Flector - Revision history

Eric Lengyel: /* Norm */

2024-07-08T01:20:42Z

Norm

← Older revision		Revision as of 01:20, 8 July 2024
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	The [[bulk norm]] of a flector $$\mathbf F$$ is given by		The [[bulk norm]] of a flector $$\mathbf F$$ is given by

	:$$\left\Vert\mathbf F\right\Vert_\unicode{x25CF} = \sqrt{\mathbf F \mathbin{\unicode{~~x25CF~~}} \mathbf{\tilde F}} = \sqrt{F_{px}^2 + F_{py}^2 + F_{pz}^2 + F_{gw}^2}$$ ,		:$$\left\Vert\mathbf F\right\Vert_\unicode["segoe ui symbol"]{x25CF} = \sqrt{\mathbf F \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf{\tilde F}} = \sqrt{F_{px}^2 + F_{py}^2 + F_{pz}^2 + F_{gw}^2}$$ ,

	and its [[weight norm]] is given by		and its [[weight norm]] is given by

	:$$\left\Vert\mathbf F\right\Vert_\unicode{x25CB} = \sqrt{\mathbf F \mathbin{\unicode{~~x25CB~~}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{F}}}\vphantom{\mathbf{\tilde F}}} = {\large\unicode{x1D7D9}}\sqrt{F_{gx}^2 + F_{gy}^2 + F_{gz}^2 + F_{pw}^2}$$ .		:$$\left\Vert\mathbf F\right\Vert_\unicode["segoe ui symbol"]{x25CB} = \sqrt{\mathbf F \mathbin{\unicode["segoe ui symbol"]{x2218}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{F}}}\vphantom{\mathbf{\tilde F}}} = {\large\unicode{x1D7D9}}\sqrt{F_{gx}^2 + F_{gy}^2 + F_{gz}^2 + F_{pw}^2}$$ .

	The [[geometric norm]] of a flector $$\mathbf F$$ is thus		The [[geometric norm]] of a flector $$\mathbf F$$ is thus

Eric Lengyel at 23:51, 13 April 2024

2024-04-13T23:51:36Z

← Older revision		Revision as of 23:51, 13 April 2024
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	$$h'_w = h_w + 2[(\mathbf h_{xyz} \times \mathbf p - F_{gw}\mathbf h_{xyz}) \cdot \mathbf g + F_{pw}(\mathbf p \cdot \mathbf h_{xyz})]$$		$$h'_w = h_w + 2[(\mathbf h_{xyz} \times \mathbf p - F_{gw}\mathbf h_{xyz}) \cdot \mathbf g + F_{pw}(\mathbf p \cdot \mathbf h_{xyz})]$$
	\|}		\|}

			== In the Book ==

			* General reflection operators (flectors) are discussed in Section 3.7.

	== See Also ==		== See Also ==

Eric Lengyel: /* Flector Transformations */

2024-04-08T05:12:10Z

Flector Transformations

← Older revision		Revision as of 05:12, 8 April 2024
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	== Flector Transformations ==		== Flector Transformations ==

	[[Point \| Points]], [[Line \| lines]], and [[Plane \| planes]] are transformed by a [[unitized]] flector $$\mathbf F$$ as shown in the following table.		[[Point \| Points]], [[Line \| lines]], and [[Plane \| planes]] are transformed by a [[unitized]] flector $$\mathbf F$$ as shown in the following table, where $$\mathbf p = (F_{px}, F_{py}, F_{pz})$$ and $$\mathbf g = (F_{gx}, F_{gy}, F_{gz})$$.

	{\| class="wikitable"		{\| class="wikitable"
Line 130:		Line 130:

	$$\mathbf q = q_x \mathbf e_1 + q_y \mathbf e_2 + q_z \mathbf e_3 + q_w \mathbf e_4$$		$$\mathbf q = q_x \mathbf e_1 + q_y \mathbf e_2 + q_z \mathbf e_3 + q_w \mathbf e_4$$
	\| style="padding: 12px;" \| $$\~~begin{split}~~\mathbf F \~~mathbin{~~\~~unicode~~{~~x27C7}~~} \mathbf ~~q \mathbin{\unicode{x27C7}} \smash{~~\mathbf{~~\underset{\Large\unicode{x7E}}{F}}~~} =\~~, &\left[(2F_{gy}^2 + 2F_~~{gz}~~^2 - 1)q_x~~ + 2(~~F_{gz}~~F_{pw} ~~- F_{gx}F_{gy})q_y - 2(F_{gy}F_{pw} + F_{gz}F_{gx})q_z + 2(F_{gy}F_{pz} - F_{gz}F_{py}~~ + ~~F_{px}F_{pw} - F_{gx}F_{gw})q_w\right]~~\mathbf ~~e_1~~ \\ +~~\, &\left[(2F_{gz}^2 + 2F_{gx}^2 - 1)q_y + 2(F_{gx}F_{pw} - F_{gy}F_{gz})q_z - 2(F_{gz}F_{pw} + F_{gx}F_{gy})q_x + 2(F_{gz}F_{px} - F_{gx}F_{pz} + F_{py}F_{pw} - F_{gy}~~F_{gw})q_w~~\right]~~\mathbf ~~e_2 \\ +\, &\left[(2F_{gx}^2 + 2F_{gy}^2 - 1~~)~~q_z + 2(F_{gy}F_{pw} - F_{gz}F_{gx})q_x - 2(F_{gx}F_{pw} + F_{gy}F_{gz})q_y + 2(F_{gx}F_{py} - F_{gy}F_{px} + F_{pz}F_{pw}~~ - ~~F_{gz}F_{gw})~~q_w~~\right]\mathbf e_3 \\ +\, &q_w\mathbf e_4\end{split}~~$$		\| style="padding: 12px;" \| $$\mathbf a = \mathbf g \times \mathbf q_{xyz} - q_w\mathbf p$$

			$$\mathbf q'_{xyz} = \mathbf q_{xyz} + 2(F_{pw}\mathbf a + \mathbf g \times \mathbf a + F_{gw}q_w\mathbf g)$$

			$$q'_w = -q_w$$
	\|-		\|-
	\| style="padding: 12px;" \| [[Line]]		\| style="padding: 12px;" \| [[Line]]

	$$\begin{split}\~~boldsymbol l~~ =\, &l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} \\ +\, &l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}\end{split}$$		$$\begin{split}\mathbf k =\, &k_{vx} \mathbf e_{41} + k_{vy} \mathbf e_{42} + k_{vz} \mathbf e_{43} \\ +\, &k_{mx} \mathbf e_{23} + k_{my} \mathbf e_{31} + k_{mz} \mathbf e_{12}\end{split}$$
	\| style="padding: 12px;" \| $$\~~begin{split}~~\mathbf F \~~mathbin~~{\~~unicode{x27C7}~~} \~~boldsymbol l~~ \~~mathbin{~~\~~unicode{x27C7}}~~ \~~smash~~{\mathbf{\~~underset{\Large\unicode{x7E}}{F}}}~~ =\~~, &~~\~~left[(1 - 2F_{gy}^2 - 2F_{gz}^2)l_{vx} + 2(F_{gx}F_{gy} - F_{gz}F_{pw})l_{vy} + 2(F_{gz}F_{gx} + F_{gy}F_{pw})l_{vz}\right]~~\mathbf e_{~~41}~~ \~~\ +\, &\left[(1 - 2F_{gz}^2 - 2F_{gx}^2)l_{vy} + 2(F_{gy}F_{gz} - F_{gx}F_{pw})l_{vz} + 2(F_{gx}F_{gy} + F_{gz}F_{pw})l_{vx~~}~~\right]~~\mathbf e_{~~42}~~ \~~\ +\, &\left[(1 - 2F_{gx~~}~~^2 - 2F_{gy}^2)l_{vz} +~~ 2(~~F_{gz}F_{gx} - F_{gy}F_{pw})l_{vx} + 2(F_{gy}F_{gz} + F_{gx}F_{pw})l_{vy}\right]~~\mathbf ~~e_{43}~~ \\ ~~+\, &\left[~~-~~4(F_{gy}F_{py} + F_{gz}F_{pz})l_{vx} + 2(F_{gx}F_{py} + F_{gy}F_{px})l_{vy} + 2(F_{gx}F_{pz} + F_{gz}F_{px} + F_{gy}F_{gw} + F_{py}~~F_{pw})~~l_{vz} + (2F_{gy}^2 + 2F_{gz}^2~~ - ~~1)l_~~{~~mx} + 2(F_{gz}F_{pw} - F_{gx}F_{gy})l_{my} - 2(F_{gy}F_{pw} + F_{gz}F_{gx})l_{mz~~}~~\right]~~\mathbf e_{~~23}~~ \~~\ +\, &\left[-4(F_{gz~~}~~F_{pz} + F_{gx}F_{px})l_{vy} +~~ 2~~(F_{gy}F_{pz} + F_{gz}F_{py})l_{vz} + 2(F_{gy}F_{px} + F_{gx}F_{py} + F_{gz}~~F_{gw} + ~~F_{pz}~~F_{pw}~~)l_{vx} +~~ (~~2F_{gz}^2 + 2F_{gx}^2~~ - 1)~~l_{my}~~ + 2(~~F_{gx}F_{pw}~~ - ~~F_{gy}F_{gz}~~)~~l_{mz} - 2(F_{gz}F_{pw}~~ + ~~F_{gx}F_{gy})l_{mx}\right]~~\mathbf ~~e_{31}~~ \\ ~~+\, &\left[-4(F_{gx}F_{px} + F_{gy}F_{py})l_{vz} + 2(F_{gz}F_{px} + F_{gx}F_{pz})l_{vx} + 2(F_{gz}F_{py} + F_{gy}F_{pz} + F_{gx}F_{gw} + F_{px}F_{pw})l_{vy} + (2F_{gx}^2 + 2F_{gy}^2~~ - ~~1)l_{mz} + 2(F_{gy}F_{pw} - F_{gz}F_{gx})l_{mx} - 2(F_{gx}F_{pw} + F_{gy}F_{gz})l_{my}\right]~~\mathbf e_{~~12}~~\~~end{split~~}$$		\| style="padding: 12px;" \| $$\mathbf a = \mathbf g \times \mathbf k_{\mathbf v}$$

			$$\mathbf b = \mathbf g \times \mathbf k_{\mathbf m}$$

			$$\mathbf c = \mathbf p \times \mathbf k_{\mathbf v}$$

			$$\mathbf k'_{\mathbf v} = 2(\mathbf a \times \mathbf g - F_{pw}\mathbf a) - \mathbf k_{\mathbf v}$$

			$$\mathbf k'_{\mathbf m} = 2[F_{gw}\mathbf a + F_{pw}(\mathbf c - \mathbf b) + \mathbf g \times (\mathbf c - \mathbf b) + \mathbf p \times \mathbf a] - \mathbf k_{\mathbf m}$$
	\|-		\|-
	\| style="padding: 12px;" \| [[Plane]]		\| style="padding: 12px;" \| [[Plane]]

	$$\mathbf h = h_x \mathbf e_{423} + h_y \mathbf e_{431} + h_z \mathbf e_{412} + h_w \mathbf e_{321}$$		$$\mathbf h = h_x \mathbf e_{423} + h_y \mathbf e_{431} + h_z \mathbf e_{412} + h_w \mathbf e_{321}$$
	\| style="padding: 12px;" \| $$\~~begin{split}~~\mathbf F \~~mathbin{~~\~~unicode~~{~~x27C7}~~} \mathbf h ~~\mathbin{\unicode{x27C7}} \smash{\mathbf{\underset~~{~~\Large\unicode{x7E}}{F}}~~} =~~\, &\left[(1 - 2F_{gy}^2 - 2F_{gz}^2)h_x + 2(F_{gx}F_{gy} - F_{gz}F_{pw})h_y +~~ 2(~~F_{gz}F_{gx} + F_{gy}F_{pw})h_z\right]~~\mathbf ~~e_{423}~~ \\ ~~+\, &\left[(1 - 2F_{gz}^2 - 2F_{gx}^2)h_y + 2(F_{gy}F_{gz}~~ - ~~F_{gx}~~F_{pw}~~)h_z + 2(F_{gx}F_{gy} + F_{gz}F_{pw})h_x\right]~~\mathbf ~~e_{431} \\ +\, &\left[(1 - 2F_{gx}^2 - 2F_{gy}^2~~)~~h_z + 2(F_{gz}F_{gx}~~ - F_{gy}~~F_{pw})h_x~~ + 2(~~F_{gy}F_{gz} + F_{gx}F_{pw})h_y\right]~~\mathbf e_{~~412~~} \\ ~~+\, &\left[2(F_{gy}F_{pz}~~ - ~~F_{gz}F_{py} + F_{gx}~~F_{gw} ~~- F_{px}F_~~{pw})~~h_x + 2(F_{gz}F_{px} - F_{gx}F_{pz}~~ + ~~F_{gy}F_{gw} - F_{py}~~F_{pw}~~)h_y + 2~~(~~F_{gx}F_{py} - F_{gy}F_{px} + F_{gz}F_{gw} - F_{pz}F_{pw})h_z - h_w~~\~~right]~~\mathbf e_{~~321}\end{split~~}$$		\| style="padding: 12px;" \| $$\mathbf a = \mathbf g \times \mathbf h_{xyz}$$

			$$\mathbf h'_{xyz} = 2(\mathbf a \times \mathbf g - F_{pw}\mathbf a) - \mathbf h_{xyz}$$

			$$h'_w = h_w + 2[(\mathbf h_{xyz} \times \mathbf p - F_{gw}\mathbf h_{xyz}) \cdot \mathbf g + F_{pw}(\mathbf p \cdot \mathbf h_{xyz})]$$
	\|}		\|}

Eric Lengyel at 01:02, 9 February 2024

2024-02-09T01:02:05Z

← Older revision		Revision as of 01:02, 9 February 2024
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	:$$\mathbf F = F_{px} \mathbf e_1 + F_{py} \mathbf e_2 + F_{pz} \mathbf e_3 + F_{pw} \mathbf e_4 + F_{gx} \mathbf e_{423} + F_{gy} \mathbf e_{431} + F_{gz} \mathbf e_{412} + F_{gw} \mathbf e_{321}$$ ,		:$$\mathbf F = F_{px} \mathbf e_1 + F_{py} \mathbf e_2 + F_{pz} \mathbf e_3 + F_{pw} \mathbf e_4 + F_{gx} \mathbf e_{423} + F_{gy} \mathbf e_{431} + F_{gz} \mathbf e_{412} + F_{gw} \mathbf e_{321}$$ ,

	which is the sum of a [[point]] $$\mathbf p$$ and a [[plane]] $$\mathbf g$$. To ~~possess~~ the [[geometric ~~property~~]], the components of $$\mathbf F$$ must satisfy the equation		which is the sum of a [[point]] $$\mathbf p$$ and a [[plane]] $$\mathbf g$$. To satisfy the [[geometric constraint]], the components of $$\mathbf F$$ must satisfy the equation

	:$$F_{px} F_{gx} + F_{py} F_{gy} + F_{pz} F_{gz} + F_{pw} F_{gw} = 0$$ ,		:$$F_{px} F_{gx} + F_{py} F_{gy} + F_{pz} F_{gz} + F_{pw} F_{gw} = 0$$ ,

Eric Lengyel: Created page with "'''Figure 1.''' A flector represents an improper Euclidean isometry, which can always be regarded as a rotation about a line $$\boldsymbol l$$ and a reflection across a plane perpendicular to the same line. A ''flector'' is an operator that performs an improper isometry in Euclidean space. Such isometries encompass all possible combinations of an odd number of reflections, inversions, transflections, and rotorefle..."

2023-07-15T06:06:10Z

Created page with "400px|thumb|right|'''Figure 1.''' A flector represents an improper Euclidean isometry, which can always be regarded as a rotation about a line $$\boldsymbol l$$ and a reflection across a plane perpendicular to the same line. A ''flector'' is an operator that performs an improper isometry in Euclidean space. Such isometries encompass all possible combinations of an odd number of reflections, inversions, transflections, and rotorefle..."

New page

[[Image:improper_isom.svg|400px|thumb|right|'''Figure 1.''' A flector represents an improper Euclidean isometry, which can always be regarded as a rotation about a line $$\boldsymbol l$$ and a reflection across a plane perpendicular to the same line.]]
A ''flector'' is an operator that performs an improper isometry in Euclidean space. Such isometries encompass all possible combinations of an odd number of [[reflections]], [[inversions]], [[transflections]], and rotoreflections. The name flector is a portmanteau of ''reflection operator''. Flectors cannot perform proper isometries that do not include [[reflections]]; those are instead performed by operators called [[motors]].

In the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$, a flector $$\mathbf F$$ has the general form

:$$\mathbf F = F_{px} \mathbf e_1 + F_{py} \mathbf e_2 + F_{pz} \mathbf e_3 + F_{pw} \mathbf e_4 + F_{gx} \mathbf e_{423} + F_{gy} \mathbf e_{431} + F_{gz} \mathbf e_{412} + F_{gw} \mathbf e_{321}$$ ,

which is the sum of a [[point]] $$\mathbf p$$ and a [[plane]] $$\mathbf g$$. To possess the [[geometric property]], the components of $$\mathbf F$$ must satisfy the equation

:$$F_{px} F_{gx} + F_{py} F_{gy} + F_{pz} F_{gz} + F_{pw} F_{gw} = 0$$ ,

and this means that the [[point]] $$\mathbf p$$ must lie in the [[plane]] $$\mathbf g$$.

An element $$\mathbf x$$ is transformed by a flector $$\mathbf F$$ through the operation $$\mathbf x' = -\mathbf F \mathbin{\unicode{x27C7}} \mathbf x \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{F}}}$$, where $$\unicode{x27C7}$$ is the [[geometric antiproduct]].

The set of all flectors is the coset of the set $$\unicode{x1D544}$$ of all [[motors]] in a geometric algebra. As such, the set of all flectors is $$\{\mathbf Q \mathbin{\unicode{x27C7}} \mathbf F \mid \mathbf Q \in \unicode{x1D544}\}$$, where $$\mathbf F$$ is any fixed flector. In particular, since $$\mathbf e_4$$ is a flector (representing [[inversion]] through the origin), any flector can be written as $$\mathbf Q \mathbin{\unicode{x27C7}} \mathbf e_4$$ for some motor $$\mathbf Q$$.

== Norm ==

The [[bulk norm]] of a flector $$\mathbf F$$ is given by

:$$\left\Vert\mathbf F\right\Vert_\unicode{x25CF} = \sqrt{\mathbf F \mathbin{\unicode{x25CF}} \mathbf{\tilde F}} = \sqrt{F_{px}^2 + F_{py}^2 + F_{pz}^2 + F_{gw}^2}$$ ,

and its [[weight norm]] is given by

:$$\left\Vert\mathbf F\right\Vert_\unicode{x25CB} = \sqrt{\mathbf F \mathbin{\unicode{x25CB}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{F}}}\vphantom{\mathbf{\tilde F}}} = {\large\unicode{x1D7D9}}\sqrt{F_{gx}^2 + F_{gy}^2 + F_{gz}^2 + F_{pw}^2}$$ .

The [[geometric norm]] of a flector $$\mathbf F$$ is thus

:$$\widehat{\left\Vert\mathbf F\right\Vert} = \sqrt{\dfrac{F_{px}^2 + F_{py}^2 + F_{pz}^2 + F_{gw}^2}{F_{gx}^2 + F_{gy}^2 + F_{gz}^2 + F_{pw}^2}}$$ ,

and this is equal to half the distance that the origin is moved by the operator.

A flector is [[unitized]] when $$F_{gx}^2 + F_{gy}^2 + F_{gz}^2 + F_{pw}^2 = 1$$.

== Trigonometric Form ==

A general flector $$\mathbf F$$ can be expressed in terms of a [[unitized]] [[point]] $$\mathbf p$$ and a [[unitized]] [[plane]] $$\mathbf g$$ as

:$$\mathbf F = \mathbf p\sin\phi + \mathbf g\cos\phi$$ .

This can be interpreted as performing a rotoreflection consisting of a reflection through the plane $$\mathbf g$$ and a rotation by twice the angle $$\phi$$ about an axis perpendicular to $$\mathbf g$$ passing through the point $$\mathbf p$$. All combinations of a reflection, a rotation, and a translation, even if the original rotation axis is not perpendicular to the original reflection plane, can be formulated as a rotoreflection with respect to some plane.

== Factorization ==

Any [[unitized]] flector $$\mathbf F$$ for which $$F_{pw} \neq \pm1$$ can be factored into the product of a simple [[motor]] $$\mathbf Q$$ and a [[plane]] by calculating

:$$\mathbf Q = \dfrac{1}{\sqrt{1 - F_{pw}^2}}\left(\mathbf F \mathbin{\unicode{x27C7}} \mathbf{\underset{\Large\unicode{x7E}}{g}}\right)$$ ,

where the division unitizes the plane $$\mathbf g = F_{gx} \mathbf e_{423} + F_{gy} \mathbf e_{431} + F_{gz} \mathbf e_{412} + F_{gw} \mathbf e_{321}$$. The motor $$\mathbf Q$$ is then given by

:$$\mathbf Q = \dfrac{1}{\sqrt{1 - F_{pw}^2}}\left[-F_{gx} F_{pw} \mathbf e_{41} - F_{gy} F_{pw} \mathbf e_{42} - F_{gz} F_{pw} \mathbf e_{43} + (F_{gy} F_{pz} - F_{gz} F_{py})\mathbf e_{23} + (F_{gz} F_{px} - F_{gx} F_{pz})\mathbf e_{31} + (F_{gx} F_{py} - F_{gy} F_{px})\mathbf e_{12} + (1 - F_{pw}^2){\large\unicode{x1d7d9}}\right]$$ ,

which is always unitized. The original flector $$\mathbf F$$ can now be expressed as

:$$\mathbf F = \mathbf Q \mathbin{\unicode{x27C7}} \dfrac{\mathbf g}{\sqrt{1 - F_{pw}^2}}$$ .

== Conversion from Flector to Matrix ==

Given a specific [[Unitization | unitized]] flector $$\mathbf G$$, define the matrices

:$$\mathbf A = \begin{bmatrix}2(F_{gy}^2 + F_{gz}^2) - 1 & -2F_{gx}F_{gy} & -2F_{gz}F_{gx} & 2(F_{px}F_{pw} - F_{gx}F_{gw}) \\ -2F_{gx}F_{gy} & 2(F_{gz}^2 + F_{gx}^2) - 1 & -2F_{gy}F_{gz} & 2(F_{py}F_{pw} - F_{gy}F_{gw}) \\ -2F_{gz}F_{gx} & -2F_{gy}F_{gz} & 2(F_{gx}^2 + F_{gy}^2) - 1 & 2(F_{pz}F_{pw} - F_{gz}F_{gw}) \\ 0 & 0 & 0 & 1\end{bmatrix}$$

and

:$$\mathbf B = \begin{bmatrix}0 & 2F_{gz}F_{pw} & -2F_{gy}F_{pw} & 2(F_{gy}F_{pz} - F_{gz}F_{py}) \\ -2F_{gz}F_{pw} & 0 & 2F_{gx}F_{pw} & 2(F_{gz}F_{px} - F_{gx}F_{pz}) \\ 2F_{gy}F_{pw} & -2F_{gx}F_{pw} & 0 & 2(F_{gx}F_{py} - F_{gy}F_{px}) \\ 0 & 0 & 0 & 0\end{bmatrix}$$ .

Then the corresponding 4×4 matrix $$\mathbf M$$ that transforms a [[point]] $$\mathbf q$$, regarded as a column matrix, as $$\mathbf q' = \mathbf{Mq}$$ is given by

:$$\mathbf M = \mathbf A + \mathbf B$$ .

The inverse of $$\mathbf M$$, which transforms a [[plane]] $$\mathbf h$$, regarded as a row matrix, as $$\mathbf h' = \mathbf{hM^{-1}}$$ is given by

:$$\mathbf M^{-1} = \mathbf A - \mathbf B$$ .

== Conversion from Matrix to Flector ==

Let $$\mathbf M$$ be an orthogonal 4×4 matrix with determinant −1 having the form

:$$\mathbf M = \begin{bmatrix} M_{00} & M_{01} & M_{02} & M_{03} \\ M_{10} & M_{11} & M_{12} & M_{13} \\ M_{20} & M_{21} & M_{22} & M_{23} \\ 0 & 0 & 0 & 1\end{bmatrix}$$ .

Then, by equating the entries of $$\mathbf M$$ to the entries of $$\mathbf A + \mathbf B$$ from above, we have the following four relationships based on the diagonal entries of $$\mathbf M$$:

:$$1 - M_{00} + M_{11} + M_{22} = 4F_{gx}^2$$

:$$1 - M_{11} + M_{22} + M_{00} = 4F_{gy}^2$$

:$$1 - M_{22} + M_{00} + M_{11} = 4F_{gz}^2$$

:$$1 - M_{00} - M_{11} - M_{22} = 4(1 - F_{gx}^2 - F_{gy}^2 - F_{gz}^2) = 4F_{pw}^2$$

And we have the following six relationships based on the off-diagonal entries of $$\mathbf M$$:

:$$M_{21} + M_{12} = -4F_{gy}F_{gz}$$

:$$M_{02} + M_{20} = -4F_{gz}F_{gx}$$

:$$M_{10} + M_{01} = -4F_{gx}F_{gy}$$

:$$M_{21} - M_{12} = -4F_{gx}F_{pw}$$

:$$M_{02} - M_{20} = -4F_{gy}F_{pw}$$

:$$M_{10} - M_{01} = -4F_{gz}F_{pw}$$

If $$M_{00} + M_{11} + M_{22} \leq 0$$, then we calculate

:$$F_{pw} = \pm \dfrac{1}{2}\sqrt{1 - M_{00} - M_{11} - M_{22}}$$ ,

where either sign can be chosen. In this case, we know $$|F_{pw}|$$ is at least $$1/2$$, so we can safely divide by $$-4F_{pw}$$ in the last three off-diagonal relationships above to solve for $$F_{gx}$$, $$F_{gy}$$, and $$F_{gz}$$. Otherwise, if $$M_{00} + M_{11} + M_{22} > 0$$, then we select one of the first three diagonal relationships based on the smallest diagonal entry $$M_{00}$$, $$M_{11}$$, or $$M_{22}$$. After calculating $$F_{gx}$$, $$F_{gy}$$, or $$F_{gz}$$, we plug its value into two of the first three off-diagonal relationships to solve for the other two values of $$F_{gx}$$, $$F_{gy}$$, and $$F_{gz}$$. Finally, we plug it into one of the last three off-diagonal relationships to solve for $$F_{pw}$$.

Setting $$t_x = M_{03}$$, $$t_y = M_{13}$$, and $$t_z = M_{23}$$, the values of $$F_{px}$$, $$F_{py}$$, $$F_{pz}$$, and $$F_{gw}$$ are given by

$$F_{px} = \dfrac{1}{2}(F_{pw}t_x + F_{gz}t_y - F_{gy}t_z)$$

$$F_{py} = \dfrac{1}{2}(F_{pw}t_y + F_{gx}t_z - F_{gz}t_x)$$

$$F_{pz} = \dfrac{1}{2}(F_{pw}t_z + F_{gy}t_x - F_{gx}t_y)$$

$$F_{gw} = -\dfrac{1}{2}(F_{gx}t_x + F_{gy}t_y + F_{gz}t_z)$$ .

== Flector Transformations ==

[[Point | Points]], [[Line | lines]], and [[Plane | planes]] are transformed by a [[unitized]] flector $$\mathbf F$$ as shown in the following table.

{| class="wikitable"
! Type || Transformation
|-
| style="padding: 12px;" | [[Point]]

$$\mathbf q = q_x \mathbf e_1 + q_y \mathbf e_2 + q_z \mathbf e_3 + q_w \mathbf e_4$$
| style="padding: 12px;" | $$\begin{split}\mathbf F \mathbin{\unicode{x27C7}} \mathbf q \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{F}}} =\, &\left[(2F_{gy}^2 + 2F_{gz}^2 - 1)q_x + 2(F_{gz}F_{pw} - F_{gx}F_{gy})q_y - 2(F_{gy}F_{pw} + F_{gz}F_{gx})q_z + 2(F_{gy}F_{pz} - F_{gz}F_{py} + F_{px}F_{pw} - F_{gx}F_{gw})q_w\right]\mathbf e_1 \\ +\, &\left[(2F_{gz}^2 + 2F_{gx}^2 - 1)q_y + 2(F_{gx}F_{pw} - F_{gy}F_{gz})q_z - 2(F_{gz}F_{pw} + F_{gx}F_{gy})q_x + 2(F_{gz}F_{px} - F_{gx}F_{pz} + F_{py}F_{pw} - F_{gy}F_{gw})q_w\right]\mathbf e_2 \\ +\, &\left[(2F_{gx}^2 + 2F_{gy}^2 - 1)q_z + 2(F_{gy}F_{pw} - F_{gz}F_{gx})q_x - 2(F_{gx}F_{pw} + F_{gy}F_{gz})q_y + 2(F_{gx}F_{py} - F_{gy}F_{px} + F_{pz}F_{pw} - F_{gz}F_{gw})q_w\right]\mathbf e_3 \\ +\, &q_w\mathbf e_4\end{split}$$
|-
| style="padding: 12px;" | [[Line]]

$$\begin{split}\boldsymbol l =\, &l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} \\ +\, &l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}\end{split}$$
| style="padding: 12px;" | $$\begin{split}\mathbf F \mathbin{\unicode{x27C7}} \boldsymbol l \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{F}}} =\, &\left[(1 - 2F_{gy}^2 - 2F_{gz}^2)l_{vx} + 2(F_{gx}F_{gy} - F_{gz}F_{pw})l_{vy} + 2(F_{gz}F_{gx} + F_{gy}F_{pw})l_{vz}\right]\mathbf e_{41} \\ +\, &\left[(1 - 2F_{gz}^2 - 2F_{gx}^2)l_{vy} + 2(F_{gy}F_{gz} - F_{gx}F_{pw})l_{vz} + 2(F_{gx}F_{gy} + F_{gz}F_{pw})l_{vx}\right]\mathbf e_{42} \\ +\, &\left[(1 - 2F_{gx}^2 - 2F_{gy}^2)l_{vz} + 2(F_{gz}F_{gx} - F_{gy}F_{pw})l_{vx} + 2(F_{gy}F_{gz} + F_{gx}F_{pw})l_{vy}\right]\mathbf e_{43} \\ +\, &\left[-4(F_{gy}F_{py} + F_{gz}F_{pz})l_{vx} + 2(F_{gx}F_{py} + F_{gy}F_{px})l_{vy} + 2(F_{gx}F_{pz} + F_{gz}F_{px} + F_{gy}F_{gw} + F_{py}F_{pw})l_{vz} + (2F_{gy}^2 + 2F_{gz}^2 - 1)l_{mx} + 2(F_{gz}F_{pw} - F_{gx}F_{gy})l_{my} - 2(F_{gy}F_{pw} + F_{gz}F_{gx})l_{mz}\right]\mathbf e_{23} \\ +\, &\left[-4(F_{gz}F_{pz} + F_{gx}F_{px})l_{vy} + 2(F_{gy}F_{pz} + F_{gz}F_{py})l_{vz} + 2(F_{gy}F_{px} + F_{gx}F_{py} + F_{gz}F_{gw} + F_{pz}F_{pw})l_{vx} + (2F_{gz}^2 + 2F_{gx}^2 - 1)l_{my} + 2(F_{gx}F_{pw} - F_{gy}F_{gz})l_{mz} - 2(F_{gz}F_{pw} + F_{gx}F_{gy})l_{mx}\right]\mathbf e_{31} \\ +\, &\left[-4(F_{gx}F_{px} + F_{gy}F_{py})l_{vz} + 2(F_{gz}F_{px} + F_{gx}F_{pz})l_{vx} + 2(F_{gz}F_{py} + F_{gy}F_{pz} + F_{gx}F_{gw} + F_{px}F_{pw})l_{vy} + (2F_{gx}^2 + 2F_{gy}^2 - 1)l_{mz} + 2(F_{gy}F_{pw} - F_{gz}F_{gx})l_{mx} - 2(F_{gx}F_{pw} + F_{gy}F_{gz})l_{my}\right]\mathbf e_{12}\end{split}$$
|-
| style="padding: 12px;" | [[Plane]]

$$\mathbf h = h_x \mathbf e_{423} + h_y \mathbf e_{431} + h_z \mathbf e_{412} + h_w \mathbf e_{321}$$
| style="padding: 12px;" | $$\begin{split}\mathbf F \mathbin{\unicode{x27C7}} \mathbf h \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{F}}} =\, &\left[(1 - 2F_{gy}^2 - 2F_{gz}^2)h_x + 2(F_{gx}F_{gy} - F_{gz}F_{pw})h_y + 2(F_{gz}F_{gx} + F_{gy}F_{pw})h_z\right]\mathbf e_{423} \\ +\, &\left[(1 - 2F_{gz}^2 - 2F_{gx}^2)h_y + 2(F_{gy}F_{gz} - F_{gx}F_{pw})h_z + 2(F_{gx}F_{gy} + F_{gz}F_{pw})h_x\right]\mathbf e_{431} \\ +\, &\left[(1 - 2F_{gx}^2 - 2F_{gy}^2)h_z + 2(F_{gz}F_{gx} - F_{gy}F_{pw})h_x + 2(F_{gy}F_{gz} + F_{gx}F_{pw})h_y\right]\mathbf e_{412} \\ +\, &\left[2(F_{gy}F_{pz} - F_{gz}F_{py} + F_{gx}F_{gw} - F_{px}F_{pw})h_x + 2(F_{gz}F_{px} - F_{gx}F_{pz} + F_{gy}F_{gw} - F_{py}F_{pw})h_y + 2(F_{gx}F_{py} - F_{gy}F_{px} + F_{gz}F_{gw} - F_{pz}F_{pw})h_z - h_w\right]\mathbf e_{321}\end{split}$$
|}

== See Also ==

* [[Motor]]
* [[Reflection]]
* [[Inversion]]
* [[Transflection]]