Geometric Methods and Applications: For Computer Science and EngineeringSpringer Science & Business Media, 2001 - 565 Seiten 1 Introduction.- 1.1 Geometries: Their Origin, Their Uses.- 1.2 Prerequisit es and Notation.- 2 Basics of Affine Geometry.- 2.1 Affine Spaces.- 2.2 Examples of Affine Spaces.- 2.3 Chasles's Identity.- 2.4 Affine Combinations, Barycenters.- 2.5 Affine Subspaces.- 2.6 Affine Independence and Affine Frames.- 2.7 Affine Maps.- 2.8 Affine Groups.- 2.9 Affine Geometry: A Glimpse.- 2.10 Affine Hyperplanes.- 2.11 Intersection of Affine Spaces.- 2.12 Problems.- 3 Properties of Convex Sets: A Glimpse.- 3.1 Convex Sets.- 3.2 Carathéodory's Theorem.- 3.3 Radon's and Helly's Theorems.- 3.4 Problems.- 4 Embedding an Affine Space in a Vector Space.- 4.1 The "Hat Construction," or Homogenizing.- 4.2 Affine Frames of E and Bases of Ê.- 4.3 Another Construction of Ê.- 4.4 Extending Affine Maps to Linear Maps.- 4.5 Problems.- 5 Basics of Projective Geometry.- 5.1 Why Projective Spaces?.- 5.2 Projective Spaces.- 5.3 Projective Subspaces.- 5.4 Projective Frames.- 5.5 Projective Maps.- 5.6 Projective Completion of an Affine Space, Affine Patches.- 5.7 Making Good Use of Hyperplanes at Infinity.- 5.8 The Cross-Ratio.- 5.9 Duality in Projective Geometry.- 5.10 Cross-Ratios of Hyperplanes.- 5.11 Complexification of a Real Projective Space.- 5.12 Similarity Structures on a Projective Space.- 5.13 Some Applications of Projective Geometry.- 5.14 Problems.- 6 Basics of Euclidean Geometry.- 6.1 Inner Products, Euclidean Spaces.- 6.2 Orthogonality, Duality, Adjoint of a Linear Map.- 6.3 Linear Isometries (Orthogonal Transformations).- 6.4 The Orthogonal Group, Orthogonal Matrices.- 6.5 QR-Decomposition for Invertible Matrices.- 6.6 Some Applications of Euclidean Geometry.- 6.7 Problems.- 7 The Cartan-Dieudonné Theorem.- 7.1 Orthogonal Reflections.- 7.2 The Cartan-Dieudonné Theorem for Linear Isometries.- 7.3 QR-Decomposition Using Householder Matrices.- 7.4 Affine Isometries (Rigid Motions).- 7.5 Fixed Points of Affine Maps.- 7.6 Affine Isometries and Fixed Points.- 7.7 The Cartan-Dieudonné Theorem for Affine Isometries.- 7.8 Orientations of a Euclidean Space, Angles.- 7.9 Volume Forms, Cross Products.- 7.10 Problems.- 8 The Quaternions and the Spaces S3, SU(2), SO(3), and ?P3.- 8.1 The Algebra ? of Quaternions.- 8.2 Quaternions and Rotations in SO(3).- 8.3 Quaternions and Rotations in SO(4).- 8.4 Applications of Euclidean Geometry to Motion Interpolation.- 8.5 Problems.- 9 Dirichlet-Voronoi Diagrams and Delaunay Triangulations.- 9.1 Dirichlet-Voronoi Diagrams.- 9.2 Simplicial Complexes and Triangulations.- 9.3 Delaunay Triangulations.- 9.4 Delaunay Triangulations and Convex Hulls.- 9.5 Applications of Voronoi Diagrams and Delaunay Triangulations.- 9.6 Problems.- 10 Basics of Hermitian Geometry.- 10.1 Sesquilinear and Hermitian Forms, Pre-Hilbert Spaces and Hermitian Spaces.- 10.2 Orthogonality, Duality, Adjoint of a Linear Map.- 10.3 Linear Isometries (Also Called Unitary Transformations).- 10.4 The Unitary Group, Unitary Matrices.- 10.5 Problems.- 11 Spectral Theorems in Euclidean and Hermitian Spaces.- 11.1 Introduction: What's with Lie Groups and Lie Algebras?.- 11.2 Normal Linear Maps.- 11.3 Self-Adjoint, Skew Self-Adjoint, and Orthogonal Linear Maps.- 11.4 Normal, Symmetric, Skew Symmetric, Orthogonal, Hermitian, Skew Hermitian, and Unitary Matrices.- 11.5 Problems.- 12 Singular Value Decomposition (SVD) and Polar Form.- 12.1 Polar Form.- 12.2 Singular Value Decomposition (SVD).- 12.3 Problems.- 13 Applications of Euclidean Geometry to Various Optimization Problems.- 13.1 Applications of the SVD and QR-Decomposition to Least Squares Problems.- 13.2 Minimization of Quadratic Functions Using Lagrange Multipliers.- 13.3 Problems.- 14 Basics of Classical Lie Groups: The Exponential Map, Lie Groups, and Lie Algebras.- 14.1 The Exponential Map.- 14.2 The Lie Groups GL(n, ?), SL(n, ?), O(n), SO(n), the Lie Algebras gl(n, ?), sl(n, ?), o(n), so(n), and the Exponential Map.- 14.3 Symmetric Matrices, Symmetric Positive Definite Matrices, and the Expo |
Inhalt
II | 5 |
IV | 9 |
V | 10 |
VIII | 18 |
IX | 20 |
X | 24 |
XI | 30 |
XII | 35 |
LXXIV | 285 |
LXXV | 287 |
LXXVI | 289 |
LXXVII | 291 |
LXXVIII | 298 |
LXXIX | 302 |
LXXX | 303 |
LXXXI | 306 |
XIII | 42 |
XIV | 44 |
XV | 48 |
XVI | 49 |
XVII | 51 |
XVIII | 66 |
XXI | 68 |
XXII | 72 |
XXIII | 74 |
XXV | 82 |
XXVI | 84 |
XXVII | 87 |
XXVIII | 90 |
XXIX | 91 |
XXXI | 95 |
XXXII | 99 |
XXXIII | 102 |
XXXIV | 109 |
XXXV | 115 |
XXXVI | 122 |
XXXVII | 125 |
XXXVIII | 130 |
XXXIX | 133 |
XL | 134 |
XLI | 136 |
XLII | 141 |
XLIII | 145 |
XLIV | 166 |
XLVI | 172 |
XLVII | 184 |
XLVIII | 189 |
XLIX | 192 |
L | 193 |
LI | 201 |
LIV | 206 |
LV | 217 |
LVI | 221 |
LVII | 223 |
LVIII | 225 |
LIX | 231 |
LX | 235 |
LXI | 239 |
LXII | 243 |
LXIII | 252 |
LXVI | 257 |
LXVII | 264 |
LXVIII | 268 |
LXIX | 269 |
LXX | 271 |
LXXII | 279 |
LXXIII | 283 |
LXXXII | 313 |
LXXXV | 314 |
LXXXVI | 322 |
LXXXVII | 328 |
LXXXVIII | 332 |
LXXXIX | 337 |
XCII | 343 |
XCIII | 353 |
XCIV | 356 |
XCVI | 360 |
XCVII | 368 |
XCVIII | 371 |
CI | 380 |
CII | 384 |
CIII | 387 |
CIV | 389 |
CV | 391 |
CVI | 395 |
CVII | 409 |
CVIII | 410 |
CIX | 419 |
CXI | 424 |
CXII | 428 |
CXIII | 430 |
CXIV | 443 |
CXV | 444 |
CXVI | 446 |
CXVII | 448 |
CXVIII | 453 |
CXIX | 454 |
CXX | 456 |
CXXI | 458 |
CXXII | 469 |
CXXV | 471 |
CXXVI | 477 |
CXXVII | 482 |
CXXVIII | 487 |
CXXIX | 491 |
CXXX | 498 |
CXXXI | 504 |
CXXXII | 507 |
CXXXIV | 510 |
CXXXV | 515 |
CXXXVI | 522 |
CXXXVII | 523 |
CXXXVIII | 534 |
535 | |
CXLI | 539 |
549 | |
555 | |
Andere Ausgaben - Alle anzeigen
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Häufige Begriffe und Wortgruppen
affine frame affine isometries affine map affine space affine subspace angle arc length assume barycenter basis e1 bijection called center of curvature circle class CP collinear compute convex hull cross-ratio defined Delaunay triangulation denoted distinct eigenvalues eigenvectors equation equivalent Euclidean space example exponential map finite dimension fixed point function fundamental form geodesic Given GL(n h₁ h₂ Hermitian space hyperplane H infinity inner product intersection invertible isometry isomorphic Ker f Lie algebras Lie groups linear form linear map M₁ manifold map f map ƒ nonnull normal null orthogonal orthogonal matrix orthonormal basis polynomial positive definite Problem projective frame projective geometry projective space proof properties Prove quaternion rigid motion rotation RP¹ scalar self-adjoint skew symmetric sl(n SO(n subgroup subset theorem topology u₁ uk+1 unique unit vector vector space Voronoi diagram ΕΙ
Beliebte Passagen
Seite 2 - That if a straight line meet two other straight lines, so as to make the two interior angles on the same side of it, taken together, less than two right angles...
Seite 535 - AH Barr, B. Currin, S. Gabriel, and JF Hughes, "Smooth Interpolation of Orientations with Angular Velocity Constraints using Quaternions," SIGGRAPH
Seite xiii - Ars Combinatoria' (genetics) ? It was a particularly cruel fate that imposed upon him 'the yoke of a foreign tongue that was not sung at my cradle'.
Verweise auf dieses Buch
Handbook of Computer Aided Geometric Design G. Farin,J. Hoschek,M.-S. Kim Eingeschränkte Leseprobe - 2002 |