Geometric constraint and Errata: Difference between pages

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An element $$\mathbf x$$ of a geometric algebra possesses the ''geometric property'' if and only if the [[geometric product]] between $$\mathbf u$$ and its own reverse is a scalar, which is given by the [[dot product]], and the [[geometric antiproduct]] between $$\mathbf u$$ and its own antireverse is an antiscalar, which is given by the [[antidot product]]. That is,
This is the errata page for the ''Projective Geometric Algebra Illuminated''. To find out which printing you have, look on the copyright page.


:$$\mathbf u \mathbin{\unicode{x27D1}} \mathbf{\tilde u} = \mathbf u \mathbin{\unicode{x25CF}} \mathbf u$$
== Second Printing ==


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


:$$\mathbf u \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{u}}} = \mathbf u \mathbin{\unicode{x25CB}} \mathbf u$$ .
== First Printing ==


The set of all elements possessing the geometric property is closed under both the [[geometric product]] and [[geometric antiproduct]].
* '''Page 84.''' In Table 2.16, both bulk duals of $$\mathbf e_4$$ should be 0.


The following table lists the requirements that must be satisfied for the main types in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ to possess the geometric property. Points and planes do not have any requirements—they all possess the geometric property.
* '''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.


{| class="wikitable"
* '''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.
! Type !! Definition !! Requirement
|-
| style="padding: 12px;" | [[Magnitude]]
| style="padding: 12px;" | $$\mathbf z = x \mathbf 1 + y {\large\unicode{x1d7d9}}$$
| style="padding: 12px;" | $$xy = 0$$
|-
| style="padding: 12px;" | [[Point]]
| style="padding: 12px;" | $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$
| style="padding: 12px;" | —
|-
| style="padding: 12px;" | [[Line]]
| style="padding: 12px;" | $$\boldsymbol l = l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} + l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}$$
| style="padding: 12px;" | $$l_{vx} l_{mx} + l_{vy} l_{my} + l_{vz} l_{mz} = 0$$
|-
| style="padding: 12px;" | [[Plane]]
| style="padding: 12px;" | $$\mathbf g = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + g_w \mathbf e_{321}$$
| style="padding: 12px;" | —
|-
| style="padding: 12px;" | [[Motor]]
| style="padding: 12px;" | $$\mathbf Q = Q_{vx} \mathbf e_{41} + Q_{vy} \mathbf e_{42} + Q_{vz} \mathbf e_{43} + Q_{vw} {\large\unicode{x1d7d9}} + Q_{mx} \mathbf e_{23} + Q_{my} \mathbf e_{31} + Q_{mz} \mathbf e_{12} + Q_{mw} \mathbf 1$$
| style="padding: 12px;" | $$Q_{vx} Q_{mx} + Q_{vy} Q_{my} + Q_{vz} Q_{mz} + Q_{vw} Q_{mw} = 0$$
|-
| style="padding: 12px;" | [[Flector]]
| style="padding: 12px;" | $$\mathbf F = F_{px} \mathbf e_1 + F_{py} \mathbf e_2 + F_{pz} \mathbf e_3 + F_{pw} \mathbf e_4 + F_{gx} \mathbf e_{423} + F_{gy} \mathbf e_{431} + F_{gz} \mathbf e_{412} + F_{gw} \mathbf e_{321}$$
| style="padding: 12px;" | $$F_{px} F_{gx} + F_{py} F_{gy} + F_{pz} F_{gz} + F_{pw} F_{gw} = 0$$
|}
 
== See Also ==
 
* [[Geometric norm]]

Revision as of 20:30, 12 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.
  • 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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