Affine Transformation
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Aﬃne transformation From Wikipedia, the free encyclopedia Contents 1 2 × 2 real matrices 1 1.1 Proﬁle ................................................. 1 1.2 Equi-areal mapping .......................................... 2 1.3 Functions of 2 × 2 real matrices .................................... 2 1.4 2 × 2 real matrices as complex numbers ............................... 3 1.5 References ............................................... 4 2 3D projection 5 2.1 Orthographic projection ........................................ 5 2.2 Weak perspective projection ..................................... 5 2.3 Perspective projection ......................................... 6 2.4 Diagram ................................................ 8 2.5 See also ................................................ 8 2.6 References ............................................... 9 2.7 External links ............................................. 9 2.8 Further reading ............................................ 9 3 Aﬃne coordinate system 10 3.1 See also ................................................ 10 4 Aﬃne geometry 11 4.1 History ................................................. 12 4.2 Systems of axioms ........................................... 12 4.2.1 Pappus’ law .......................................... 12 4.2.2 Ordered structure ....................................... 13 4.2.3 Ternary rings ......................................... 13 4.3 Aﬃne transformations ......................................... 14 4.4 Aﬃne space .............................................. 14 4.5 Projective view ............................................ 15 4.6 See also ................................................ 15 4.7 References ............................................... 15 4.8 Further reading ............................................ 16 4.9 External links ............................................. 16 i ii CONTENTS 5 Aﬃne group 17 5.1 Relation to general linear group .................................... 17 5.1.1 Construction from general linear group ............................ 17 5.1.2 Stabilizer of a point ...................................... 17 5.2 Matrix representation ......................................... 18 5.3 Planar aﬃne group ........................................... 18 5.4 Other aﬃne groups .......................................... 18 5.4.1 General case ......................................... 18 5.4.2 Special aﬃne group ...................................... 19 5.4.3 Projective subgroup ...................................... 19 5.4.4 Poincaré group ........................................ 19 5.5 See also ................................................ 19 5.6 References ............................................... 19 6 Aﬃne space 20 6.1 Informal descriptions ......................................... 21 6.2 Deﬁnition ............................................... 21 6.2.1 Subtraction and Weyl’s axioms ................................ 22 6.2.2 Aﬃne combinations ...................................... 22 6.3 Examples ............................................... 22 6.4 Aﬃne subspaces ........................................... 23 6.5 Aﬃne combinations and aﬃne dependence .............................. 23 6.6 Geometric objects as points and vectors ................................ 24 6.7 Axioms ................................................ 24 6.8 Relation to projective spaces ..................................... 24 6.9 See also ................................................ 25 6.10 Notes ................................................. 25 6.11 References .............................................. 25 7 Aﬃne transformation 27 7.1 Mathematical deﬁnition ........................................ 27 7.1.1 Alternative deﬁnition ..................................... 29 7.2 Representation ............................................. 29 7.2.1 Augmented matrix ...................................... 29 7.3 Properties ............................................... 31 7.4 Aﬃne transformation of the plane ................................... 31 7.5 Examples of aﬃne transformations .................................. 32 7.5.1 Aﬃne transformations over the real numbers ......................... 32 7.5.2 Aﬃne transformation over a ﬁnite ﬁeld ............................ 32 7.5.3 Aﬃne transformation in plane geometry ........................... 33 7.6 See also ................................................ 33 7.7 Notes ................................................. 33 CONTENTS iii 7.8 References .............................................. 34 7.9 External links ............................................. 35 8 Augmented matrix 37 8.1 Examples ............................................... 37 8.1.1 Matrix inverse ......................................... 37 8.1.2 Existence and number of solutions .............................. 38 8.1.3 Solution of a linear system .................................. 38 8.2 References ............................................... 39 9 Barycenter 40 9.1 Two-body problem .......................................... 41 9.1.1 Primary–secondary examples ................................ 41 9.1.2 Inside or outside the Sun? ................................... 41 9.2 Gallery ................................................ 43 9.3 Relativistic corrections ........................................ 43 9.4 Selected barycentric orbital elements ................................. 43 9.5 See also ................................................ 43 9.6 References .............................................. 44 10 Bent function 45 10.1 Walsh transform ........................................... 45 10.2 Deﬁnition and properties ....................................... 46 10.3 Applications .............................................. 47 10.4 Generalizations ............................................ 47 10.5 References .............................................. 48 10.6 Further reading ............................................ 48 11 Cartesian coordinate system 50 11.1 History ................................................. 52 11.2 Description .............................................. 52 11.2.1 One dimension ........................................ 52 11.2.2 Two dimensions ....................................... 52 11.2.3 Three dimensions ....................................... 53 11.2.4 Higher dimensions ...................................... 54 11.2.5 Generalizations ........................................ 54 11.3 Notations and conventions ....................................... 55 11.3.1 Quadrants and octants ..................................... 55 11.4 Cartesian formulae for the plane .................................... 55 11.4.1 Distance between two points ................................. 55 11.4.2 Euclidean transformations ................................... 56 11.5 Orientation and handedness ...................................... 59 11.5.1 In two dimensions ....................................... 59 iv CONTENTS 11.5.2 In three dimensions ...................................... 60 11.6 Representing a vector in the standard basis .............................. 62 11.7 Applications .............................................. 62 11.8 See also ................................................ 63 11.9 Notes ................................................. 63 11.10References ............................................... 63 11.11Sources ................................................ 63 11.12Further reading ............................................ 64 11.13External links ............................................. 64 12 Collinearity 65 12.1 Points on a line ............................................ 65 12.2 Examples in Euclidean geometry ................................... 65 12.2.1 Triangles ........................................... 65 12.2.2 Quadrilaterals ......................................... 66 12.2.3 Hexagons ........................................... 66 12.2.4 Conic sections ......................................... 66 12.2.5 Cones ............................................. 67 12.2.6 Tetrahedrons ......................................... 67 12.3 Algebra ................................................ 67 12.3.1 Collinearity of points whose coordinates are given ...................... 67 12.3.2 Collinearity of points whose pairwise distances are given ................... 67 12.4 Number theory ............................................ 68 12.5 Concurrency (plane dual) ....................................... 68 12.6 Collinearity graph ........................................... 68 12.7 Usage in statistics and econometrics .................................. 68 12.8 Usage in other areas .......................................... 68 12.8.1 Antenna arrays ........................................ 69 12.8.2 Photography .......................................... 70 12.9 See also ................................................ 70 12.10Notes ................................................. 70 12.11References ............................................... 70 13 Endianness 71 13.1 Illustration ............................................... 71 13.2 History ................................................. 71 13.2.1 Etymology .......................................... 72 13.3 Hardware ............................................... 72 13.3.1 Bi-endian hardware ...................................... 74 13.3.2 Floating-point ......................................... 74 13.4 Optimization