# Glossary¶

- Affine matrix
- A matrix implementing an affine transformation in 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 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 unit vector. The angle is \(\theta\). The 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:

\[\|\mathbf{x}\| := \sqrt{x_1^2 + \cdots + x_n^2}\]Pure Pythagoras.

- Euler angles
- See: wikipedia Euler angles and Mathworld Euler angles.
- Gimbal lock
- See 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 Euler angles but do not suffer from 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 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 axis angle rotation. The angle \(\theta\) and unit vector axis \(\boldsymbol{\hat{u}}\) are stored in a

*rotation vector*\(\boldsymbol{u}\), such that:\[ \begin{align}\begin{aligned}\theta = \|\boldsymbol{u}\| \,\\\boldsymbol{\hat{u}} = \frac{\boldsymbol{u}}{\|\boldsymbol{u}\|}\end{aligned}\end{align} \]where \(\|\boldsymbol{u}\|\) is the 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 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.