of them is known to build quadric surfaces from control points. This paper presents a novel Geometric Algebra framework, the Geometric Algebra of $${\mathbb {R}}^{9,6}$$ , to deal with quadric surfaces where an arbitrary quadric surface is constructed by the mere wedge of nine points. The proposed framework enables us not only to intuitively represent quadric surfaces but also to construct objects using Conformal Geometric Algebra. Our proposed framework also provides the computation of the intersection of quadric surfaces, the normal vector, and the tangent plane of a quadric surface.

Easter R.B., Hitzer E.

This paper introduces the Double Conformal/Darboux Cyclide Geometric Algebra (DCGA), based in the $$\mathcal {G}_{8, 2}$$ Clifford geometric algebra. DCGA is an extension of CGA and has entities representing points and general (quartic) Darboux cyclide surfaces in Euclidean 3D space, including circular tori and all quadrics, and all surfaces formed by their inversions in spheres. Dupin cyclides are quartic surfaces formed by inversions in spheres of torus, cylinder, and cone surfaces. Parabolic cyclides are cubic surfaces formed by inversions in spheres that are centered on points of other surfaces. All DCGA entities can be transformed by versors, and reflected in spheres and planes.

Gregory A.L., Lasenby J., Agarwal A.

We present a novel derivation of the elastic theory of shells. We use the language of geometric algebra, which allows us to express the fundamental laws in component-free form, thus aiding physical interpretation. It also provides the tools to express equations in an arbitrary coordinate system, which enhances their usefulness. The role of moments and angular velocity, and the apparent use by previous authors of an unphysical angular velocity, has been clarified through the use of a bivector representation. In the linearized theory, clarification of previous coordinate conventions which have been the cause of confusion is provided, and the introduction of prior strain into the linearized theory of shells is made possible.

Goldman R., Mann S.

We investigate the efficacy of the Clifford algebra R(4, 4) as a computational framework for contemporary 3-dimensional computer graphics. We give explicit rotors in R(4, 4) for all the standard affine and projective transformations in the graphics pipeline, including translation, rotation, reflection, uniform and nonuniform scaling, classical and scissor shear, orthogonal and perspective projection, and pseudoperspective. We also explain how to represent planes by vectors and quadric surfaces by bivectors in R(4, 4), and we show how to apply rotors in R(4, 4) to these vectors and bivectors to transform planes and quadric surfaces by affine transformations.

Richter-Gebert J.

Dorst L., Fontijne D., Mann S.

This chapter provides an introduction to geometric algebra, a powerful computational system used to describe and solve geometrical problems. The main features of geometric algebra includes that vectors can be used to represent aspects of geometry, but the precise correspondence is a modeling choice. Geometric algebra offers three increasingly powerful models for Euclidean geometry. Geometric algebra has products to combine vectors to new elements of computation. They represent oriented subspaces of any dimension, and they have rich geometric interpretations within the models. A linear transformation on the vector space dictates how subspaces transform; this augments the power of linear algebra in a structural manner to the extended elements. Geometric objects and operators are represented on a par and are exchangeable: objects can act as operators, and operators can be transformed such as geometrical objects. Geometric algebra focuses on the subspaces of a vector space as elements of computation. It constructs these systematically from the underlying vector space and extends the matrix techniques to transform them, even supplanting those completely when the transformations are orthogonal. In the vector space model and the conformal model, orthogonal transformations are used to represent basic motions of Euclidean geometry. This makes that type of linear transformation fundamental to doing geometry in the models.

Buchholz S., Tachibana K., Hitzer E.M.

Neural computation in Clifford algebras, which include familiar complex numbers and quaternions as special cases, has recently become an active research field. As always, neurons are the atoms of computation. The paper provides a general notion for the Hessian matrix of Clifford neurons of an arbitrary algebra. This new result on the dynamics of Clifford neurons then allows the computation of optimal learning rates. A thorough discussion of error surfaces together with simulation results for different neurons is also provided. The presented contents should give rise to very efficient second–order training methods for Clifford Multi-layer perceptrons in the future.

Hartley R., Zisserman A.