2. Generalized Homogeneous Coordinates for Computational Geometryy (original) (raw)
Conformal Geometry, Euclidean Space and Geometric Algebra
Uncertainty in Geometric Computations, 2002
Projective geometry provides the preferred framework for most implementations of Euclidean space in graphics applications. Translations and rotations are both linear transformations in projective geometry, which helps when it comes to programming complicated geometrical operations. But there is a fundamental weakness in this approach-the Euclidean distance between points is not handled in a straightforward manner. Here we discuss a solution to this problem, based on conformal geometry. The language of geometric algebra is best suited to exploiting this geometry, as it handles the interior and exterior products in a single, unified framework. A number of applications are discussed, including a compact formula for reflecting a line off a general spherical surface.
Aspects of Geometric Algebra in Euclidean, Projective and Conformal Space
2003
This report is meant to be a script of a tutorial on Clifford (or Geometric) algebra. It is therefore not complete in the description of the algebra and neither completely rigorous. The reader is also not likely to be able to perform arbitrary calculations with Clifford algebra after reading this script. The goal of this text is to give the reader a feeling for what Clifford algebra is about and how it may be used. It is attempted to convey the basic ideas behind the use of Clifford algebra in the description of geometry in Euclidean, projective and conformal space
2004
CLUCalc is a user friendly frontend to these libraries. It is used in the "Interactive Introduction. .. " and is available for download from [27]. In CLUCalc you can type your equations in a simple script language, called CLUScript and visualize the results immediately with OpenGL graphics. The program comes with a manual in HTML form and a number of example scripts. There is also an online version of the manual under: http://www.perwass.de/CLU/CLUCalcDoc/ CLUCalc should serve as a good accompaniment to this script, helping you to understand the concepts behind Geometric algebra visually. The CLUScripts used in chapter three can also be downloaded through the following link: www.dgm.informatik.tu-darmstadt.de/staff/dietmar/ By the way, CLUCalc was also used to create all of the 2d and 3d graphics in this script. You can use it for the same purpose, illustrating your publications or web-pages, from the version 3.0 onwards, which is now available. Some other features of CLUCalc v3.0.0 are: • render and display LaTeX text and formulas to annotate your graphics, or to create slides for presentations, • prepare presentations with user interactive 3D-graphics included in your slides, • draw 2D-surfaces, including the surface generated by a set of circles, • do structured programming with if-clauses and loops, • do error propagation in Clifford algebra, • and much more...
The claim that Clifford algebra should be regarded as a universal geometric algebra is strengthened by showing that the algebra is applicable to nonmetrical as well as metrical geometry. Clifford algebra is used to develop a coordinate-free algebraic formulation of projective geometry. Major theorems of projective geometry are reduced to algebraic identities which apply as well to metrical geometry. Improvements in the formulation of linear algebra are suggested to simplify its intimate relation to projective geometry. Relations among Clifford algebras of different dimensions are interpreted geometrically as "projective and conformal splits." The conformal split is employed to simplify and elucidate the pin and spin representations of the conformal group for arbitrary dimension and signature.
VERSOR: Spatial Computing with Conformal Geometric Algebra
This Master's thesis investigates new computer graphics synthesis techniques made possible by the Euclidean, spherical, and hyperbolic transformational capacities of conformal geometric algebra [CGA]. An explication of the mathematical system's geometric elements is followed by documentation of Versor -an integrated CGA software library for immersive 3D visualizations and dynamic simulations.
Treatise of Plane Geometry Through Geometric Algebra
2004
Inverse and quotient of two vectors, 7.-Priority of algebraic operations, 8.-Geometric algebra of the vector plane, 9.-Exercises, 9. 2. A vector basis for the Euclidean plane Linear combination of two vectors, 10.-Basis and components, 10.-Orthonormal bases, 11.-Applications of formulae for products, 11.-Exercises, 12. 3. Complex numbers The subalgebra of complex numbers, 13.-Binomial, polar and trigonometric form of a complex number, 13.-Algebraic operations with complex numbers, 14.-Permutation of complex numbers and vectors, 17.-The complex plane, 18.-Complex analytic functions, 19.-Fundamental theorem of algebra, 24.-Exercises, 26. 4. Transformations of vectors Rotations, 27.-Axial symmetries, 28.-Inversions, 29.-Dilations, 30.-Exercises, 30 Second Part: Geometry of the Euclidean plane 5. Points and straight lines Translations, 31.-Coordinate systems, 31.-Barycentric coordinates, 33.-Distance between two points and area, 33.-Condition of collinearity of three points, 35.-Cartesian coordinates, 36.-Vectorial and parametric equations of a line, 36.-Algebraic equation and distance from a point to a line, 37.-Slope and intercept equations of a line, 40.-Polar equation of a line, 41.-Intersection of two lines and pencil of lines, 41.-Dual coordinates, 43.-Desargues's theorem, 48.-Exercises, 50. 6. Angles and elemental trigonometry Sum of the angles of a polygon, 53.-Definition of trigonometric functions and fundamental identities, 54.-Angle inscribed in a circle and double-angle identities, 55.-Addition of vectors and sum of trigonometric functions, 56.-Product of vectors and addition identities, 57.-Rotations and de Moivre's identity, 58.-Inverse trigonometric functions, 59.-Exercises, 60. 7. Similarities and simple ratio Direct similarity (similitude), 61.-Opposite similarity, 62.-Menelaus' theorem, 63.-Ceva's theorem, 64.-Homothety and simple ratio, 65.-Exercises, 67. 8. Properties of triangles Area of a triangle, 68.-Medians and centroid, 69.-Perpendicular bisectors and circumcentre, 70.-Angle bisectors and incentre, 72.-Altitudes and orthocentre, 73.-Euler's line, 76.-Fermat's theorem, 77.-Exercises, 78. XIII 9. Circles Algebraic and Cartesian equations, 80.-Intersections of a line with a circle, 80.-Power of a point with respect to a circle, 82.-Polar equation, 82.-Inversion with respect to a circle, 83.-The nine-point circle, 85.-Cyclic and circumscribed quadrilaterals, 87.-Angle between circles, 89.-Radical axis of two circles, 89.-Exercises, 91. 10. Cross ratios and related transformations Complex cross ratio, 92.-Harmonic characteristic and ranges, 94.-Homography (Möbius transformation), 96.-Projective cross ratio, 99.-Points at infinity and homogeneous coordinates, 102.-Perspectivity and projectivity, 103.-Projectivity as a tool for theorem demonstrations, 108.-Homology, 110.-Exercises, 115. 11. Conics Conic sections, 117.-Two foci and two directrices, 120.-Vectorial equation, 121.-Chasles' theorem, 122.-Tangent and perpendicular to a conic, 124.-Central equations for ellipse and hyperbola, 126.-Diameters and Apollonius' theorem, 128.-Conic passing through five points, 131.-Pencil of conics passing through four points, 133.-Conic equation in barycentric coordinates and dual conic, 133.-Polarities, 135.-Reduction of the conic matrix to diagonal form, 136.-Exercises, 137. Third part: Pseudo-Euclidean geometry 12. Matrix representation and hyperbolic numbers Rotations and the representation of complex numbers, 139.-The subalgebra of hyperbolic numbers, 140.-Hyperbolic trigonometry, 141.-Hyperbolic exponential and logarithm, 143.-Polar form, powers and roots of hyperbolic numbers, 144.-Hyperbolic analytic functions, 147.-Analyticity and square of convergence of power series, 150.-About the isomorphism of Clifford algebras, 152.-Exercises, 153. 13. The hyperbolic or pseudo-Euclidean plane Hyperbolic vectors, 154.-Inner and outer products of hyperbolic vectors, 155.-Angles between hyperbolic vectors, 156.-Congruence of segments and angles, 158.-Isometries, 158.-Theorems about angles, 160.-Distance between points, 160.-Area in the hyperbolic plane, 161.-Diameters of the hyperbola and Apollonius' theorem, 163.-The law of sines and cosines, 164.-Hyperbolic similarity, 167.-Power of a point with respect to a hyperbola with constant radius, 168.-Exercises, 169. Fourth part: Plane projections of three-dimensional spaces 14. Spherical geometry in the Euclidean space The geometric algebra of the Euclidean space, 170.-Spherical trigonometry, 172.-The dual spherical triangle of a given triangle, 175.-Right spherical triangles and Napier's rule, 176.-Area of a spherical triangle, 176.-Properties of the projections of the spherical surface, 177.-Central or gnomonic projection, 177.-Stereographic projection, 180.-Orthographic projection, 181.-Lambert's azimuthal equivalent projection, 182.-Spherical coordinates and cylindrical equidistant (plate carré) projection, 183.-Mercator XIV projection, 184.-Cylindrical equivalent projection, 184.-Conic projections, 185.-Exercises, 186. 15. Hyperboloidal geometry in the pseudo-Euclidean space The geometric algebra of the pseudo-Euclidean space, 189.-The hyperboloid of two sheets (Lobachevskian surface), 191.-Central projection (Beltrami disk), 192.-Lobachevskian trigonometry, 197.-Stereographic projection (Poincaré disk), 199.-Azimuthal equivalent projection, 201.-Weierstrass coordinates and cylindrical equidistant projection, 202.-Cylindrical conformal projection, 203.-Cylindrical equivalent projection, 204.-Conic projections, 204.-About the congruence of geodesic triangles, 206.-The hyperboloid of one sheet, 206.-Central projection and arc length on the one-sheeted hyperboloid, 207.-Cylindrical projections, 208.-Cylindrical central projection, 209.-Cylindrical equidistant projection, 210.-Cylindrical equivalent projection, 210.-Cylindrical conformal projection, 210.-Area of a triangle on the onesheeted hyperboloid, 211.-Trigonometry of right triangles, 214.-Hyperboloidal trigonometry, 215.-Dual triangles, 218.-Summary, 221.-Comment about the names of the non-Euclidean geometry, 222.-Exercises, 222. 16. Solutions to the proposed exercises 1. Euclidean vectors and their operations, 224.-2. A vector basis for the Euclidean plane, 225.-3. Complex numbers, 227.-4. Transformations of vectors, 230.-5. Points and straight lines, 231.-6. Angles and elemental trigonometry, 241.-7. Similarities and simple ratio, 244.-8. Properties of triangles, 246.-9. Circles, 257.-10. Cross ratios and related transformations, 262.-11. Conics, 266.-12. Matrix representation and hyperbolic numbers, 273.-13. The hyperbolic or pseudo-Euclidean plane, 275.-14. Spherical geometry in the Euclidean space, 278.-15. Hyperboloidal geometry in the pseudo-Euclidean space, 284. Bibliography, 293. Internet bibliography, 296. Index, 298. Chronology of the geometric algebra, 305.
Geometric algebra and its application to computer graphics
2004
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