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Using Geometric Algebra for Navigation in Riemannian and Hard Disc Space Using Geometric Algebra for Navigation in Riemannian and Hard Disc Space Werner Benger Andrew Hamilton Mike Folk/Quincey Koziol Center for Computation & Center for Astrophysics and The HDF Group Technology Space Astronomy 1901 So. First St. Suite C-2 Louisiana State University JILA, University of Colorado Champaign, IL 61820. USA 239 Johnston Hall Boulder, CO 80309, USA [email protected] Baton Rouge, LA 70803, USA Andrew.Hamilton [email protected] @colorado.edu Simon Su Erik Schnetter Marcel Ritter/Georg Ritter Princeton Institute for Center for Computation & Department of Computer Computational Science and Technology Science Engineering Dept. of Physics & Astronomy University of Innsbruck 345 Peter B. Lewis Library Louisiana State University Technikerstrasse 21a Princeton, NJ 08544, USA Baton Rouge, LA 70803, USA A-6020 Innsbruck, Austria [email protected] [email protected] [email protected] ABSTRACT where Geometric Algebra is used for navigating the cam- A \vector" in 3D computer graphics is commonly under- era position in space and time. Another application exam- stood as a triplet of three floating point numbers, eventually ple is given by a simulation code solving Einstein's equa- equipped with a set of functions operating on them. This tion in general relativity numerically on supercomputers, hides the fact that there are actually different kinds of vec- outputting the Newman-Penrose pseudo scalars as primary tors, each of them with different algebraic properties and quantities of interest to study gravitational waves, both for consequently different sets of functions. Differential Geome- visualization and observational verification. try (DG) and Geometric Algebra (GA) are the appropriate mathematical theories to describe these different types of Geometric Algebra moreover provides means to describe \vectors". They consistently define the proper set of opera- how the metadata information required per \vector" can be tions attached to each class of \floating point triplet" and al- provided in persistent storage. Given large datasets that are low to derive what meta-information is required to uniquely expensively collected or generated by simulations requiring identify a specific type of vector in addition to its purely millions of CPU hours, it is increasingly important and diffi- numerical values. We shortly review the various types of cult to be able to share and correctly interpret such datasets \vectors" in 3D computer graphics, their relations to rota- years after their generation, across different research groups tions and quaternions, and connect these to the terminology from different fields of science. A unique, standardized, of co-vectors and bi-vectors in DG and GA. Not only in extensible identification of the geometric properties of the 3D, but also in 4D, the elegant formulations of GA yield dataset elements is a necessary pre-requisite for this. simi- to more clarity, which will be demonstrated on behalf of lar to the way in which the IEEE standard for floating point the use of bi-quaternions in relativity, allowing for instance values enables sharing floating point values. We utilize the a more insightful formulation to determine the Newman- mechanisms as provided by the HDF5 library here, a generic Penrose pseudo scalars from the Weyl tensor. self-describing file format developed for large datasets as used in high performance computing. It allows specifying 1. INTRODUCTION metadata in addition to the purely numerical data, providing Geometric Algebra [14] and the sometimes mystified concept an abstraction layer for specifying the mathematical prop- of spinors eases implementation and intuition significantly, erties on top of the lower-level binary layout. It is therefore both in computer graphics and in physics [15]. We demon- desirable to us the functionality of this powerful I/O library strate the concrete application of these concepts in two in- to express the semantics of vector quantities as they arise in dependently developed computer graphic software packages, Geometric Algebra. This will be discussed in section 5. 2. VECTOR SPACES A vector space over a field F (such as R) is a set V together with two binary operations vector addition + : V × V ! V and scalar multiplication ◦ : F × V ! V. The elements of V are called vectors. A vector space is closed under the operations + and ◦, i.e., for all elements u; v 2 V and all el- ements λ 2 F there is u + v 2 V and λ ◦ u 2 V (vector space axioms). The vector space axioms allow computing the dif- ferences of vectors and therefore defining the derivative of a vector-valued function v(s): R !V as d v(s + ds) − v(s) v(s) := lim : (1) ds ds!0 ds 2.1 Tangential Vectors In differential geometry, a tangential vector on a manifold d M is the operator ds that computes the derivative along a curve q(s): R ! M for an arbitrary scalar-valued function f : M ! R: d df (q(s)) f := : (2) ds q(s) ds Tangential vectors fulfill the vector space axioms and can therefore be expressed as linear combinations of deriva- µ tives along the n coordinate functions x : M ! R with µ = 0 : : : n − 1, which define a basis of the tangential space Tq(s)(M) on the n-dimensional manifold M at each point q(s) 2 M: n−1 µ n−1 d X dx (q(s)) @ X µ f = f =: q_ @ f (3) ds ds @xµ µ µ=0 µ=0 µ d whereq _ are the components of the tangential vector ds in µ the chart fx g and f@µg are the basis vectors of the tangen- tial space in this chart. We will use the Einstein sum conven- Figure 1: Vector transformation under shrinking the tion in the following text, which assumes implicit summation height coordinate by a factor of two: tangential vec- over indices occurring on the same side of an equation. Of- tors (differences between two points) shrink in their ten tangential vectors are used synonymous with the term height component by a factor two as well, whereas \vectors" in computer graphics when a direction vector from surface normal vectors (co-vectors) grow by a fac- point A to point B is meant. A tangential vector on an tor two in height, see the vertical components of the n-dimensional manifold is represented by n numbers in a vector and co-vector shown on the right hand side chart. in the figure. 2.2 Co-Vectors three-dimensional space, a plane is equivalently described by The set of operations df : T (M) ! R that map tangential a \normal vector", which is orthogonal to the plane. While vectors v 2 T (M) to a scalar value v(f) for any function \normal vectors" are frequently symbolized via a vector ar- f : M ! R defines another vector space which is dual to the row, like tangential vectors, they are not the same, rather tangential vectors. Its elements are called co-vectors. they are dual to tangential vectors. It is more appropri- @f ate to visually symbolize them as a plane. This visual is < df; v >= df(v) := v(f) = vµ@ f = vµ (4) µ @xµ also supported by (5), which can be interpreted as the to- tal differential of a function f: a co-vector describes how a Co-vectors fulfill the vector space axioms and can be written scalar function advances in space, which can be visualized as linear combination of co-vector basis functions dxµ: as surfaces of constant function value (\isosurface"). On an @f df =: dxµ (5) n-dimensional manifold a co-vector is correspondingly sym- @xµ bolized by an (n − 1)-dimensional subspace. with the dual basis vectors fulfilling the duality relation ( 2.3 Tensors ν µ = ν : 1 l < dx ;@µ >= (6) A tensor T of rank l × m is a multi-linear map of l vectors µ 6= ν : 0 m and m co-vectors to a scalar The space of co-vectors is called the co-tangential space l ∗ ∗ ∗ Tm : T (M) × :::T (M) × T (M) × :::T (M) ! R : (7) Tp (M). A co-vector on an n-dimensional manifold is repre- | {z } | {z } sented by n numbers in a chart, same as a tangential vector. l m However, co-vector transforms inverse to tangential vectors Tensors are elements of a vector space themselves and form when changing coordinate systems, as is directly obvious 0 the tensor algebra. They are represented relative to a coor- from eq. (6) in the one-dimensional case: As < dx ;@0 >= 1 l+m 0 dinate system by a set of k numbers for a k-dimensional must be sustained under coordinate transformation, dx manifold. The construction of an tensor of higher rank from must shrink by the same amount as @0 grows when an- lower rank is called the outer product (also known as tensor, other coordinate scale is used to represent these vectors. dyadic or Kronecker product), denoted by ⊗: In higher dimensions this is expressed by an inverse trans- µν µ ν µ ν formation matrix, as demonstrated in Fig. 1. In Euclidean T ≡ T @µ ⊗ @ν = v u @µ ⊗ @ν = v @µ ⊗ u @ν = v ⊗ u (8) Tensors of rank 2 may be represented using matrix notation. The new vector space is denoted Λ2(V). With the exterior 0 Tensors of type T1 are equivalent to co-vectors and called product, v ^ u = −u ^ v 8u; v 2 V , which consequently co-variant, in matrix notation (relative to a chart) they cor- results in v ^ v = 0 8 v 2 V. The exterior product defines 1 respond to rows. Tensors of type T0 are equivalent to a tan- an algebra on its elements, the exterior algebra (or Grass- gential vector and are called contra-variant, corresponding man algebra) [9, 5].
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