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| | [[Image:plane.svg|400px|thumb|right|'''Figure 1.''' A plane is the intersection of a 4D trivector with the 3D subspace where $$w = 1$$.]] |
| A ''quaternion'' is an operator that performs a rotation about the origin in 3D space. Conventionally, a quaternion $$\mathbf q$$ is written as
| | In the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$, a ''plane'' $$\mathbf f$$ is a trivector having the general form |
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| :$$\mathbf q = q_w + q_x \mathbf i + q_y \mathbf j + q_z \mathbf k$$ , | | :$$\mathbf f = f_x \mathbf e_{234} + f_y \mathbf e_{314} + f_z \mathbf e_{124} + f_w \mathbf e_{321}$$ . |
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| where the "imaginary" units $$\mathbf i$$, $$\mathbf j$$, and $$\mathbf k$$ all square to $$-1$$ and multiply according to the rules
| | All planes possess the [[geometric property]]. |
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| :$$\mathbf{ij} = -\mathbf{ji} = \mathbf k$$
| | The [[bulk]] of a plane is given by its $$w$$ coordinate, and the [[weight]] of a plane is given by its $$x$$, $$y$$, and $$z$$ coordinates. A plane is [[unitized]] when $$f_x^2 + f_y^2 + f_z^2 = 1$$. |
| :$$\mathbf{jk} = -\mathbf{kj} = \mathbf i$$
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| :$$\mathbf{ki} = -\mathbf{ik} = \mathbf j$$ .
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| A ''unit quaternion'' is one for which $$q_w^2 + q_x^2 + q_y^2 + q_z^2 = 1$$.
| | When used as an operator in the sandwich with the [[geometric antiproduct]], a plane is a specific kind of [[flector]] that performs a [[reflection]] through itself. |
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| == Quaternions in 4D Rigid Geometric Algebra ==
| | A [[dual translation]] operator $$\mathbf T$$ that moves a plane $$\mathbf f$$ to the horizon is given by |
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| Because quaternions keep the origin fixed, they are part of the group SO(3) where the special Euclidean group SE(3) and dual special Euclidean group DSE(3) intersect. Consequently, the quaternions have two different representations in the four-dimensional rigid geometric algebra $$\mathcal G_{3,0,1}$$.
| | :$$\mathbf T = \underline{\mathbf f} \wedge \mathbf e_{4} + 2\mathbf f \vee \mathbf e_4 = f_{x\vphantom{y}} \mathbf e_{41} + f_y \mathbf e_{42} + f_{z\vphantom{y}} \mathbf e_{43} + 2f_w$$ . |
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| === Quaternions as Motors === | | == Plane at Infinity == |
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| 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
| | If the weight of a plane is zero (i.e., its $$x$$, $$y$$, and $$z$$ coordinates are all zero), then the plane lies at infinity in all directions. Such a plane is normalized when $$f_w = \pm 1$$. This is the ''horizon'' of three-dimensional space. |
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| :$$\mathbf i = \mathbf e_{41}$$
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| :$$\mathbf j = \mathbf e_{42}$$
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| :$$\mathbf k = \mathbf e_{43}$$ .
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| A quaternion can then be written as
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| :$$\mathbf q = q_x \mathbf e_{41} + q_y \mathbf e_{42} + q_z \mathbf e_{43} + q_w {\large\unicode{x1D7D9}}$$ ,
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| and any object $$\mathbf a$$ (such as a [[point]], [[line]], or [[plane]]) is rotated about the origin through the sandwich product
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| :$$\mathbf a' = \mathbf q \mathbin{\unicode{x27C7}} \mathbf a \mathbin{\unicode{x27C7}} \mathbf{\underset{\Large\unicode{x7E}}{q}}$$ ,
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| using the [[geometric antiproduct]].
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| A unit quaternion can also be written as
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| :$$\mathbf q = \mathbf a \sin\phi + {\large\unicode{x1D7D9}}\cos\phi$$ ,
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| 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.
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| === Quaternions as Dual Motors ===
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| 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
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| :$$\mathbf i = -\mathbf e_{23}$$
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| :$$\mathbf j = -\mathbf e_{31}$$
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| :$$\mathbf k = -\mathbf e_{12}$$ .
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| A quaternion can then be written as
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| :$$\mathbf q = q_w - q_x \mathbf e_{23} - q_y \mathbf e_{31} - q_z \mathbf e_{12}$$ ,
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| and any object $$\mathbf a$$ (such as a [[point]], [[line]], or [[plane]]) is rotated about the origin through the sandwich product
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| :$$\mathbf a' = \mathbf q \mathbin{\unicode{x27D1}} \mathbf a \mathbin{\unicode{x27D1}} \mathbf{\tilde q}$$ ,
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| using the [[geometric product]].
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| A unit quaternion can also be written as
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| :$$\mathbf q = \cos\phi - \mathbf a \sin\phi$$ ,
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| 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.
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| == See Also == | | == See Also == |
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| * [[Motor]] | | * [[Point]] |
| * [[Flector]] | | * [[Line]] |
Figure 1. A plane is the intersection of a 4D trivector with the 3D subspace where $$w = 1$$.
In the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$, a plane $$\mathbf f$$ is a trivector having the general form
- $$\mathbf f = f_x \mathbf e_{234} + f_y \mathbf e_{314} + f_z \mathbf e_{124} + f_w \mathbf e_{321}$$ .
All planes possess the geometric property.
The bulk of a plane is given by its $$w$$ coordinate, and the weight of a plane is given by its $$x$$, $$y$$, and $$z$$ coordinates. A plane is unitized when $$f_x^2 + f_y^2 + f_z^2 = 1$$.
When used as an operator in the sandwich with the geometric antiproduct, a plane is a specific kind of flector that performs a reflection through itself.
A dual translation operator $$\mathbf T$$ that moves a plane $$\mathbf f$$ to the horizon is given by
- $$\mathbf T = \underline{\mathbf f} \wedge \mathbf e_{4} + 2\mathbf f \vee \mathbf e_4 = f_{x\vphantom{y}} \mathbf e_{41} + f_y \mathbf e_{42} + f_{z\vphantom{y}} \mathbf e_{43} + 2f_w$$ .
Plane at Infinity
If the weight of a plane is zero (i.e., its $$x$$, $$y$$, and $$z$$ coordinates are all zero), then the plane lies at infinity in all directions. Such a plane is normalized when $$f_w = \pm 1$$. This is the horizon of three-dimensional space.
See Also
