Difference between pages "Point" and "Dual translation"

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m (Eric Lengyel moved page Antitranslation to Dual translation without leaving a redirect)
 
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[[Image:point.svg|400px|thumb|right|'''Figure 1.''' A point is the intersection of a 4D vector with the 3D subspace where $$w = 1$$.]]
An ''antitranslation'' is a proper isometry of dual Euclidean space.
In the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$, a ''point'' $$\mathbf p$$ is a vector having the general form


:$$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$ .
The specific kind of [[antimotor]]


All points possess the [[geometric property]].
:$$\mathbf T = t_x \mathbf e_{41} + t_y \mathbf e_{42} + t_z \mathbf e_{43} + 1$$


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 $$p_w^2 = 1$$.
performs a perspective projection. The exact antitranslation calculations for points, lines, and planes are shown in the following table.


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


A [[translation]] operator $$\mathbf T$$ that moves a point $$\mathbf p$$ to the origin is given by
$$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$
| style="padding: 12px;" | $$\mathbf T \mathbin{\unicode{x27D1}} \mathbf p \mathbin{\unicode{x27D1}} \mathbf{\tilde T} = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + (2t_xp_x + 2t_yp_y + 2t_zp_z + p_w) \mathbf e_4$$
|-
| style="padding: 12px;" | [[Line]]


:$$\mathbf T = \dfrac{1}{2}\underline{\mathbf p} \vee \mathbf e_{321} + \underline{\mathbf p} \wedge \mathbf e_4 = -\dfrac{p_{x\vphantom{y}}}{2} \mathbf e_{23} - \dfrac{p_y}{2} \mathbf e_{31} - \dfrac{p_{z\vphantom{y}}}{2} \mathbf e_{12} + p_w {\large\unicode{x1D7D9}}$$ .
$$\begin{split}\mathbf L =\, &v_x \mathbf e_{41} + v_y \mathbf e_{42} + v_z \mathbf e_{43} \\ +\, &m_x \mathbf e_{23} + m_y \mathbf e_{31} + m_z \mathbf e_{12}\end{split}$$
| style="padding: 12px;" | $$\mathbf T \mathbin{\unicode{x27D1}} \mathbf L \mathbin{\unicode{x27D1}} \mathbf{\tilde T} = (v_x - 2t_ym_z + 2t_zm_y)\mathbf e_{41} + (v_y - 2t_zm_x + 2t_xm_z)\mathbf e_{42} + (v_z - 2t_xm_y - 2t_ym_x)\mathbf e_{43} + m_x \mathbf e_{23} + m_y \mathbf e_{31} + m_z \mathbf e_{12}$$
|-
| style="padding: 12px;" | [[Plane]]


== Points at Infinity ==
$$\mathbf f = f_x \mathbf e_{234} + f_y \mathbf e_{314} + f_z \mathbf e_{124} + f_w \mathbf e_{321}$$
| style="padding: 12px;" | $$\mathbf T \mathbin{\unicode{x27D1}} \mathbf f \mathbin{\unicode{x27D1}} \mathbf{\tilde T} = (f_x - 2t_xf_w) \mathbf e_{234} + (f_y - 2t_yf_w) \mathbf e_{314} + (f_z - 2t_zf_w) \mathbf e_{124} + f_w \mathbf e_{321}$$
|}


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 ==
== See Also ==


* [[Line]]
* [[Antirotation]]
* [[Plane]]
* [[Antireflection]]

Revision as of 05:21, 19 June 2022

An antitranslation is a proper isometry of dual Euclidean space.

The specific kind of antimotor

$$\mathbf T = t_x \mathbf e_{41} + t_y \mathbf e_{42} + t_z \mathbf e_{43} + 1$$

performs a perspective projection. The exact antitranslation calculations for points, lines, and planes are shown in the following table.

Type Antitranslation
Point

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

$$\mathbf T \mathbin{\unicode{x27D1}} \mathbf p \mathbin{\unicode{x27D1}} \mathbf{\tilde T} = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + (2t_xp_x + 2t_yp_y + 2t_zp_z + p_w) \mathbf e_4$$
Line

$$\begin{split}\mathbf L =\, &v_x \mathbf e_{41} + v_y \mathbf e_{42} + v_z \mathbf e_{43} \\ +\, &m_x \mathbf e_{23} + m_y \mathbf e_{31} + m_z \mathbf e_{12}\end{split}$$

$$\mathbf T \mathbin{\unicode{x27D1}} \mathbf L \mathbin{\unicode{x27D1}} \mathbf{\tilde T} = (v_x - 2t_ym_z + 2t_zm_y)\mathbf e_{41} + (v_y - 2t_zm_x + 2t_xm_z)\mathbf e_{42} + (v_z - 2t_xm_y - 2t_ym_x)\mathbf e_{43} + m_x \mathbf e_{23} + m_y \mathbf e_{31} + m_z \mathbf e_{12}$$
Plane

$$\mathbf f = f_x \mathbf e_{234} + f_y \mathbf e_{314} + f_z \mathbf e_{124} + f_w \mathbf e_{321}$$

$$\mathbf T \mathbin{\unicode{x27D1}} \mathbf f \mathbin{\unicode{x27D1}} \mathbf{\tilde T} = (f_x - 2t_xf_w) \mathbf e_{234} + (f_y - 2t_yf_w) \mathbf e_{314} + (f_z - 2t_zf_w) \mathbf e_{124} + f_w \mathbf e_{321}$$

See Also

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