File:Antispace4.svg and Complements: Difference between pages

Revision as of 01:27, 19 October 2023

Complements are unary operations in geometric algebra that perform a specific type of dualization.

Every basis element $$\mathbf x$$ has a right complement, which we denote by $$\overline{\mathbf x}$$, that satisfies the equation

$$\mathbf x \wedge \overline{\mathbf x} = {\large\unicode{x1D7D9}}$$ .

There is also a left complement, which we denote by $$\underline{\mathbf x}$$, that satisfies the equation

$$\underline{\mathbf x} \wedge \mathbf x = {\large\unicode{x1D7D9}}$$ .

Complements exchange full and empty dimensions, and the left and right complements can differ only by sign according to the relationship

$$\underline{\mathbf x} = (-1)^{\operatorname{gr}(\mathbf x)\operatorname{ag}(\mathbf x)}\,\overline{\mathbf x}$$ .

This shows that the left and right complements of an element $$\mathbf x$$ are always the same if either its grade $$\operatorname{gr}(\mathbf x)$$ or its antigrade $$\operatorname{ag}(\mathbf x)$$ is even. If the number of dimensions is odd, then it is always true that one of these is even, so left and right complements are the same for all elements in an odd-dimensional algebra. As shown in the table below, applying the right or left complement twice can negate the operand in even numbers of dimensions. However, the right and left complements are inverse operations, so we always have $$\overline{\underline{\mathbf x}} = \mathbf x$$.

The right and left complements under the wedge product are also the right and left complements under the antiwedge product, so we can write

$$\mathbf x \vee \overline{\mathbf x} = \mathbf 1$$

$$\underline{\mathbf x} \vee\mathbf x = \mathbf 1$$ .

To extend the complements to all elements of an algebra, we simply require that they are linear operations. For any basis elements $$\mathbf x$$ and $$\mathbf y$$, and for any scalars $$a$$ and $$b$$, we must have, for the right complement,

$$\overline{(a\mathbf x + b\mathbf y)} = a\overline{\mathbf x} + b\overline{\mathbf y}$$ ,

and similarly for the left complement.

The following table lists the left and right complements for all of the basis elements in the 4D geometric algebra $$\mathcal G_{3,0,1}$$.

Weight Complements

The complement of an element's weight is particularly useful because it extracts the attitude of a geometric object as an element expressed on an orthogonal basis. These arise naturally in projections, which make use of the interior product.

The following table lists the weight left complement for the main types in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$.

Type	Definition	Weight Left Complement	Description
Point	$$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$	$$\underline{\mathbf p_\smash{\unicode{x25CB}}} = -p_w \mathbf e_{321}$$	Plane at infinity.
Line	$$\boldsymbol l = l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} + l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}$$	$$\underline{\boldsymbol l_\smash{\unicode{x25CB}}} = -l_{vx} \mathbf e_{23} - l_{vy} \mathbf e_{31} - l_{vz} \mathbf e_{12}$$	Line at infinity perpendicular to line $$\boldsymbol l$$.
Plane	$$\mathbf g = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + g_w \mathbf e_{321}$$	$$\underline{\mathbf g_\smash{\unicode{x25CB}}} = g_x \mathbf e_1 + g_y \mathbf e_2 + g_z \mathbf e_3$$	Normal vector, or point at infinity in direction perpendicular to plane $$\mathbf g$$.
Motor	$$\mathbf Q = Q_{vx} \mathbf e_{41} + Q_{vy} \mathbf e_{42} + Q_{vz} \mathbf e_{43} + Q_{vw} {\large\unicode{x1d7d9}} + Q_{mx} \mathbf e_{23} + Q_{my} \mathbf e_{31} + Q_{mz} \mathbf e_{12} + Q_{mw}$$	$$\underline{\mathbf Q_\smash{\unicode{x25CB}}} = -Q_{vx} \mathbf e_{23} - Q_{vy} \mathbf e_{31} - Q_{vz} \mathbf e_{12} + Q_{vw}$$	Conventional quaternion $$\mathbf q = Q_{vx} \mathbf i + Q_{vy} \mathbf j + Q_{vz} \mathbf k + Q_{vw} = (a_x \mathbf i + a_y \mathbf j + a_z \mathbf k)\sin\phi + \cos\phi$$, which is the 3D position-free counterpart of a motor. The sandwich product $$\mathbf q \mathbin{\unicode{x27D1}} \mathbf u \mathbin{\unicode{x27D1}} \mathbf{\tilde q}$$ rotates the vector $$\mathbf u$$ through the angle $$2\phi$$ about the axis $$\mathbf a$$.
Flector	$$\mathbf F = F_{px} \mathbf e_1 + F_{py} \mathbf e_2 + F_{pz} \mathbf e_3 + F_{pw} \mathbf e_4 + F_{gx} \mathbf e_{423} + F_{gy} \mathbf e_{431} + F_{gz} \mathbf e_{412} + F_{gw} \mathbf e_{321}$$	$$\underline{\mathbf F_\smash{\unicode{x25CB}}} = F_{gx} \mathbf e_1 + F_{gy} \mathbf e_2 + F_{gz} \mathbf e_3 - F_{gw} \mathbf e_{321}$$	3D position-free counterpart of a flector having the form $$\mathbf f = (a_x \mathbf e_1 + a_y \mathbf e_2 + a_z \mathbf e_3)\cos\phi + \mathbf e_{123}\sin\phi$$. The sandwich product $$-\mathbf f \mathbin{\unicode{x27D1}} \mathbf u \mathbin{\unicode{x27D1}} \mathbf{\tilde f}$$ rotates the vector $$\mathbf u$$ through the angle $$2\phi$$ about the axis $$\mathbf a$$ and reflects it along the direction of $$\mathbf a$$.

File history

Click on a date/time to view the file as it appeared at that time.

	Date/Time	Thumbnail	Dimensions	User	Comment
current	03:56, 5 August 2023		512 × 257 (857 KB)	Eric Lengyel (talk \| contribs)

You cannot overwrite this file.

File usage

The following page uses this file:

Space-Antispace Transform Correspondence in Projective Geometric Algebra

Metadata

This file contains additional information, probably added from the digital camera or scanner used to create or digitize it.

If the file has been modified from its original state, some details may not fully reflect the modified file.

Width	100%
Height	100%

@@ Line 1: / Line 1: @@
+''Complements'' are unary operations in geometric algebra that perform a specific type of dualization.
+Every basis element $$\mathbf x$$ has a ''right complement'', which we denote by $$\overline{\mathbf x}$$, that satisfies the equation
+:$$\mathbf x \wedge \overline{\mathbf x} = {\large\unicode{x1D7D9}}$$ .
+There is also a ''left complement'', which we denote by $$\underline{\mathbf x}$$, that satisfies the equation
+:$$\underline{\mathbf x} \wedge \mathbf x = {\large\unicode{x1D7D9}}$$ .
+Complements exchange full and empty dimensions, and the left and right complements can differ only by sign according to the relationship
+:$$\underline{\mathbf x} = (-1)^{\operatorname{gr}(\mathbf x)\operatorname{ag}(\mathbf x)}\,\overline{\mathbf x}$$ .
+This shows that the left and right complements of an element $$\mathbf x$$ are always the same if either its [[grade]] $$\operatorname{gr}(\mathbf x)$$ or its [[antigrade]] $$\operatorname{ag}(\mathbf x)$$ is even. If the number of dimensions is odd, then it is always true that one of these is even, so left and right complements are the same for all elements in an odd-dimensional algebra. As shown in the table below, applying the right or left complement twice can negate the operand in even numbers of dimensions. However, the right and left complements are inverse operations, so we always have $$\overline{\underline{\mathbf x}} = \mathbf x$$.
+The right and left complements under the [[wedge product]] are also the right and left complements under the [[antiwedge product]], so we can write
+:$$\mathbf x \vee \overline{\mathbf x} = \mathbf 1$$
+:$$\underline{\mathbf x} \vee\mathbf x = \mathbf 1$$ .
+To extend the complements to all elements of an algebra, we simply require that they are linear operations. For any basis elements $$\mathbf x$$ and $$\mathbf y$$, and for any scalars $$a$$ and $$b$$, we must have, for the right complement,
+:$$\overline{(a\mathbf x + b\mathbf y)} = a\overline{\mathbf x} + b\overline{\mathbf y}$$ ,
+and similarly for the left complement.
+The following table lists the left and right complements for all of the basis elements in the 4D geometric algebra $$\mathcal G_{3,0,1}$$.
+[[Image:Complements.svg|900px]]
+== Weight Complements ==
+The complement of an element's weight is particularly useful because it extracts the attitude of a geometric object as an element expressed on an orthogonal basis. These arise naturally in [[projections]], which make use of the [[interior product]].
+The following table lists the weight left complement for the main types in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$.
+{| class="wikitable"
+! Type !! Definition !! Weight Left Complement !! Description
+|-
+| 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;" | $$\underline{\mathbf p_\smash{\unicode{x25CB}}} = -p_w \mathbf e_{321}$$
+| style="padding: 12px;" | Plane at infinity.
+|-
+| 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;" | $$\underline{\boldsymbol l_\smash{\unicode{x25CB}}} = -l_{vx} \mathbf e_{23} - l_{vy} \mathbf e_{31} - l_{vz} \mathbf e_{12}$$
+| style="padding: 12px;" | Line at infinity perpendicular to line $$\boldsymbol l$$.
+|-
+| 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;" | $$\underline{\mathbf g_\smash{\unicode{x25CB}}} = g_x \mathbf e_1 + g_y \mathbf e_2 + g_z \mathbf e_3$$
+| style="padding: 12px;" | Normal vector, or point at infinity in direction perpendicular to plane $$\mathbf g$$.
+|-
+| style="padding: 12px;" | [[Motor]]
+| style="padding: 12px;" | $$\mathbf Q = Q_{vx} \mathbf e_{41} + Q_{vy} \mathbf e_{42} + Q_{vz} \mathbf e_{43} + Q_{vw} {\large\unicode{x1d7d9}} + Q_{mx} \mathbf e_{23} + Q_{my} \mathbf e_{31} + Q_{mz} \mathbf e_{12} + Q_{mw}$$
+| style="padding: 12px;" | $$\underline{\mathbf Q_\smash{\unicode{x25CB}}} = -Q_{vx} \mathbf e_{23} - Q_{vy} \mathbf e_{31} - Q_{vz} \mathbf e_{12} + Q_{vw}$$
+| style="padding: 12px;" | Conventional [[quaternion]] $$\mathbf q = Q_{vx} \mathbf i + Q_{vy} \mathbf j + Q_{vz} \mathbf k + Q_{vw} = (a_x \mathbf i + a_y \mathbf j + a_z \mathbf k)\sin\phi + \cos\phi$$, which is the 3D position-free counterpart of a [[motor]].
+The sandwich product $$\mathbf q \mathbin{\unicode{x27D1}} \mathbf u \mathbin{\unicode{x27D1}} \mathbf{\tilde q}$$ rotates the vector $$\mathbf u$$ through the angle $$2\phi$$ about the axis $$\mathbf a$$.
+|-
+| 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;" | $$\underline{\mathbf F_\smash{\unicode{x25CB}}} = F_{gx} \mathbf e_1 + F_{gy} \mathbf e_2 + F_{gz} \mathbf e_3 - F_{gw} \mathbf e_{321}$$
+| style="padding: 12px;" | 3D position-free counterpart of a [[flector]] having the form $$\mathbf f = (a_x \mathbf e_1 + a_y \mathbf e_2 + a_z \mathbf e_3)\cos\phi + \mathbf e_{123}\sin\phi$$.
+The sandwich product $$-\mathbf f \mathbin{\unicode{x27D1}} \mathbf u \mathbin{\unicode{x27D1}} \mathbf{\tilde f}$$ rotates the vector $$\mathbf u$$ through the angle $$2\phi$$ about the axis $$\mathbf a$$ and reflects it along the direction of $$\mathbf a$$.
+|}
+== See Also ==
+* [[Duality]]
+* [[Grade and antigrade]]
+* [[Bulk and weight]]
+* [[Reverses]]