Finding Fixed Points

An isometry on a metric space is a one-to-one distance-preserving transformation on the space. The Euclidean group ${E(n)}$ is the group of isometries of ${n}$ -dimensional Euclidean space. These are all the transformations that preserve the distance between any two points. The group depends on the dimension of the space. For the Euclidean plane ${\mathbb{E}^2}$ , we have the group ${E(2)}$ , comprising all combinations of translations, rotations and reflections of the plane.

Euclid allowed triangles to be “flipped over” either by reflection in a line or rotation out of the plane. The group ${E(2)}$ includes these, but if reflections, which change the orientation of figures, are disallowed, we have the special Euclidean group ${SE(2)}$ of rigid motions in the plane, which comprise arbitrary combinations of translations and rotations, but not reflections.

Augmented matrix

In vector algebra we use matrix multiplication to represent rotations and vector addition to represent translations. If the rotation is represented by a orthogonal matrix ${\mathbf{R}}$ and the translation as the addition of a vector ${\mathbf{p}}$ , the composition of these is

$\displaystyle \mathbf {y} = \mathbf {R x} +\mathbf{p}$

This cannot be represented by a ${2\times2}$ matrix, since the origin is moved.

Transformations of the 2D Euclidean plane in ${SE(2)}$ ) can be represented by a single 3D matrix multiplication. Vectors are augmented with a `1′ at the end, and matrices are augmented with an extra row of zeros at the bottom, an extra column — the translation vector ${\mathbf{p}}$ — and a `1′ in the lower right corner. Thus, ${\mathbf {y} =\mathbf {R x} +\mathbf{p}}$ is represented as

$\displaystyle {\begin{bmatrix}\mathbf {y} \\1\end{bmatrix}} = \left[\begin{array}{c|c} \mathbf{R} & \mathbf{p} \\ \hline \mathbf{0} & 1 \end{array}\right] {\begin{bmatrix}\mathbf {x} \\1\end{bmatrix}}$

The augmented matrix is a special case of an affine transformation matrix.

All elements of ${SE(2)}$ can be represented as a translation followed by a rotation, or symbolically as

$\displaystyle SE(2) = SO(2) \ltimes T(2) \,,$

the semidirect product of the special orthogonal group ${SO(2)}$ and the group ${T(2)}$ of translations in the plane. Elements of ${SE(2)}$ are pairs ${(R,\mathbf{p})}$ , where ${\mathbf{R}}$ is an ${n\times n}$ matrix and ${\mathbf{p}}$ is an ${n\times 1}$ vector in ${\mathbb{R}^2}$ . Each element can be represented by ${3\times 3}$ matrix

$\displaystyle \mathbf{M} = \left[\begin{array}{c|c} \mathbf{R} & \mathbf{p} \\ \hline \mathbf{0} & 1 \end{array}\right] \,,$

which transforms ${(\mathbf{r},1)^{\mathrm T}}$ to ${(\mathbf{R r}+\mathbf{p},1)^{\mathrm T}}$ , a rotation followed by a translation. This matrix factors into a pure rotation followed by a pure translation:

$\displaystyle \left[\begin{array}{c|c} \mathbf{R} & \mathbf{p} \\ \hline \mathbf{0} & 1 \end{array}\right] = \left[\begin{array}{c|c} \mathbf{I} & \mathbf{p} \\ \hline \mathbf{0} & 1 \end{array}\right] \times \left[\begin{array}{c|c} \mathbf{R} & \mathbf{0} \\ \hline \mathbf{0} & 1 \end{array}\right] \,.$

The set of matrices of form ${\mathbf{M}}$ forms a group, a matrix representation of the group ${SE(2)}$ . Elements of ${SE(2)}$ are particular forms of affine transformations: the origin gets moved, and rotations may be about any point, not just the origin as for rotations in a vector space.

Finding the centre of rotation: Regular Points

For pure translations, every point in the plane is moved to a different point; there are no fixed points. However, for any transformation that is not a pure translation, there is a fixed point. Obviously, a pure rotation about a point ${\mathbf{p}}$ has a fixed point at ${\mathbf{p}}$ . A pure rotation about a point ${\mathbf{p}}$ can be considered as a composition of three transformations: a translation of the origin to ${\mathbf{p}}$ , a rotation about this origin, and a reverse translation of the origin to the original position.

Points regularly distributed in a square region before rotation (left) and after rotation (right).

Superimposition of regularly distributed points before rotation (black) and after rotation (blue).

Suppose we have a rotation about some point in the plane, and we wish to find the centre of rotation. We might consider a square region with a grid of points, and examine how they move under the rotation. The figure above shows a set of points regularly distributed in a square region before rotation (left panel) and after rotation (right panel). The panels may overlap or may be remote from each other. in the figure to the right, we superimpose them and a pattern of circles appears. The details of this Moire’ pattern are unimportant. The important point is that the pattern repeats indefinitely: if we showed a larger region, there would be more circles, but no further information about the fixed point would emerge.

Finding the centre of rotation: Random Points

Points randomly distributed in a square region before rotation (left) and after rotation (right).

Superimposition of randomly distributed points before rotation (black) and after rotation (blue)

The regular grid has not enabled us to find the centre of rotation. We now select a set of points randomly distributed throughout the square region and see how they move under a rotation. The figure above shows a set of points regularly distributed in a square region before rotation (left panel) and after rotation (right panel). The panels may overlap or may be remote from each other.

In the figure here, we superimpose them and a clear pattern emerges, indicating circular motion about a definite point. If we showed a larger region, the rotation about a single point would still be evident.

Sources

${\bullet}$ Nishiyama, Yutaka, 2013: An elegant solution for drawing a fixed point. Chapter 3 in The Mysterious Number 6174. Gendai Sugakusha, Co. ISBN: 978-4-7687-6174-8.

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