Plane - Revision history

Eric Lengyel at 00:49, 22 December 2024

2024-12-22T00:49:53Z

← Older revision		Revision as of 00:49, 22 December 2024
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			__NOTOC__
	[[Image:plane.svg\|400px\|thumb\|right\|'''Figure 1.''' A plane is the intersection of a 4D trivector with the 3D subspace where $$w = 1$$.]]		[[Image:plane.svg\|400px\|thumb\|right\|'''Figure 1.''' A plane is the intersection of a 4D trivector with the 3D subspace where $$w = 1$$.]]
	In the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$, a ''plane'' $$\mathbf g$$ is a trivector having the general form		In the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$, a ''plane'' $$\mathbf g$$ is a trivector having the general form

Eric Lengyel at 01:16, 8 July 2024

2024-07-08T01:16:47Z

← Older revision		Revision as of 01:16, 8 July 2024
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	When used as an operator in the sandwich with the [[geometric antiproduct]], a plane is a specific kind of [[flector]] that performs a [[reflection]] through itself.		When used as an operator in the sandwich with the [[geometric antiproduct]], a plane is a specific kind of [[flector]] that performs a [[reflection]] through itself.

	A [[~~dual~~ translation]] operator $$\mathbf T$$ that moves a plane $$\mathbf g$$ to the horizon is given by		A [[complement translation]] operator $$\mathbf T$$ that moves a plane $$\mathbf g$$ to the horizon is given by

	:$$\mathbf T = \underline{\mathbf g} \wedge \mathbf e_{4} + 2\mathbf g \vee \mathbf e_4 = g_{x\vphantom{y}} \mathbf e_{41} + g_y \mathbf e_{42} + g_{z\vphantom{y}} \mathbf e_{43} + 2g_w$$ .		:$$\mathbf T = \underline{\mathbf g} \wedge \mathbf e_{4} + 2\mathbf g \vee \mathbf e_4 = g_{x\vphantom{y}} \mathbf e_{41} + g_y \mathbf e_{42} + g_{z\vphantom{y}} \mathbf e_{43} + 2g_w$$ .

Eric Lengyel at 23:50, 13 April 2024

2024-04-13T23:50:25Z

← Older revision		Revision as of 23:50, 13 April 2024
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	<br clear="right" />		<br clear="right" />
			== In the Book ==

			* Homogeneous planes are discussed in Section 2.4.3.

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

	* [[Point]]		* [[Point]]
	* [[Line]]		* [[Line]]

Eric Lengyel at 06:03, 5 April 2024

2024-04-05T06:03:06Z

← Older revision		Revision as of 06:03, 5 April 2024
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	:$$\mathbf g = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + g_w \mathbf e_{321}$$ .		:$$\mathbf g = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + g_w \mathbf e_{321}$$ .

	All planes possess the [[geometric ~~property~~]].		All planes possess the [[geometric constraint]].

	The [[bulk]] of a plane is given by its $$w$$ coordinate, and the [[weight]] of a plane is given by its $$x$$, $$y$$, and $$z$$ coordinates. A plane is [[unitized]] when $$g_x^2 + g_y^2 + g_z^2 = 1$$. The [[attitude]] of a plane is the bivector $$g_x \mathbf e_{23} + g_y \mathbf e_{31} + g_z \mathbf e_{12}$$ corresponding to its normal.		The [[bulk]] of a plane is given by its $$w$$ coordinate, and the [[weight]] of a plane is given by its $$x$$, $$y$$, and $$z$$ coordinates. A plane is [[unitized]] when $$g_x^2 + g_y^2 + g_z^2 = 1$$. The [[attitude]] of a plane is the bivector $$g_x \mathbf e_{23} + g_y \mathbf e_{31} + g_z \mathbf e_{12}$$ corresponding to its normal.

Eric Lengyel: Created page with "'''Figure 1.''' A plane is the intersection of a 4D trivector with the 3D subspace where $$w = 1$$. In the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$, a ''plane'' $$\mathbf g$$ is a trivector having the general form :$$\mathbf g = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + g_w \mathbf e_{321}$$ . All planes possess the geometric property. The bulk of a plane is given by its $$w$$ coordinate, a..."

2023-07-15T06:16:58Z

Created page with "400px|thumb|right|'''Figure 1.''' A plane is the intersection of a 4D trivector with the 3D subspace where $$w = 1$$. In the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$, a ''plane'' $$\mathbf g$$ is a trivector having the general form :$$\mathbf g = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + g_w \mathbf e_{321}$$ . All planes possess the geometric property. The bulk of a plane is given by its $$w$$ coordinate, a..."

New page

[[Image:plane.svg|400px|thumb|right|'''Figure 1.''' A plane is the intersection of a 4D trivector with the 3D subspace where $$w = 1$$.]]
In the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$, a ''plane'' $$\mathbf g$$ is a trivector having the general form

:$$\mathbf g = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + g_w \mathbf e_{321}$$ .

All planes possess the [[geometric property]].

The [[bulk]] of a plane is given by its $$w$$ coordinate, and the [[weight]] of a plane is given by its $$x$$, $$y$$, and $$z$$ coordinates. A plane is [[unitized]] when $$g_x^2 + g_y^2 + g_z^2 = 1$$. The [[attitude]] of a plane is the bivector $$g_x \mathbf e_{23} + g_y \mathbf e_{31} + g_z \mathbf e_{12}$$ corresponding to its normal.

When used as an operator in the sandwich with the [[geometric antiproduct]], a plane is a specific kind of [[flector]] that performs a [[reflection]] through itself.

A [[dual translation]] operator $$\mathbf T$$ that moves a plane $$\mathbf g$$ to the horizon is given by

:$$\mathbf T = \underline{\mathbf g} \wedge \mathbf e_{4} + 2\mathbf g \vee \mathbf e_4 = g_{x\vphantom{y}} \mathbf e_{41} + g_y \mathbf e_{42} + g_{z\vphantom{y}} \mathbf e_{43} + 2g_w$$ .

== Plane at Infinity ==

If the weight of a plane is zero (i.e., its $$x$$, $$y$$, and $$z$$ coordinates are all zero), then the plane lies at infinity in all directions. Such a plane is normalized when $$g_w = \pm 1$$. This is the ''horizon'' of three-dimensional space.

<br clear="right" />
== See Also ==

* [[Point]]
* [[Line]]