# Dilation

A dilation is a conformal transformation of Euclidean space performed by the operator

$$\mathbf D = \dfrac{1 - \sigma}{2} (c_x \mathbf e_{235} + c_y \mathbf e_{315} + c_z \mathbf e_{125} - \mathbf e_{321}) + \dfrac{1 + \sigma}{2} {\large\unicode{x1d7d9}}$$ .

This operator scales an object $$\mathbf x$$ by the factor $$\sigma$$ about the center point $$\mathbf c = (c_x, c_y, c_z)$$ when used with the sandwich antiproduct $$\mathbf D \mathbin{\unicode{x27C7}} \mathbf x \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{D}}}$$. This dilation operator is scaled so that the round weight of $$\mathbf x$$ remains the same after the dilation is applied.

## Exponential Form

A dilation by a scale factor $$\sigma$$ about the center of a unitized, positively oriented sphere $$\mathbf s$$ can be expressed as an exponential of the sphere's attitude as

$$\mathbf D = \exp_\unicode{x27C7}\left(-\dfrac{1}{2} \delta \operatorname{att}(\mathbf s)\right) = -\operatorname{att}(\mathbf s) \sinh \dfrac{\delta}{2} + {\large\unicode{x1d7d9}} \cosh \dfrac{\delta}{2}$$ ,

where $$\delta = \log \sigma$$. Expanding the $$\sinh$$ and $$\cosh$$ functions, we can rewrite this as

$$\mathbf D = \dfrac{e^{-\delta/2} - e^{\delta/2}}{2} \operatorname{att}(\mathbf s) + \dfrac{e^{\delta/2} + e^{-\delta/2}}{2} {\large\unicode{x1d7d9}}$$ .

Homogeneous multiplication by $$e^{\delta/2}$$ gives us

$$\mathbf D = \dfrac{1 - e^\delta}{2} \operatorname{att}(\mathbf s) + \dfrac{e^\delta + 1}{2} {\large\unicode{x1d7d9}}$$ ,

and replacing $$e^\delta$$ with $$\sigma$$ produces

$$\mathbf D = \dfrac{1 - \sigma}{2} \operatorname{att}(\mathbf s) + \dfrac{1 + \sigma}{2} {\large\unicode{x1d7d9}}$$ .

## Matrix Form

When a dilation $$\mathbf D$$ is applied to a round point, it is equivalent to premultiplying the point by the $$5 \times 5$$ matrix

$$\begin{bmatrix} \sigma & 0 & 0 & c_x (1 - \sigma) & 0 \\ 0 & \sigma & 0 & c_y (1 - \sigma) & 0 \\ 0 & 0 & \sigma & c_z (1 - \sigma) & 0 \\ 0 & 0 & 0 & 1 & 0 \\ c_x \sigma (1 - \sigma) & c_y \sigma (1 - \sigma) & c_z \sigma (1 - \sigma) & \dfrac{c^2 (1 - \sigma)^2}{2} & \sigma^2 \end{bmatrix}$$ .