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$Eikonal Rendering: Efficient Light Transport in Refractive Objects$

Geometric Product • William Kingdon Clifford (1845-79): • Combine inner and outer product to defined the geometric product AB (1878): AB := A B A B • Result is not a vector, but the sum of a scalar + bivector! • Operates on “multivectors” • Subset of the tensoralgebra • Geometric Product is invertible! Das Geometrische Produkt 22
Multi-vector components • R 2 : A = A 0 + A 1 e 0 + A 2 e 1 + A 3 e 0 e 1 • R 3 : A = + A1 e0 + A 0 + A 2 e 1 + A 3 e 2 A 4 e 0 e 1+A 5 e 1 e 2+A 6 e 0 e 2 + A 7 e 0 e 1 e 2 Struktur von Multivektoren 2.7819… + + + + + + + 23
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 10 and 11: Geometry and Vectors • Geometric
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 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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