Duality, Barycentric Coordinates and Intersection Computation in Projective Space with GPU support

This paper presents solution of selected problems using principle of duality and projective space representation. It will be shown that alternative formulation in the projective space offers quite surprisingly simple solutions that lead to more robust and faster algorithms which are convenient for use within parallel architectures as GPU (Graphical Processor Units-NVIDIA-TESLA/Fermi) or SCC (Intel-Single-chip Cloud Computing), which can speed up solutions of numerical problems in magnitude of 10-100. There are many geometric algorithms based on computation of intersection of lines, planes etc. Sometimes, very complex mathematical notations are used to express simple mathematical solutions, even if their formulation in the projective space offers much more simple solution. It is shown that a solution of a system of linear equations is equivalent to generalized cross product, which leads with the duality principle to new algorithms. This is presented on a new formulation of a line in 3D given as intersection of two planes which is robust and fast, based on duality of Plücker coordinates. The presented approach can be used also for reformulation of barycentric coordinates computations on parallel architectures. The presented approach for intersection computation is well suited especially for applications where robustness is required, e.g. large GIS/CAD/CAM systems etc

Solvability of Equations in Clifford Algebras

University of Minnesota M.S. thesis. October 2016. Major: Applied and Computational Mathematics. Advisor: Joseph Gallian. 1 computer file (PDF); vii, 71 pages.In this paper, we are studying selected types of quadratic equations in Clifford algebra, using methods developed for solving analogous equations in quaternions. Our goal is to classify the solutions in order to build a solid foundation for the study of Minkowski Pythagorean hodograph curves

Articulating Space: Geometric Algebra for Parametric Design -- Symmetry, Kinematics, and Curvature

To advance the use of geometric algebra in practice, we develop computational methods for parameterizing spatial structures with the conformal model. Three discrete parameterizations – symmetric, kinematic, and curvilinear – are employed to generate space groups, linkage mechanisms, and rationalized surfaces. In the process we illustrate techniques that directly benefit from the underlying mathematics, and demonstrate how they might be applied to various scenarios. Each technique engages the versor – as opposed to matrix – representation of transformations, which allows for structure-preserving operations on geometric primitives. This covariant methodology facilitates constructive design through geometric reasoning: incidence and movement are expressed in terms of spatial variables such as lines, circles and spheres. In addition to providing a toolset for generating forms and transformations in computer graphics, the resulting expressions could be used in the design and fabrication of machine parts, tensegrity systems, robot manipulators, deployable structures, and freeform architectures. Building upon existing algorithms, these methods participate in the advancement of geometric thinking, developing an intuitive spatial articulation that can be creatively applied across disciplines, ranging from time-based media to mechanical and structural engineering, or reformulated in higher dimensions

Geometric Algebra and its Application to Computer Graphics

Early in the development of computer graphics it was realized that projective geometry is suited quite well to represent points and transformations. Now, maybe another change of paradigm is lying ahead of us based on Geometric Algebra. If you already use quaternions or Lie algebra in additon to the well-known vector algebra, then you may already be familiar with some of the algebraic ideas that will be presented in this tutorial. In fact, quaternions can be represented by Geometric Algebra, next to a number of other algebras like complex numbers, dual-quaternions, Grassmann algebra and Grassmann-Cayley algebra. In this half day tutorial we will emphasize that Geometric Algebra\ud • is a unified language for a lot of mathematical systems used in Computer Graphics,\ud • can be used in an easy and geometrically intuitive way in Computer Graphics.\ud \ud We will focus on the (5D) Conformal Geometric Algebra.It is an extension of the 4D projective geometric algebra. For example, spheres and circles are simply represented by algebraic objects. To represent a circle you only have to intersect two spheres (or a sphere and a plane), which can be done with a basic algebraic operation. Alternatively you can simply combine three points (using another product in the algebra) to obtain the circle through these three points.\ud \ud Next to the construction of algebraic entities, kinematics can also be expressed in Geometric Algebra. For example, the inverse kinematics of a robot can be computed in an easy way. The geometrically intuitive operations of Geometric Algebra make it easy to compute the joint angles of a robot which need to be set in order for the robot to reach its goal