1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
|
==========
Glossary
==========
.. glossary::
Affine matrix
A matrix implementing an :term:`affine transformation` in
:term:`homogenous coordinates`. For a 3 dimensional transform, the
matrix is shape 4 by 4.
Affine transformation
See `wikipedia affine`_ definition. An affine transformation is a
:term:`linear transformation` followed by a translation.
Axis angle
A representation of rotation. See: `wikipedia axis angle`_ .
From Euler's rotation theorem we know that any rotation or
sequence of rotations can be represented by a single rotation
about an axis. The axis $\boldsymbol{\hat{u}}$ is a :term:`unit
vector`. The angle is $\theta$. The :term:`rotation vector` is a
more compact representation of $\theta$ and
$\boldsymbol{\hat{u}}$.
Euclidean norm
Also called Euclidean length, or L2 norm. The Euclidean norm
$\|\mathbf{x}\|$ of a vector $\mathbf{x}$ is given by:
.. math::
\|\mathbf{x}\| := \sqrt{x_1^2 + \cdots + x_n^2}
Pure Pythagoras.
Euler angles
See: `wikipedia Euler angles`_ and `Mathworld Euler angles`_.
Gimbal lock
See :ref:`gimbal-lock`
Homogenous coordinates
See `wikipedia homogenous coordinates`_
Linear transformation
A linear transformation is one that preserves lines - that is, if
any three points are on a line before transformation, they are
also on a line after transformation. See `wikipedia linear
transform`_. Rotation, scaling and shear are linear
transformations.
Quaternion
See: `wikipedia quaternion`_. An extension of the complex numbers
that can represent a rotation. Quaternions have 4 values, $w, x,
y, z$. $w$ is the *real* part of the quaternion and the vector
$x, y, z$ is the *vector* part of the quaternion. Quaternions are
less intuitive to visualize than :term:`Euler angles` but do not
suffer from :term:`gimbal lock` and are often used for rapid
interpolation of rotations.
Reflection
A transformation that can be thought of as transforming an object
to its mirror image. The mirror in the transformation is a plane.
A plan can be defined with a point and a vector normal to the
plane. See `wikipedia reflection`_.
Rotation matrix
See `wikipedia rotation matrix`_. A rotation matrix is a matrix
implementing a rotation. Rotation matrices are square and
orthogonal. That means, that the rotation matrix $R$ has columns
and rows that are :term:`unit vector`, and where $R^T R = I$ ($R^T$ is
the transpose and $I$ is the identity matrix). Therefore $R^T =
R^{-1}$ ($R^{-1}$ is the inverse). Rotation matrices also have a
determinant of $1$.
Rotation vector
A representation of an :term:`axis angle` rotation. The angle
$\theta$ and unit vector axis $\boldsymbol{\hat{u}}$ are stored in a
*rotation vector* $\boldsymbol{u}$, such that:
.. math::
\theta = \|\boldsymbol{u}\| \,
\boldsymbol{\hat{u}} = \frac{\boldsymbol{u}}{\|\boldsymbol{u}\|}
where $\|\boldsymbol{u}\|$ is the :term:`Euclidean norm` of
$\boldsymbol{u}$
Shear matrix
Square matrix that results in shearing transforms - see
`wikipedia shear matrix`_.
Unit vector
A vector $\boldsymbol{\hat{u}}$ with a :term:`Euclidean norm`
of 1. Normalized vector is a synonym. The "hat" over the
$\boldsymbol{\hat{u}}$ is a convention to express the fact that it
is a unit vector.
.. include:: links_names.inc
|
