Exponential Curves on the Similarity Group

\(\operatorname{SIM}(2)\) is the similarity group on \(\mathbb{R}^2\) consists of transformations which preserve similarity and orientation: if you apply a transformation in \(\operatorname{SIM}(2)\) to a triangle, you get a similar triangle (equal angles) in the same orientation. We can see that \(\operatorname{SIM}(2) \cong \mathbb{R}^2 \rtimes (\operatorname{SO}(2) \times \mathbb{R}_{> 0})\): orientation-preserving similarity transformations are precisely translations, rotations, and isotropic scalings. Because of this, we may denote elements in SIM(2) by a tuple of a translation, rotation, and scaling \[ (x, R, a) \in \mathbb{R}^2 \times \operatorname{SO}(2) \times \mathbb{R}_{> 0}. \] The \(\operatorname{SIM}(2)\) group product is however not a direct product of the group products on \(\mathbb{R}^2, \operatorname{SO}(2), \mathbb{R}_{> 0}\) (this is the meaning \(\rtimes\)), instead: \[ (x_1, R_1, a_1) (x_2, R_2, a_2) \coloneqq (x_1 + a_1 R_1 x_2, R_1 R_2, a_1 a_2). \] Equivalently, we can work with the following matrix representation of \(\operatorname{SIM}(2)\): \[ (x, R, a) \leftrightarrow \begin{pmatrix} a R & x \\ 0 & 1 \end{pmatrix}. \] One can confirm that the matrix product behaves like the \(\operatorname{SIM}(2)\) group product for these matrices. It is natural to let \(\operatorname{SIM}(2)\) act on \(\mathbb{R}^2\): the obvious action is \[ (x, R, a) y \coloneqq x + a R y, \] equivalently written with homogeneous coordinates as \[ \begin{pmatrix} a R & x \\ 0 & 1 \end{pmatrix} \begin{pmatrix} y \\ 1 \end{pmatrix} \coloneqq \begin{pmatrix} x + a R y \end{pmatrix}. \] Finally, since \(\operatorname{SO}(2) \cong S^1\) (as smooth manifolds), we can replace a rotation \(R\) by its angle \(\theta\).

Click on the canvas to place points; three points will form a triangle. The sliders determine the similarity transform that will be applied to the triangle; by default we have the identity element \(((x, y), \theta, a) = ((0, 0), 0, 1)\). To start over, simply add a new point by clicking on the canvas again.

Exponential curves can be thought of as the natural generalisations of straight lines to Lie groups. We have already seen them before, e.g. in "Generators for Pairs of Position-Orientations". Spatial projections of \(\operatorname{SIM}(2)\), exponential curves, i.e. \(t \mapsto \gamma(t) y\) for some exponential curve \(\gamma\), are nice. We can compute the spatial projection using the fact that the matrix representation maps the Lie group exponential to the matrix exponential.

In particular, we consider the exponential curve generated by some infinitesimal scaling and rotation. These generators are given by \[ \begin{aligned} A & \coloneqq \left. \partial_t \begin{pmatrix} (a + t) \cos(\theta) & -(a + t) \sin(\theta) & x \\ (a + t) \sin(\theta) & (a + t) \cos(\theta) & y \\ 0 & 0 & 1 \end{pmatrix} \right|_{(x, \theta, a) = (0, 0, 1)} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \textrm{ and} \\ \Theta & \coloneqq \left. \partial_t \begin{pmatrix} a \cos(\theta + t) & -a \sin(\theta + t) & x \\ a \sin(\theta + t) & a \cos(\theta + t) & y \\ 0 & 0 & 1 \end{pmatrix} \right|_{(x, \theta, a) = (0, 0, 1)} = \begin{pmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}. \end{aligned} \] It turns out that \(\lambda (c^A A + c^\Theta \Theta)\) generates the same exponential curve as \(c^A A + c^\Theta \Theta\) (up to reparametrisation). We therefore consider the generator \(\omega \coloneqq c^A A + \Theta\) and corresponding exponential curve \(\gamma(t) \coloneqq \exp(t \omega)\). We project spatially using the reference vector \(y = (1, 0) \in \mathbb{R}^2\): \[ \gamma(t) y = \exp(t \omega) y = \begin{pmatrix} e^{t c^A} \cos(t) & -e^{t c^A} \sin(t) & 0 \\ e^{t c^A} \sin(t) & e^{t c^A} \cos(t) & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} e^{t c^A} \cos(t) \\ e^{t c^A} \sin(t) \\ 1 \end{pmatrix}. \] Hence, the spatial projection is the logarithmic spiral \(t \mapsto (e^{t c^A} \cos(t), e^{t c^A} \sin(t))\). To see this, define the radius \(r(t) \coloneqq \Vert \gamma(t) y \Vert^2 = e^{t c^A}\); it follows that \(r(t + \pi/2) / r(t) = e^{\frac{\pi}{2} c^A}\), which is constant for all \(t \in \mathbb{R}\). Choosing \(c^A\) such that \(e^{\frac{\pi}{2} c^A} = \varphi\), i.e. \(c^A = \frac{2}{\pi} \log(\phi)\), with \(\phi \coloneqq \frac{1 + \sqrt{5}}{2}\) the golden ratio, this curve becomes a so-called golden spiral.

The spatial projection of the exponential curve is plotted below, where \(c^A\) can be controlled using the slider. To get the width of the curve to vary, we apply the exponential curve to a pair of vectors, where one is slightly longer than the other.