Quaternion and Geometric constraint: Difference between pages

From Rigid Geometric Algebra
(Difference between pages)
Jump to navigation Jump to search
(Created page with "__NOTOC__ A ''quaternion'' is an operator that performs a rotation about the origin in 3D space. Conventionally, a quaternion $$\mathbf q$$ is written as :$$\mathbf q = q_w + q_x \mathbf i + q_y \mathbf j + q_z \mathbf k$$ , where the "imaginary" units $$\mathbf i$$, $$\mathbf j$$, and $$\mathbf k$$ all square to $$-1$$ and multiply according to the rules :$$\mathbf{ij} = -\mathbf{ji} = \mathbf k$$ :$$\mathbf{jk} = -\mathbf{kj} = \mathbf i$$ :$$\mathbf{ki} = -\mathbf{...")
 
No edit summary
 
Line 1: Line 1:
__NOTOC__
An element $$\mathbf x$$ of a geometric algebra possesses the ''geometric constraint'' 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,
A ''quaternion'' is an operator that performs a rotation about the origin in 3D space. Conventionally, a quaternion $$\mathbf q$$ is written as


:$$\mathbf q = q_w + q_x \mathbf i + q_y \mathbf j + q_z \mathbf k$$ ,
:$$\mathbf u \mathbin{\unicode{x27D1}} \mathbf{\tilde u} = \mathbf u \mathbin{\unicode{x25CF}} \mathbf u$$


where the "imaginary" units $$\mathbf i$$, $$\mathbf j$$, and $$\mathbf k$$ all square to $$-1$$ and multiply according to the rules
and


:$$\mathbf{ij} = -\mathbf{ji} = \mathbf k$$
:$$\mathbf u \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{u}}} = \mathbf u \mathbin{\unicode{x25CB}} \mathbf u$$ .
:$$\mathbf{jk} = -\mathbf{kj} = \mathbf i$$
:$$\mathbf{ki} = -\mathbf{ik} = \mathbf j$$ .


A ''unit quaternion'' is one for which $$q_w^2 + q_x^2 + q_y^2 + q_z^2 = 1$$.
The set of all elements satisfying the geometric constraint is closed under both the [[geometric product]] and [[geometric antiproduct]].


== Quaternions in 4D Rigid Geometric Algebra ==
The following table lists the requirements that must be satisfied for the main types in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ due to the geometric constraint. Points and planes do not have any requirements—they have no constraints.


Because quaternions keep the origin fixed, they are part of the group SO(3) where the special Euclidean group SE(3) and reciprocal special Euclidean group RSE(3) intersect. Consequently, the quaternions have two different representations in the four-dimensional rigid geometric algebra $$\mathcal G_{3,0,1}$$.
{| class="wikitable"
! 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;" | $$F_{px} F_{gx} + F_{py} F_{gy} + F_{pz} F_{gz} + F_{pw} F_{gw} = 0$$
|}


=== Quaternions as Motors ===
== In the Book ==


First, the quaternions are exactly the subset of [[motors]] that perform pure rotations about the origin without any translation. In this case, the units $$\mathbf i$$, $$\mathbf j$$, and $$\mathbf k$$ are identified as
* The geometric constraint is discussed in Section 3.4.3.
 
:$$\mathbf i = \mathbf e_{41}$$
:$$\mathbf j = \mathbf e_{42}$$
:$$\mathbf k = \mathbf e_{43}$$ .
 
A quaternion can then be written as
 
:$$\mathbf q = q_x \mathbf e_{41} + q_y \mathbf e_{42} + q_z \mathbf e_{43} + q_w {\large\unicode{x1D7D9}}$$ ,
 
and any object $$\mathbf x$$ (such as a [[point]], [[line]], or [[plane]]) is rotated about the origin through the sandwich product
 
:$$\mathbf x' = \mathbf q \mathbin{\unicode{x27C7}} \mathbf x \mathbin{\unicode{x27C7}} \mathbf{\underset{\Large\unicode{x7E}}{q}}$$ ,
 
using the [[geometric antiproduct]].
 
A unit quaternion can also be written as
 
:$$\mathbf q = \mathbf a \sin\phi + {\large\unicode{x1D7D9}}\cos\phi$$ ,
 
where $$\mathbf a = a_x \mathbf e_{41} + a_y \mathbf e_{42} + a_z \mathbf e_{43}$$ is a unit bivector representing the axis of rotation, and $$\phi$$ is half the angle of rotation.
 
=== Quaternions as Dual Motors ===
 
Second, the quaternions are exactly the subset of [[dual motors]] for which the directrix lies in the horizon. In this case, the units $$\mathbf i$$, $$\mathbf j$$, and $$\mathbf k$$ are identified as
 
:$$\mathbf i = -\mathbf e_{23}$$
:$$\mathbf j = -\mathbf e_{31}$$
:$$\mathbf k = -\mathbf e_{12}$$ .
 
A quaternion can then be written as
 
:$$\mathbf q = -q_x \mathbf e_{23} - q_y \mathbf e_{31} - q_z \mathbf e_{12} + q_w \mathbf 1$$ ,
 
and any object $$\mathbf x$$ (such as a [[point]], [[line]], or [[plane]]) is rotated about the origin through the sandwich product
 
:$$\mathbf x' = \mathbf q \mathbin{\unicode{x27D1}} \mathbf x \mathbin{\unicode{x27D1}} \mathbf{\tilde q}$$ ,
 
using the [[geometric product]].
 
A unit quaternion can also be written as
 
:$$\mathbf q = -\mathbf a \sin\phi + \mathbf 1\cos\phi$$ ,
 
where $$\mathbf a = a_x \mathbf e_{23} + a_y \mathbf e_{31} + a_z \mathbf e_{12}$$ is a unit bivector representing the axis of rotation, and $$\phi$$ is half the angle of rotation.


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


* [[Motor]]
* [[Geometric norm]]
* [[Flector]]

Latest revision as of 23:33, 13 April 2024

An element $$\mathbf x$$ of a geometric algebra possesses the geometric constraint 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,

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

and

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

The set of all elements satisfying the geometric constraint is closed under both the geometric product and geometric antiproduct.

The following table lists the requirements that must be satisfied for the main types in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ due to the geometric constraint. Points and planes do not have any requirements—they have no constraints.

Type Definition Requirement
Magnitude $$\mathbf z = x \mathbf 1 + y {\large\unicode{x1d7d9}}$$ $$xy = 0$$
Point $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$ —
Line $$\boldsymbol l = l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} + l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}$$ $$l_{vx} l_{mx} + l_{vy} l_{my} + l_{vz} l_{mz} = 0$$
Plane $$\mathbf g = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + g_w \mathbf e_{321}$$ —
Motor $$\mathbf Q = Q_{vx} \mathbf e_{41} + Q_{vy} \mathbf e_{42} + Q_{vz} \mathbf e_{43} + Q_{vw} {\large\unicode{x1d7d9}} + Q_{mx} \mathbf e_{23} + Q_{my} \mathbf e_{31} + Q_{mz} \mathbf e_{12} + Q_{mw} \mathbf 1$$ $$Q_{vx} Q_{mx} + Q_{vy} Q_{my} + Q_{vz} Q_{mz} + Q_{vw} Q_{mw} = 0$$
Flector $$\mathbf F = F_{px} \mathbf e_1 + F_{py} \mathbf e_2 + F_{pz} \mathbf e_3 + F_{pw} \mathbf e_4 + F_{gx} \mathbf e_{423} + F_{gy} \mathbf e_{431} + F_{gz} \mathbf e_{412} + F_{gw} \mathbf e_{321}$$ $$F_{px} F_{gx} + F_{py} F_{gy} + F_{pz} F_{gz} + F_{pw} F_{gw} = 0$$

In the Book

  • The geometric constraint is discussed in Section 3.4.3.

See Also

Retrieved from "https://rigidgeometricalgebra.org/wiki/index.php?title=Quaternion&oldid=361"