Читать книгу Mathematical Basics of Motion and Deformation in Computer Graphics - Ken Anjyo - Страница 10
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ОглавлениеSymbols and Notations
| Tb | translation, 5 |
| Rθ | 2D rotation matrix, 6 |
| SO(2) | 2D rotation group (special orthogonal group), 7 |
| M(n, R) | the set of square matrices of size n with real entries, 7 |
| I, In | the identity matrix of size n, 7, 11, 30, |
| AT | transpose of a matrix A, 7 |
| SE(2) | 2D motion group (the set of non-flip rigid transformations), 8 |
| O(2) | 2D orthogonal group, 9 |
| SO(n) | special orthogonal group, 10 |
| O(n) | orthogonal group, 10 |
| SO(3) | 3D rotation group, 11 |
| Rx (θ) | 3D axis rotations, 11 |
| H | the set of quaternions, 3, 13 |
| q̄ | conjugate of a quaternion, 13 |
| Re(q) | real part of a quaternion, 14 |
| Im(q) | imaginary part of a quaternion, 14 |
| Im H | the set of imaginary quaternions, 14 |
| |q| | the absolute value of a quaternion, 14 |
| S3 | the set of unit quaternions, 15 |
| exp | exponential map, 16, 29 |
| slerp(q0, q1, t) | spherical linear interpolation, 16 |
| ε | dual number, 16 |
| M(2, H) | the set of square matrices of size 2 with entries in H, 16 |
| the set of anti-commutative dual complex numbers (DCN), 18 | |
| E(n) | rigid transformation group, 19 |
| SE(n) | n-dimensional motion group, 20 |
| GL(n) | general linear group, 23 |
| Aff(n) | affine transformations group, 23 |
| GL+(n) | general linear group with positive determinants, 23 |
| Aff+(n) | the set of orientation-preserving affine transformations, 23 |
| ⋉ | semi-direct product, 25 |
| Sym+(n) | the set of positive definite symmetric matrices, 26 |
| Diag+(n) | the set of diagonal matrices with positive diagonal entries, 27 |
| SVD | Singular Value Decomposition, 28 |
| exp(A) | exponential of a square matrix, 29 |
| C× | the set of non-zero complex numbers, 31 |
| gl(n) | Lie algebra of GL(n), 33 |
| so(n) | Lie algebra of SO(n), 33 |
| sl(n) | Lie algebra of SL(n), 33 |
| aff(n) | Lie algebra of Aff(n), 34 |
| se(n) | Lie algebra of SE(n), 34 |
| [A, B] | Lie bracket, 34 |
| Jx, Jy, Jz | basis of so(3), 37 |
| log | logarithmic map (logarithm), 31, 42 |
| AL, AP, AE | interpolant, 42 |
| EP, EF, ES, ER | error functions, 49 |
| ||·||F | Frobenius norm of a matrix, 49 |
| se(3) | Lie algebra of SE(3), 53 |
| sym(3) | the set of symmetric matrices of size three, 53 |
| ϕ, ψ | map between Aff+(3) and a vector space, 53 |
| ι | embedding M(3, R) → M(4, R), 54 |
| R̂, X̂ | element of SE(3) and se(3), 54 |
| ∇ | gradient, 58 |
| ∆ | Laplacian, 58 |
| div | divergence, 58 |
| ∂Ω | boundary, 58 |
