Eric Lengyel at 09:04, 22 December 2024

2024-12-22T09:04:23Z

← Older revision		Revision as of 09:04, 22 December 2024
Line 2:		Line 2:
	A ''dilation'' is a conformal transformation of Euclidean space performed by the operator		A ''dilation'' is a conformal transformation of Euclidean space performed by the operator

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

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

	== Exponential Form ==		== Exponential Form ==
Line 29:		Line 29:

	:$$\begin{bmatrix}		:$$\begin{bmatrix}
	\sigma & 0 & 0 & ~~c_x~~ (1 - \sigma) & 0 \\		\sigma & 0 & 0 & m_x (1 - \sigma) & 0 \\
	0 & \sigma & 0 & ~~c_y~~ (1 - \sigma) & 0 \\		0 & \sigma & 0 & m_y (1 - \sigma) & 0 \\
	0 & 0 & \sigma & ~~c_z~~ (1 - \sigma) & 0 \\		0 & 0 & \sigma & m_z (1 - \sigma) & 0 \\
	0 & 0 & 0 & 1 & 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		m_x \sigma (1 - \sigma) & m_y \sigma (1 - \sigma) & m_z \sigma (1 - \sigma) & \dfrac{\mathbf m^2 (1 - \sigma)^2}{2} & \sigma^2
	\end{bmatrix}$$ .		\end{bmatrix}$$ .

Eric Lengyel: Created page with "NOTOC 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{x27C..."

2023-08-06T03:20:07Z

Created page with "__NOTOC__ 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{x27C..."

New page

__NOTOC__
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

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

== See Also ==

* [[Translation]]
* [[Rotation]]

Dilation - Revision history

Eric Lengyel at 09:04, 22 December 2024