Complement translation and Errata: Difference between pages

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__NOTOC__
This is the errata page for the ''Projective Geometric Algebra Illuminated''. To find out which printing you have, look on the copyright page.
A ''reciprocal translation'' is a proper isometry of reciprocal Euclidean space.


The specific kind of [[reciprocal motor]]
== Second Printing ==


:$$\mathbf T = t_x \mathbf e_{41} + t_y \mathbf e_{42} + t_z \mathbf e_{43} + \mathbf 1$$
There are no known errata in the second printing. If you have found something, please email the author (lengyel@terathon.com).


performs a perspective projection in the direction of $$\mathbf t = (t_x, t_y, t_z)$$ with the focal length given by
== First Printing ==


:$$g = \dfrac{1}{2\Vert \mathbf t \Vert}$$ .
* '''Page 84.''' In Table 2.16, both bulk duals of $$\mathbf e_4$$ should be 0. The table should look like the one on the wiki page for [[duals]].


== Example ==
* '''Page 156.''' The number of multiply-adds required for transforming a line with a motor did not include the work needed to calculate $$\mathbf a$$, $$\mathbf b$$, and $$\mathbf c$$. Each cross product adds 6 multiply-adds, so the correct total for transforming a line is 54.


The left image below shows the flow field in the ''x''-''z'' plane for the translation $$\mathbf T = -\frac{1}{2} \mathbf e_{31} + {\large\unicode{x1d7d9}}$$. The right image shows the flow field in the ''x''-''z'' plane for the reciprocal translation $$\mathbf T = \frac{1}{2} \mathbf e_{42} + \mathbf 1$$. The yellow line is fixed as a whole, but points on it move to other locations on the line. All points with $$z = 0$$, represented by the blue plane, are fixed. The white plane at $$z = -1$$ represents the division between regions where the signs of projected $$z$$ coordinates are positive and negative.
* '''Page 156.''' Similarly, the number of multiply-adds required for transforming a plane with a motor did not include the work needed to calculate $$\mathbf a$$. The correct total for transforming a plane is 35.
 
[[Image:Translation.svg|480px]]
[[Image:DualTranslation.svg|480px]]
 
== Calculation ==
 
The exact reciprocal translation calculations for points, lines, and planes are shown in the following table.
 
{| class="wikitable"
! Type || Reciprocal Translation
|-
| style="padding: 12px;" | [[Point]]
 
$$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$
| 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]]
 
$$\begin{split}\mathbf L =\, &v_x \mathbf e_{41} + v_y \mathbf e_{42} + v_z \mathbf e_{43} \\ +\, &m_x \mathbf e_{23} + m_y \mathbf e_{31} + m_z \mathbf e_{12}\end{split}$$
| 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]]
 
$$\mathbf f = f_x \mathbf e_{234} + f_y \mathbf e_{314} + f_z \mathbf e_{124} + f_w \mathbf e_{321}$$
| style="padding: 12px;" | $$\mathbf T \mathbin{\unicode{x27D1}} \mathbf f \mathbin{\unicode{x27D1}} \mathbf{\tilde T} = (f_x - 2t_xf_w) \mathbf e_{234} + (f_y - 2t_yf_w) \mathbf e_{314} + (f_z - 2t_zf_w) \mathbf e_{124} + f_w \mathbf e_{321}$$
|}
 
== Reciprocal Translation to Horizon ==
 
A plane $$\mathbf f = f_x \mathbf e_{234} + f_y \mathbf e_{314} + f_z \mathbf e_{124} + f_w \mathbf e_{321}$$ is reciprocal translated to the horizon by the operator
 
:$$\mathbf T = \dfrac{f_{x\vphantom{y}}}{2f_w} \mathbf e_{41} + \dfrac{f_y}{2f_w} \mathbf e_{42} + \dfrac{f_{z\vphantom{y}}}{2f_w} \mathbf e_{43} + \mathbf 1$$ .
 
== See Also ==
 
* [[Translation]]
* [[Reciprocal rotation]]
* [[Reciprocal reflection]]

Revision as of 16:50, 13 May 2024

This is the errata page for the Projective Geometric Algebra Illuminated. To find out which printing you have, look on the copyright page.

Second Printing

There are no known errata in the second printing. If you have found something, please email the author (lengyel@terathon.com).

First Printing

  • Page 84. In Table 2.16, both bulk duals of $$\mathbf e_4$$ should be 0. The table should look like the one on the wiki page for duals.
  • Page 156. The number of multiply-adds required for transforming a line with a motor did not include the work needed to calculate $$\mathbf a$$, $$\mathbf b$$, and $$\mathbf c$$. Each cross product adds 6 multiply-adds, so the correct total for transforming a line is 54.
  • Page 156. Similarly, the number of multiply-adds required for transforming a plane with a motor did not include the work needed to calculate $$\mathbf a$$. The correct total for transforming a plane is 35.
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