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$Eikonal Rendering: Efficient Light Transport in Refractive Objects$

Geometry and Vectors • Geometric interpretation of a vector – Directed line segment or tangent • Vector-algebra in Euclidean Geometry or T p(M) • Addition / subtraction of vectors a+b • Multiplication / division by scalars a • Multiplication / Division of vectors?? Multiplication of vectors 10
Complete Vector-algebra? • Invertible product of vectors? • What means vector-division “a/b” ? • ab=C � b=a -1 C • Note: C not necessarily a vector! • Inner product (not associative): a b � Skalar – Not invertible e.g. a b =0 with a≠0, b≠0 but orthogonal • Outer product (associative): a b � Bivektor – Generalized cross-product from 3D: a b – Not invertible e.g. a b =0 with a≠0, b≠0 but parallel Multiplikation von Vectoren 11
Page 1 and 2: Geometric Algebra (GA) Werner Benge
Page 3 and 4: Motivation of GA • Unification of
Page 5 and 6: Historical Roots • Complex Plane
Page 7 and 8: Historical Roots, III • Construct
Page 12 and 13: Bivector a b Describes the plane sp
Page 14 and 15: Bivectors in R 3 • 3 Basis-elemen
Page 16 and 17: Coordinate representation of “
Page 18 and 19: Comparing the products • Inner pr
Page 20 and 21: Requirements to GP • For elements
Page 22 and 23: Geometric Product • William Kingd
Page 24 and 25: Structure of Multi-vectors Linear c
Page 26 and 27: Reflection at a Vector • Unit vec
Page 28 and 29: Geometrical Quadrate Consider (AB)
Page 30 and 31: Rotation and GA Right-multiplicatio
Page 32 and 33: Rotor in 2D • Rotor R := e i = co
Page 34 and 35: Rotation in nD Consider: R v R -1 m
Page 36 and 37: Applications Crystallography Differ
Page 38 and 39: Differential Geometry Derivative op
Page 40 and 41: Cl 3(R) & Spinors • GA in 3D can
Page 42 and 43: x Basis-Bivectors in STA k: 3 timel
Page 44 and 45: Spacetime-Rotor • Spacetime-rotor
Page 46 and 47: Dirac-Equation • Relativistic Mom
Page 48 and 49: Objects in conform 5D GA Punkt x +
Page 50: Literatur http://modelingnts.la.asu

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