Point - Revision history

Eric Lengyel at 00:49, 22 December 2024

2024-12-22T00:49:46Z

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

Eric Lengyel at 23:49, 13 April 2024

2024-04-13T23:49:41Z

← Older revision		Revision as of 23:49, 13 April 2024
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	<br clear="right" />		<br clear="right" />
			== In the Book ==

			* Homogeneous points are discussed in Section 2.4.1.

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

	* [[Line]]		* [[Line]]
	* [[Plane]]		* [[Plane]]

Eric Lengyel at 06:02, 5 April 2024

2024-04-05T06:02:32Z

← Older revision		Revision as of 06:02, 5 April 2024
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	:$$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$ .		:$$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$ .

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

	The [[bulk]] of a point is given by its $$x$$, $$y$$, and $$z$$ coordinates, and the [[weight]] of a point is given by its $$w$$ coordinate. A point is [[unitized]] when $$p_w^2 = 1$$.		The [[bulk]] of a point is given by its $$x$$, $$y$$, and $$z$$ coordinates, and the [[weight]] of a point is given by its $$w$$ coordinate. A point is [[unitized]] when $$p_w^2 = 1$$.

Eric Lengyel: Created page with "'''Figure 1.''' A point is the intersection of a 4D vector with the 3D subspace where $$w = 1$$. In the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$, a ''point'' $$\mathbf p$$ is a vector having the general form :$$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$ . All points possess the geometric property. The bulk of a point is given by its $$x$$, $$y$$, and $$z$$ coordinates, and..."

2023-07-15T06:35:28Z

Created page with "400px|thumb|right|'''Figure 1.''' A point is the intersection of a 4D vector with the 3D subspace where $$w = 1$$. In the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$, a ''point'' $$\mathbf p$$ is a vector having the general form :$$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$ . All points possess the geometric property. The bulk of a point is given by its $$x$$, $$y$$, and $$z$$ coordinates, and..."

New page

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

:$$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$ .

All points possess the [[geometric property]].

The [[bulk]] of a point is given by its $$x$$, $$y$$, and $$z$$ coordinates, and the [[weight]] of a point is given by its $$w$$ coordinate. A point is [[unitized]] when $$p_w^2 = 1$$.

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

A [[translation]] operator $$\mathbf T$$ that moves a point $$\mathbf p$$ to the origin is given by

:$$\mathbf T = \underline{\mathbf p} \vee \mathbf e_{321} + 2\mathbf p \wedge \mathbf e_{321} = -p_{x\vphantom{y}} \mathbf e_{23} - p_y \mathbf e_{31} - p_{z\vphantom{y}} \mathbf e_{12} + 2p_w {\large\unicode{x1D7D9}}$$ .

== Points at Infinity ==

If the weight of a point is zero (i.e., its $$w$$ coordinate is zero), then the point is contained in the horizon infinitely far away in the direction $$(x, y, z)$$, and it cannot be unitized. A point with zero weight can also be interpreted as a direction vector, and it is normalized to unit length by dividing by its [[bulk norm]].

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

* [[Line]]
* [[Plane]]