cylindrical

Module: core

Category: mat3dvaluator

Type string: "cylindrical"

Parameters

Name	Description	Default	Units
center	center	{0,0,0}	[]
axis	axis	{0,0,1}	[]
vector	vector	{1,0,0}	[]

Description

The cylindrical mat3d valuator generates an orthonormal matrix based on cylindrical coordinates.

First, three perpendicular unit vectors are calculated. The first vector is generated using the parameters defined above. Let \(\mathbf{r}\) be the position vector at which we want to generate a mat3d. Let \(\mathbf{c}\) be the center parameter, \(\mathbf{a}\) the axis parameter and \(\mathbf{t}\) the vector parameter. Then, the vector from \(\mathbf{r}\) to the axis is,

\[ \mathbf{b}= \left( \mathbf{r} - \mathbf{c} \right) - \mathbf{a} \left( \mathbf{a} \cdot \left( \mathbf{r} - \mathbf{c}\right) \right) \]

Then, the corresponding unit vector \(\hat{\mathbf{b}}\) is found from,

\[ \hat{\mathbf{b}} = \frac{\mathbf{b}}{\left||\mathbf{b}\right||} \]

Next, we find the rotation matrix \(\mathbf{R}\) that rotates \(\mathbf{e}_1=\left(1,0,0 \right)\) to \(\hat{\mathbf{b}}\).

The first unit vector is then defined by

\[ \mathbf{a}_1 = \mathbf{R}\,\hat{\mathbf{t}} \]

To find the third vector, we first calculate \(\mathbf{d}=\mathbf{R}\,\mathbf{e}_2\). (If that produces a vector that is too close to \(\mathbf{a}\), we choose instead \(\mathbf{d}=\mathbf{R}\,\mathbf{e}_2\).)

The third vector is then defined by,

\[ \mathbf{a}_3 = \frac{\mathbf{a}_1\times\mathbf{d}}{\left|| \mathbf{a}_1\times\mathbf{d} \right||} \]

Finally, the second vector is found from,

\[ \mathbf{a}_2 = \mathbf{a}_3\times\mathbf{a}_1 \]

The orthonormal matrix is then defined as the matrix,

\[ \mathbf{Q}=\left[ \begin{matrix} {\mathbf{a}_{1}} & {\mathbf{a}_{2}} & {\mathbf{a}_{3}} \end{matrix} \right] \]