DOCSLIB.ORG

Color Image Representation Using Multivector

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Color Image Representation Using Multivector 2014 Fifth International Conference on Intelligent Systems, Modelling and Simulation Color Image Representation Using Multivector Somasis Roy Anirban Mitra Sanjit Kumar Setua Dept. of Computer Science Dept. of Computer Science Dept. of Computer Science University of Calcutta Academy of Technology, WBUT University of Calcutta Abstract – This paper discusses about Clifford Algebra and its scalar that’s why a RGB color image will use three times significance in a color image representation. A color image as much memory as a gray-scale image of the same pixel following RGB color model is described by a multivector in dimensions. The main objective is representation of an 3D space. The color blades in the multivector define the image as independent of coordinate system. colors. Different shades of color can be obtained by applying There are several mathematical tools independent of rotor operator on those blades. Clifford color space is introduced to define each color against a pixel of an image in coordinate system to represent an image. By the form of a color blade. A multivector function is derived from traditional approach of linear algebra, the concept of this space for explaining the color image. In simpler way, it is homogeneous coordinate is successful for developing proposed that a color image is stored in the Clifford color computer graphics packages [2]. The dissimilarity space. This paper is also useful in the reduction of between matrix computation and geometric computation time for color image processing as every Grade – interpretation leads a problem in this approach. As k vectors are colors. mentioned earlier the safest and efficient way to get rid – of coordinate system is the adoption of vector. A color Keywords Clifford algebra, Wedge, Multivector, Color blade, image is defined as the summation of vectors against Grade, Color vector, Clifford color space each pixel exists in that image [3]. It is observed that I. INTRODUCTION higher grade of vector is also responsible to define a color of a pixel in an image. In that case, Quaternion An image is defined as the finite set of spatial algebra [4] and Clifford algebra shows their power as a coordinates with attributes. It is described by the geometric mathematical tool in the development of computer shapes and their relations using the coordinates. Each graphics. Both of these algebra is different from coordinate is represented by a two- traditional linear algebra and both carefully handle the dimensional function at , where the function returns subspaces of any dimensions. A subspace may act as a a value known as the intensity of the coordinate. Each point (scalar), line (vector), plane (bivector), volume coordinate holds data of an image in various forms such as (trivector) etc. It is developed in different vector space binary, gray-scale or color form. A digital image which is for representing a certain geometric object. Clifford described by color is said to be a color image. It holds data algebra provides a strong framework to describe a color in color form, and contains the highest level of information in RGB model by using subspaces [5]. compare to other forms [1]. A color model or a color William K. Clifford (1845-1879) introduced Clifford representation system is needed to analyze the data present Algebra, also called Geometric Algebra where geometric in color image. product is the combination of the dot product and the There are several existing models widely used in outer product. Clifford’s motivation was to combine Computer Graphics like RGB model, HSI model, CMY Grassman’s “associativity” and Hamilton’s model etc. Every distinct color model has the distinguished “anticommutativity” features into a single matter. nature to define a pixel of an image. The most convenient Clifford also united the several advantages of quaternion color model used in digital image processing is the RGB algebra by introducing the subspaces. Clifford Algebra is color model. It is the additive color model whereas CMY used as a powerful mathematical tool in areas as (Cyan Magenta Yellow) is the subtractive color model. computer vision, computer graphics, robotics, etc. As it HSI (Hue-Saturation-Intensity) model provides best color is a coordinate-free tool it is easier to model geometric description for human interpretation. In RGB model each objects and their several transformations. pixel is represented by three values, the amount of Red, There are many color spaces by which an image is Green and Blue. It is spanned over three dimensional represented but RGB color space is considered here. Euclidean space and every coordinate of this model defines Geometrically, this RGB space is represented by a cube a color. A color in this color model is expressed as a where each of the axis representing unique color vector instead of coordinate system. In case of a gray-scale components [4]. According to pure quaternion every image a pixel holds a scalar (gray) value derived from the pixel in a color image contains three components RGB vector. Mathematically, a pixel of an image holds therefore the function is defined as: data in form of a vector or scalar to make coordinate free. A vector holds higher amount of actual data compare to 2166-0662/14 $31.00 © 2014 IEEE 357 DOI 10.1109/ISMS.2014.66 where , and are the imaginary part of the quaternion is empowered by two powerful operations like reflection [6]. The imaginary parts are actually mathematical and rotation. A rotor is responsible for representing a color at a particular pixel. Rotor edge detection based on representation of bivector. In Clifford algebra for - dimension Euclidean space number of subspace are RGB color space replaces traditional edge detection formed. The function for the color of a pixel techniques effectively [9]. An edge detection algorithm was introduced that accepts color value triples as vectors in three [10]. Clifford algebra framework is used because it dimensional Euclidean space where , and are extends the traditional convolution and Fourier transform the unit vectors. As the function shows each of the pixel is to vector fields. represented by a vector, therefore to represent an image number of vectors are needed, where and are III. CLIFFORD ALGEBRA the dimensions of the image. The wedge operator of A. Definition Clifford Algebra has a significant role in handling n A vector space is is considered for dimension . A subspaces. Each of these subspaces is defined as a wedge set of orthonormal basis vectors of n is { }. product or outer product. According to Clifford algebra every color is a subspace in RGB vector space. The A new element can be obtained from the geometric motivation of this paper is to define each of the color in product of the basis elements. Assuming any two basis RGB space by appropriate subspaces and representation of vectors and , the geometric product is formed color image in form of a multivector, combination of and it is anticommutative. Mathematically, it is subspaces. formulated as, = = - = - , This paper gives an idea to represent an image in form Squaring on the basis vectors results +1, -1 or 0. It of a multivector by using Clifford algebra. The rest of the suggests that there are nonnegative integers and paper is organized as follows. Section II discusses about such that = + + and the Clifford Algebra applications as the related works. Clifford Algebra is introduced in section III. In section IV, the geometrical transformations like reflection and rotation = is discussed. Section V explores about image representation by homogenous vector. In section VI we discuss about the proposal, how to represent a color image These operations give proof of the associativity of by using multivector. Finally section VII contains linear algebra with identity to define Clifford Algebra conclusions and future scope from this work. ( ) of dimension = + + generated by the vector space n. II. RELATED WORK The elements belong to the Clifford Algebra or Clifford Algebra is applied in Computer Science to Geometric Algebra is called multivectors. The space show new horizons in research and applications. It is a which contains all the dimensional multivectors is coordinate free algebra providing efficient framework to called the dimensional Clifford Algebra ( ). A those applications. Image processing is one of the applied multivector is generated as an entity containing several areas of Clifford Algebra. Color image representation and grades of elements of the basis set of . Such as, edge detection are the most inevitable part of color image processing. Several measures and approaches were + (1) proposed for the improvement of both the cases. This algebra authorizes number of operators to represent a color is expressed as the ג image effectively. Generally, image representation using where the multivectors RGB color space increases the dimensionality, non summation of as 0-vector (scalar), as 1- linearity, redundancy and reduces the efficiency of color vector (vector), as 2-vector (bivector), as 3- image processing [1]. Clifford algebra resolves these vector (trivector) and so on up to n-vector constraints while considering RGB color space by (pseudoscalar). introducing vector and multivector. It has invoked the vector concept to describe a color [7]. The colors in three B. Two dimensional Approach dimensional RGB color space are manipulated and the Vectors are the elements of the dimensional vector color components are the vector parts in the multivector space . In Clifford Algebra, the higher dimensional [3]. It solves real life problems such as removal of facial oriented subspaces are the basic elements of computation make up disturbances where a color is a multivector [8]. [12]. is represented as an extension of the Clifford Algebra is used to define different color spaces and act as a natural unified language to solve multicolor dimensional Euclidean vector space . For = 2, the image processing and pattern recognition problems [5].
Load more

Recommended publications
  • Quaternions and Cli Ord Geometric Algebras
    Quaternions and Cliord Geometric Algebras Robert Benjamin Easter First Draft Edition (v1) (c) copyright 2015, Robert Benjamin Easter, all rights reserved. Preface As a rst rough draft that has been put together very quickly, this book is likely to contain errata and disorganization. The references list and inline citations are very incompete, so the reader should search around for more references. I do not claim to be the inventor of any of the mathematics found here. However, some parts of this book may be considered new in some sense and were in small parts my own original research. Much of the contents was originally written by me as contributions to a web encyclopedia project just for fun, but for various reasons was inappropriate in an encyclopedic volume. I did not originally intend to write this book. This is not a dissertation, nor did its development receive any funding or proper peer review. I oer this free book to the public, such as it is, in the hope it could be helpful to an interested reader. June 19, 2015 - Robert B. Easter. (v1) [email protected] 3 Table of contents Preface . 3 List of gures . 9 1 Quaternion Algebra . 11 1.1 The Quaternion Formula . 11 1.2 The Scalar and Vector Parts . 15 1.3 The Quaternion Product . 16 1.4 The Dot Product . 16 1.5 The Cross Product . 17 1.6 Conjugates . 18 1.7 Tensor or Magnitude . 20 1.8 Versors . 20 1.9 Biradials . 22 1.10 Quaternion Identities . 23 1.11 The Biradial b/a .
    [Show full text]
  • Università Degli Studi Di Trieste a Gentle Introduction to Clifford Algebra
    Università degli Studi di Trieste Dipartimento di Fisica Corso di Studi in Fisica Tesi di Laurea Triennale A Gentle Introduction to Clifford Algebra Laureando: Relatore: Daniele Ceravolo prof. Marco Budinich ANNO ACCADEMICO 2015–2016 Contents 1 Introduction 3 1.1 Brief Historical Sketch . 4 2 Heuristic Development of Clifford Algebra 9 2.1 Geometric Product . 9 2.2 Bivectors . 10 2.3 Grading and Blade . 11 2.4 Multivector Algebra . 13 2.4.1 Pseudoscalar and Hodge Duality . 14 2.4.2 Basis and Reciprocal Frames . 14 2.5 Clifford Algebra of the Plane . 15 2.5.1 Relation with Complex Numbers . 16 2.6 Clifford Algebra of Space . 17 2.6.1 Pauli algebra . 18 2.6.2 Relation with Quaternions . 19 2.7 Reflections . 19 2.7.1 Cross Product . 21 2.8 Rotations . 21 2.9 Differentiation . 23 2.9.1 Multivectorial Derivative . 24 2.9.2 Spacetime Derivative . 25 3 Spacetime Algebra 27 3.1 Spacetime Bivectors and Pseudoscalar . 28 3.2 Spacetime Frames . 28 3.3 Relative Vectors . 29 3.4 Even Subalgebra . 29 3.5 Relative Velocity . 30 3.6 Momentum and Wave Vectors . 31 3.7 Lorentz Transformations . 32 3.7.1 Addition of Velocities . 34 1 2 CONTENTS 3.7.2 The Lorentz Group . 34 3.8 Relativistic Visualization . 36 4 Electromagnetism in Clifford Algebra 39 4.1 The Vector Potential . 40 4.2 Electromagnetic Field Strength . 41 4.3 Free Fields . 44 5 Conclusions 47 5.1 Acknowledgements . 48 Chapter 1 Introduction The aim of this thesis is to show how an approach to classical and relativistic physics based on Clifford algebras can shed light on some hidden geometric meanings in our models.
    [Show full text]
  • A Guided Tour to the Plane-Based Geometric Algebra PGA
    A Guided Tour to the Plane-Based Geometric Algebra PGA Leo Dorst University of Amsterdam Version 1.15{ July 6, 2020 Planes are the primitive elements for the constructions of objects and oper- ators in Euclidean geometry. Triangulated meshes are built from them, and reflections in multiple planes are a mathematically pure way to construct Euclidean motions. A geometric algebra based on planes is therefore a natural choice to unify objects and operators for Euclidean geometry. The usual claims of `com- pleteness' of the GA approach leads us to hope that it might contain, in a single framework, all representations ever designed for Euclidean geometry - including normal vectors, directions as points at infinity, Pl¨ucker coordinates for lines, quaternions as 3D rotations around the origin, and dual quaternions for rigid body motions; and even spinors. This text provides a guided tour to this algebra of planes PGA. It indeed shows how all such computationally efficient methods are incorporated and related. We will see how the PGA elements naturally group into blocks of four coordinates in an implementation, and how this more complete under- standing of the embedding suggests some handy choices to avoid extraneous computations. In the unified PGA framework, one never switches between efficient representations for subtasks, and this obviously saves any time spent on data conversions. Relative to other treatments of PGA, this text is rather light on the mathematics. Where you see careful derivations, they involve the aspects of orientation and magnitude. These features have been neglected by authors focussing on the mathematical beauty of the projective nature of the algebra.
    [Show full text]
  • Inverse and Determinant in 0 to 5 Dimensional Clifford Algebra
    Clifford Algebra Inverse and Determinant Inverse and Determinant in 0 to 5 Dimensional Clifford Algebra Peruzan Dadbeha) (Dated: March 15, 2011) This paper presents equations for the inverse of a Clifford number in Clifford algebras of up to five dimensions. In presenting these, there are also presented formulas for the determinant and adjugate of a general Clifford number of up to five dimensions, matching the determinant and adjugate of the matrix representations of the algebra. These equations are independent of the metric used. PACS numbers: 02.40.Gh, 02.10.-v, 02.10.De Keywords: clifford algebra, geometric algebra, inverse, determinant, adjugate, cofactor, adjoint I. INTRODUCTION the grade. The possible grades are from the grade-zero scalar and grade-1 vector elements, up to the grade-(d−1) Clifford algebra is one of the more useful math tools pseudovector and grade-d pseudoscalar, where d is the di- for modeling and analysis of geometric relations and ori- mension of Gd. entations. The algebra is structured as the sum of a Structurally, using the 1–dimensional ei’s, or 1-basis, scalar term plus terms of anticommuting products of vec- to represent Gd involves manipulating the products in tors. The algebra itself is independent of the basis used each term of a Clifford number by using the anticom- for computations, and only depends on the grades of muting part of the Clifford product to move the 1-basis the r-vector parts and their relative orientations. It is ei parts around in the product, as well as using the inner- this implementation (demonstrated via the standard or- product to eliminate the ei pairs with their metric equiv- thonormal representation) that will be used in the proofs.
    [Show full text]
  • Tutorial on Fourier and Wavelet Transformations in Geometric Algebra
    Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion Tutorial on Fourier and Wavelet Transformations in Geometric Algebra E. Hitzer Department of Applied Physics University of Fukui Japan 17. August 2008, AGACSE 3 Hotel Kloster Nimbschen, Grimma, Leipzig, Germany E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion Acknowledgements I do believe in God’s creative power in having brought it all into being in the first place, I find that studying the natural world is an opportunity to observe the majesty, the elegance, the intricacy of God’s creation. Francis Collins, Director of the US National Human Genome Research Institute, in Time Magazine, 5 Nov. 2006. I thank my wife, my children, my parents. B. Mawardi, G. Sommer, D. Hildenbrand, H. Li, R. Ablamowicz Organizers of AGACSE 3, Leipzig 2008. E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion CONTENTS 1 Geometric Calculus Multivector Functions 2 Clifford GA Fourier Transformations (FT) Definition and properties of GA FTs Applications of GA FTs, discrete and fast versions 3 Quaternion Fourier Transformations (QFT) Basic facts about Quaternions and definition of QFT Properties of QFT and applications 4 Spacetime Fourier Transformation (SFT) GA of spacetime and definition of SFT Applications of SFT 5 Clifford GA Wavelets GA wavelet basics and GA wavelet transformation GA wavelets and uncertainty limits, example of GA Gabor wavelets 6 Conclusion E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion Multivector Functions Multivectors, blades, reverse, scalar product Multivector M 2 Gp;q = Clp;q; p + q = n; has k-vector parts (0 ≤ k ≤ n) scalar, vector, bi-vector, .
    [Show full text]
  • Reciprocal Frame Vectors
    Reciprocal Frame Vectors Peeter Joot March 29, 2008 1 Approach without Geometric Algebra. Without employing geometric algebra, one can use the projection operation expressed as a dot product and calculate the a vector orthogonal to a set of other vectors, in the direction of a reference vector. Such a calculation also yields RN results in terms of determinants, and as a side effect produces equations for parallelogram area, parallelopiped volume and higher dimensional analogues as a side effect (without having to employ change of basis diagonalization arguments that don’t work well for higher di- mensional subspaces). 1.1 Orthogonal to one vector The simplest case is the vector perpendicular to another. In anything but R2 there are a whole set of such vectors, so to express this as a non-set result a reference vector is required. Calculation of the coordinate vector for this case follows directly from the dot product. Borrowing the GA term, we subtract the projection to calculate the rejection. Rejuˆ (v) = v − v · uˆ uˆ 1 = (vu2 − v · uu) u2 1 = v e u u − v u u e u2 ∑ i i j j j j i i 1 vi vj = ∑ ujei u2 ui uj 1 ui uj = (u e − u e ) 2 ∑ i j j i v v u i<j i j Thus we can write the rejection of v from uˆ as: 1 1 ui uj ui uj Rej (v) = (1) uˆ 2 ∑ v v e e u i<j i j i j Or introducing some shorthand: uv ui uj Dij = vi vj ue ui uj Dij = ei ej equation 1 can be expressed in a form that will be slightly more convient for larger sets of vectors: 1 ( ) = uv ue Rejuˆ v 2 ∑ Dij Dij (2) u i<j Note that although the GA axiom u2 = u · u has been used in equations 1 and 2 above and the derivation, that was not necessary to prove them.
    [Show full text]
  • Geometric Algebra for Mathematicians
    Geometric Algebra for Mathematicians Kalle Rutanen 09.04.2013 (this is an incomplete draft) 1 Contents 1 Preface 4 1.1 Relation to other texts . 4 1.2 Acknowledgements . 6 2 Preliminaries 11 2.1 Basic definitions . 11 2.2 Vector spaces . 12 2.3 Bilinear spaces . 21 2.4 Finite-dimensional bilinear spaces . 27 2.5 Real bilinear spaces . 29 2.6 Tensor product . 32 2.7 Algebras . 33 2.8 Tensor algebra . 35 2.9 Magnitude . 35 2.10 Topological spaces . 37 2.11 Topological vector spaces . 39 2.12 Norm spaces . 40 2.13 Total derivative . 44 3 Symmetry groups 50 3.1 General linear group . 50 3.2 Scaling linear group . 51 3.3 Orthogonal linear group . 52 3.4 Conformal linear group . 55 3.5 Affine groups . 55 4 Algebraic structure 56 4.1 Clifford algebra . 56 4.2 Morphisms . 58 4.3 Inverse . 59 4.4 Grading . 60 4.5 Exterior product . 64 4.6 Blades . 70 4.7 Scalar product . 72 4.8 Contraction . 74 4.9 Interplay . 80 4.10 Span . 81 4.11 Dual . 87 4.12 Commutator product . 89 4.13 Orthogonality . 90 4.14 Magnitude . 93 4.15 Exponential and trigonometric functions . 94 2 5 Linear transformations 97 5.1 Determinant . 97 5.2 Pinors and unit-versors . 98 5.3 Spinors and rotors . 104 5.4 Projections . 105 5.5 Reflections . 105 5.6 Adjoint . 107 6 Geometric applications 107 6.1 Cramer’s rule . 108 6.2 Gram-Schmidt orthogonalization . 108 6.3 Alternating forms and determinant .
    [Show full text]
  • Young Research Forum on International Conclave On
    Mathematica Eterna Short Communication 2020 Short Communication on Introduction of Exterior algebra Mohammad Adil* Associate Professor, Department of Mathematics, Assiut University, Egypt In mathematics, the exterior product or wedge product of vectors only k-blades, but sums of k-blades; such a calculation is called a k- is an algebraic construction worn in geometry to Vector the k-blades, since they are uncomplicated products of learn areas, volumes, and their advanced dimensional analogues. vectors, are called the easy fundamentals of the algebra. The rank of The external result of two vectors u and v, denoted by u ∧ v, is any k-vector is distinct to be the fewest integer of simple called a bivector and lives in a space called the exterior square, fundamentals of which it is a sum. The exterior product extends to a vector space that is distinctive from the inventive gap of vectors. the complete exterior algebra, so that it makes sense to multiply any The degree of u ∧ v can be interpret as the area of the lozenge two elements of the algebra. Capable of with this product, the with sides u and v, which in three dimensions can also be exterior algebra is an associative algebra, which means that α ∧ compute by means of the cross product of the two vectors. Like (β ∧ γ) = (α ∧ β) ∧ γ for any elements α, β, γ. The k-vectors have the cross product, the exterior product is anticommutative, degree k, meaning that they are sums of products of k vectors. significance that When fundamentals of dissimilar degrees are multiply, the degrees add like multiplication of polynomials.
    [Show full text]
  • Quaternions: a History of Complex Noncommutative Rotation Groups in Theoretical Physics
    QUATERNIONS: A HISTORY OF COMPLEX NONCOMMUTATIVE ROTATION GROUPS IN THEORETICAL PHYSICS by Johannes C. Familton A thesis submitted in partial fulfillment of the requirements for the degree of Ph.D Columbia University 2015 Approved by ______________________________________________________________________ Chairperson of Supervisory Committee _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ Program Authorized to Offer Degree ___________________________________________________________________ Date _______________________________________________________________________________ COLUMBIA UNIVERSITY QUATERNIONS: A HISTORY OF COMPLEX NONCOMMUTATIVE ROTATION GROUPS IN THEORETICAL PHYSICS By Johannes C. Familton Chairperson of the Supervisory Committee: Dr. Bruce Vogeli and Dr Henry O. Pollak Department of Mathematics Education TABLE OF CONTENTS List of Figures......................................................................................................iv List of Tables .......................................................................................................vi Acknowledgements .......................................................................................... vii Chapter I: Introduction ......................................................................................... 1 A. Need for Study ........................................................................................
    [Show full text]
  • A Framework for N-Dimensional Visibility Computations
    A Framework for n-dimensional Visibility Computations L. Aveneau, S. Charneau, L Fuchs and F. Mora Abstract This chapter introduces global visibility computation using Grassmann Algebra. Visibility computation is a fundamental task in computer graphics, as in many other scientific domains. While it is well understood in two dimensions, this does not remain true in higher dimensional spaces. Grassmann Algebra allows to think about visibility at a high level of abstraction, and to design a framework for solving visibility problems in any n-dimensional space, for n ≥ 2. Contrary to Stolfi’s framework which allows only the representa- tion of real lines, it’s algebraic nature dealt naturally without any particular cases. This chapter shows how the space of lines can be defined as a projective space over the bivector vector space. Then line classification, a key point for the visibil- ity computation, is achieved using the exterior product. Actually, line classification turns out to be equivalent to point vs. hyperplane classification relatively to a non- degenerate bilinear form. This ensures well-defined property and computationally robustness. Using the previous result, the lines stabbing a n-dimensional convex face are characterized. This set of lines appears to be the intersection of the decomposable bivectors set (i.e. bivectors that represent a line) and a convex polytope. Moreover, this convex polytope is proved to be minimal. This property allows useful algorith- mic improvements. To illustrate the use of our framework in practice, we present the computation of soft shadows for 3-dimensional illuminated scene. Lilian Aveneau XLIM/SIC, CNRS, University of Poitiers, France, e-mail: [email protected] Sylvain Charneau XLIM/SIC, CNRS, University of Poitiers, France e-mail: [email protected] Laurent Fuchs XLIM/SIC, CNRS, University of Poitiers, France e-mail: [email protected] Fred´ eric´ Mora XLIM/SIC, CNRS, University of Limoges, France e-mail: [email protected] 1 2 L.
    [Show full text]
  • Package 'Clifford'
    Package ‘clifford’ March 8, 2020 Type Package Title Arbitrary Dimensional Clifford Algebras Version 1.0-2 Maintainer Robin K. S. Hankin <[email protected]> Description A suite of routines for Clifford algebras, using the 'Map' class of the Standard Template Library. Canonical reference: Hestenes (1987, ISBN 90-277-1673-0, ``Clifford algebra to geometric calculus''). Special cases including Lorentz transforms, quaternion multiplication, and Grassman algebra, are discussed. Conformal geometric algebra theory is implemented. License GPL (>= 2) LazyData yes Suggests knitr,testthat,onion,lorentz VignetteBuilder knitr Imports Rcpp (>= 0.12.5) LinkingTo Rcpp,BH SystemRequirements C++11 URL https://github.com/RobinHankin/clifford.git BugReports https://github.com/RobinHankin/clifford/issues NeedsCompilation yes Author Robin K. S. Hankin [aut, cre] (<https://orcid.org/0000-0001-5982-0415>) Repository CRAN Date/Publication 2020-03-08 20:20:02 UTC R topics documented: clifford-package . .2 allcliff . .4 antivector . .5 1 2 clifford-package as.vector . .6 clifford . .7 Conj.............................................8 drop............................................. 10 dual ............................................. 11 even............................................. 12 Extract.clifford . 12 getcoeffs . 14 grade . 15 homog . 16 lowlevel........................................... 17 magnitude . 18 neg.............................................. 19 numeric_to_clifford . 20 Ops.clifford . 21 print . 24 quaternion . 25 rcliff............................................
    [Show full text]
  • Geometric Algebra: a Computational Framework for Geometrical Applications (Part II: Applications)
    Geometric algebra: a computational framework for geometrical applications (part II: applications) Leo Dorst∗ and Stephen Manny Abstract Geometric algebra is a consistent computational framework in which to de- fine geometric primitives and their relationships. This algebraic approach contains all geometric operators and permits coordinate-free specification of computational constructions. It contains primitives of any dimensionality (rather than just vec- tors). This second paper on the subject uses the basic products to represent ro- tations (naturally incorporating quaternions), intersections, and differentiation. It shows how using well-chosen geometric algebra models, we can eliminate special cases in incidence relationships, yet still have the efficiency of the Plucker¨ coordi- nate intersection computations. Keywords: geometric algebra, Clifford algebra, rotation reprensentation, quaternions, dualization, meet, join, Plucker¨ coordinates, homogeneous coordinates, geometric dif- ferentiation, computational geometry 1 Introduction This is the second of two papers introducing geometric algebra. In the previous paper, we introduced blades, a computational algebraic representation of oriented subspaces, which are the basic element of computation in geometric algebra. We also looked at the geometric product, and two products derived from the geometric product, the inner and outer products. A crucial feature of the geometric product is that it is invertible. From that first paper, you should have gathered that every vector space with an in- ner product has a geometric algebra, whether you choose to use it or not. This paper shows how to call upon this structure for the definition of common geometrical con- structs, ensuring a totally consistent computational framework. The goal is to show ∗Informatics Institute, University of Amsterdam, Kruislaan 403, 1098 SJ Amsterdam, The Netherlands yComputer Science Department, University of Waterloo, Waterloo, Ontario, N2L 3G1, CANADA 0 c 2002 IEEE.
    [Show full text]