Points and vectors 2d representation

Points 2d can be represented alternatively using carthesian or polar coordinates. These two representation are linked by the following formula :

$P = \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} \rho\cos(\theta) \\ \rho\sin(\theta) \end{pmatrix}$

where :

$\rho=||P|| \in [0, \infty[$ is the L2 norm (euclidian norm) of
$\theta \in [0, 2\Pi[$ is the azimuth

Lines 2d representation

Lines 2d can be represented using polar coordinates.

$\cos{\theta} \times x + \sin{\theta} \times y = \rho$

where :

$\theta$ is the angle (in radian) between line normal and axis of abscissas
$\rho$ is the distance from line to origin

Rotations 2d

Rotations 2d are expressed with respect to direct referential orientation.
A 2d rotation with angle $\theta$ can be represented as follow :

Given a point $P_{in}=\begin{pmatrix}x_{in} & y_{in} \end{pmatrix}^t$ , its transformed is obtain by :

$\begin{pmatrix} x_{out} \\ y_{out} \end{pmatrix} = \begin{pmatrix} cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta) \end{pmatrix} \begin{pmatrix} x_{in} \\ y_{in} \end{pmatrix}$

Examples: The direction vector for axis $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ is transformed in $\begin{pmatrix} cos(\theta) \\ sin(\theta) \end{pmatrix}$ .
The direction vector for axis $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$ is transformed in $\begin{pmatrix} -sin(\theta) \\ cos(\theta) \end{pmatrix}$ .

Points and vectors 3d representation

Points 3d can be represented alternatively using carthesian or spherical coordinates. These two representation are linked by the following formula :

$P = \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} \rho\sin(\theta)\cos(\phi) \\ \rho\sin(\theta)\sin(\phi) \\ \rho\cos(\theta) \end{pmatrix}$

where :

$\rho=||P|| \in [0, \infty[$ is the L2 norm (euclidian norm) of
$\theta \in [0, \Pi[$ is the inclination
$\phi \in [0, 2\Pi]$ is the azimuth

Plans 3d representation

Plans 3d can be represented using spherical coordinates.

$\vec{\eta} = \begin{pmatrix} \sin(\theta)\cos(\phi) \\ \sin(\theta)\sin(\phi) \\ \cos(\theta) \end{pmatrix}$

where :

$\vec{\eta}$ is the orthogonal direction to plan.
$\rho$ is the distance from plan to origin

Rotations 3d

Rotations 3d are expressed with respect to direct referential orientation.
These rotations can be expressed as the product of three elemental rotations using Euler notations :

$R(\chi, \beta, \alpha) = R_z(\chi) \times R_y(\beta) \times R_x(\alpha)$

where :

$R_x(\alpha)$ is a rotation around axis

This rotation is associated to elemetary rotation matrix :
$R_x(\alpha) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & cos(\alpha) & -sin(\alpha) \\ 0 & sin(\alpha) & cos(\alpha) \\ \end{pmatrix}$
$R_y(\beta)$ is a rotation around axis

This rotation is associated to elemetary rotation matrix :
$R_y(\beta) = \begin{pmatrix} cos(\beta) & 0 & sin(\beta) \\ 0 & 1 & 0 \\ -sin(\beta) & 0 & cos(\beta) \\ \end{pmatrix}$
$R_z(\chi)$ is a rotation around axis

This rotation is associated to elemetary rotation matrix :
$R_z(\chi) = \begin{pmatrix} cos(\chi) & -sin(\chi) & 0 \\ sin(\chi) & cos(\chi) & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}$

Full rotation matrix is given by :

$R(\chi, \beta, \alpha) = R_z(\chi) \times R_y(\beta) \times R_x(\alpha) = \begin{pmatrix} cos(\chi)cos(\beta) & cos(\chi)sin(\beta)sin(\alpha) - sin(\chi)cos(\alpha) & cos(\chi)sin(\beta)cos(\alpha) + sin(\chi)sin(\alpha) \\ sin(\chi)cos(\beta) & sin(\chi)sin(\beta)sin(\alpha) + cos(\chi)cos(\alpha) & sin(\chi)sin(\beta)cos(\alpha) - cos(\chi)sin(\alpha) \\ -sin(\beta) & cos(\beta)sin(\alpha) & cos(\beta)cos(\alpha) \\ \end{pmatrix}$