polar

Module: core

Category: mat3dvaluator

Type string: "polar"

Parameters

Name	Description	Default	Units
center	center	{0,0,0}	[]
axis	axis	{0,0,1}	[]
vector1	vector1	{1,0,0}	[]
vector2	vector2	{1,0,0}	[]
radius1	radius1	0	[]
radius2	radius2	1	[]

Description

Assume \(\mathbf{p}\) the position of a point at which we want to calculate a \(3 \times 3\) matrix.

First, find the vector to the axis

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

and its length

\[ R = \left|| \mathbf{b} \right|| \]

Define the unit vector,

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

Calculate the relative radius \(w=(R - R_0)/(R_1 - R_0)\)

Define \(\mathbf{R}_{01}\) as the rotation that rotates \(\mathbf{v}_0\) to \(\mathbf{v}_1\) and \(\mathbf{R}_w\) the rotation matrix by interpolating between the identity matrix \(\mathbf{I}\) and \(\mathbf{R}_{01}\) at fraction \(w\). Then, define,

\[ \mathbf{v}=\mathbf{R}_w\,\mathbf{v}_0 \]

Define \(\mathbf{Q}\) the rotation matrix that rotates \(\mathbf{e}_x=(1,0,0)\) to \(\mathbf{b}\) and define \(\mathbf{d}=\mathbf{Q}\,\mathbf{e}_y\).

Next, construct three orthogonal vectors,

\[ \begin{align} &{\mathbf{e}_1 = \mathbf{v}} \\ &{\mathbf{e}_3 = \mathbf{e}_1 \times \mathbf{d}} \\ &{\mathbf{e}_2 = \mathbf{e}_3 \times \mathbf{e}_1} \\ \end{align} \]

Finally, construct the orthonormal matrix.

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