Geometry Tools

1. Cartesian coordinates

The cartesian landmark is noted \(\left( \mathbf{e_x}, \, \mathbf{e_y}, \, \mathbf{e_z} \right)\).

2. Cylindrical coordinates

2.1. Presentation

The function of changing variables from cylindrical to cartesian is :

\[\phi : \left| \begin{matrix} \mathbb{R}^3 &\longrightarrow & \mathbb{R}^3 \\ \begin{pmatrix} r \\ \theta \\ z \end{pmatrix} &\longmapsto & \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} r \ cos \theta \\ r \ sin \theta \\ z \end{pmatrix} \end{matrix} \right.\]

And :

\[\phi^{-1} : \left| \begin{matrix} \mathbb{R}^3 &\longrightarrow & \mathbb{R}^3 \\ \begin{pmatrix} x \\ y \\ z \end{pmatrix} &\longmapsto & \begin{pmatrix} r \\ \theta \\ z \end{pmatrix} = \begin{pmatrix} \sqrt{x^2+y^2} \\ arctan(\frac{y}{x}) \\ z \end{pmatrix} \end{matrix} \right.\]

We have the Jacobian :

\[ Jac \, \phi = \begin{pmatrix} cos \theta && -r \, sin \theta && 0 \\ sin \theta && r \, cos \theta && 0 \\ 0 && 0 && 1 \end{pmatrix}\]

And :

\[ B^e = \left( Jac \, \phi^{-1} \right)^T = \begin{pmatrix} cos \theta && sin \theta && 0 \\ - \frac{sin \theta}{r} && \frac{cos \theta}{r} && 0 \\ 0 && 0 && 1 \end{pmatrix}\]

The passage to cartesian landmark \(\left( \mathbf{e_x}, \, \mathbf{e_y}, \, \mathbf{e_z} \right)\) to cylindrical landmark \(\left( \mathbf{e_r}, \, \mathbf{e_{\theta}}, \, \mathbf{e_z} \right)\) :

\[ \left\{ \begin{matrix} \mathbf{e_r} &=& cos \theta \, \mathbf{e_x} + sin \theta \, \mathbf{e_y} \\ \mathbf{e_{\theta}} &=& -sin \theta \, \mathbf{e_x} + cos \theta \, \mathbf{e_y} \\ \mathbf{e_z} &=& \mathbf{e_z} \end{matrix} \right.\]

The passage matrix is :

\[ P = \begin{pmatrix} cos\theta && sin \theta && 0 \\ -sin \theta && cos \theta && 0 \\ 0 && 0 && 1 \end{pmatrix} \ \text{and} \ P^{-1} = P^T = \begin{pmatrix} cos\theta && -sin \theta && 0 \\ sin \theta && cos \theta && 0 \\ 0 && 0 && 1 \end{pmatrix}\]

2.2. Formula of changement of variable

Set matrix \(M = \begin{pmatrix} M_{xx} & M_{xy} & M_{xz} \\ M_{yx} & M_{yy} & M_{yz} \\ M_{zx} & M_{zy} & M_{zz} \end{pmatrix}\).

We pass in cylindric coordinates :

\[\begin{eqnarray*} M_{cyl} &=& P \, M \, P^{-1} \\ &=& \begin{pmatrix} cos\theta && sin \theta && 0 \\ -sin \theta && cos \theta && 0 \\ 0 && 0 && 1 \end{pmatrix} \begin{pmatrix} M_{xx} & M_{xy} & M_{xz} \\ M_{yx} & M_{yy} & M_{yz} \\ M_{zx} & M_{zy} & M_{zz} \end{pmatrix} \begin{pmatrix} cos\theta && -sin \theta && 0 \\ sin \theta && cos \theta && 0 \\ 0 && 0 && 1 \end{pmatrix} \\ &=& \begin{pmatrix} cos^2 \theta \, M_{xx} + cos \theta \, sin \theta \, \left( M_{xy} + M_{yx} \right) + sin^2 \theta \, M_{yy} && - cos \theta \, sin \theta \, \left( M_{xx} - M_{yy} \right) - sin^2 \theta \, M_{yx} + cos^2 \theta \, M_{xy} && cos \theta \, M_{xz} + sin \theta \, M_{yz} \\ - cos \theta \, sin \theta \, \left( M_{xx} - M_{yy} \right) + cos^2 \theta \, M_{yx} - sin^2 \theta \, M_{xy} && sin^2 \theta \, M_{xx} - cos \theta \, sin \theta \, \left( M_{xy} + M_{yx} \right) + cos^2 \theta \, M_{yy} && -sin \theta \, M_{xz} + cos \theta \, M_{yz} \\ cos \theta \, M_{zx} + sin \theta \, M_{zy} && -sin \theta \, M_{zx} + cos \theta \, M_{zy} && M_{zz} \end{pmatrix} \end{eqnarray*}\]

3. Axisymmetrical coordinates

An axisymmetric geometry is a geometry symmetrical to an axis.

If we place the cylindrical landmark such as the axis of symmetry is \(Oz\) axis, the geometry is independent by \(\theta\). The geometry can be described in two dimensions in axisymmetric coordinates \((r, z)\).

For example :

  • A cylinder becomes a rectangle

  • A cone becomes a triangle

The axisymmetric landmark can be expressed as : \((\mathbf{e_r}, \mathbf{e_z})\).

3.1. Formula of changement of coordinates

The formula of changement to cartesian to cylindric coordinates :

Thus \(u = \begin{pmatrix} u_x \\ u_y \\ u_z \end{pmatrix} = \begin{pmatrix} u_r \\ u_{\theta} \\ u_z \end{pmatrix}\) :

\[\begin{eqnarray*} u_x &=& cos \theta \, u_r \\ u_y &=& sin \theta \, u_r \\ u_z &=& u_z \end{eqnarray*}\]
\[\begin{eqnarray*} \frac{\partial u_x}{\partial x} &=& \frac{\partial u_r}{\partial r} cos^2 \theta + \frac{1}{r} u_r sin^2 \theta \\ \frac{\partial u_x}{\partial y} &=& \frac{\partial u_r}{\partial r} cos \theta sin \theta + \frac{1}{r} u_r cos \theta sin \theta \\ \frac{\partial u_x}{\partial z} &=& \frac{\partial u_z}{\partial z} cos \theta \end{eqnarray*}\]
\[\begin{eqnarray*} \frac{\partial u_y}{\partial x} &=& \frac{\partial u_r}{\partial r} cos \theta sin \theta + \frac{1}{r} u_r cos \theta sin \theta \\ \frac{\partial u_y}{\partial y} &=& \frac{\partial u_r}{\partial r} sin^2 \theta + \frac{1}{r} u_r cos^2 \theta \\ \frac{\partial u_y}{\partial z} &=& \frac{\partial u_z}{\partial z} sin \theta \end{eqnarray*}\]
\[\begin{eqnarray*} \frac{\partial u_z}{\partial x} &=& \frac{\partial u_z}{\partial z} cos \theta \\ \frac{\partial u_z}{\partial y} &=& \frac{\partial u_z}{\partial z} sin \theta \\ \frac{\partial u_z}{\partial z} &=& \frac{\partial u_x}{\partial z} \end{eqnarray*}\]