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Affine Transformation 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
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