Complements - Revision history

Eric Lengyel at 06:42, 25 August 2024

2024-08-25T06:42:06Z

← Older revision		Revision as of 06:42, 25 August 2024
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	=== Explicit Formula ===		=== Explicit Formula ===

	~~Let~~ $$~~\mathbf u~~$$ ~~be a basis~~ element ~~with grade $$k$$. Then $$\mathbf u$$ can be expressed as~~		In an $$n$$-dimensional algebra, suppose the volume element is given by

	:$$\mathbf u = \mathbf e_{a_1} \wedge \mathbf e_{a_2} \cdots \mathbf e_{a_k}$$ ,		:$${\large\unicode{x1D7D9}} = m (\mathbf e_1 \wedge \mathbf e_2 \wedge \cdots \wedge \mathbf e_n)$$ ,

			where $$m = \pm 1$$. Let $$\mathbf u$$ be a basis element with grade $$k$$. Then $$\mathbf u$$ can be expressed as

			:$$\mathbf u = \mathbf e_{a_1} \wedge \mathbf e_{a_2} \wedge \cdots \wedge \mathbf e_{a_k}$$ ,

	where each $$a_i$$ is a unique index satisfying $$1 \le a_i \le n$$. The right complement of $$\mathbf u$$ is given by		where each $$a_i$$ is a unique index satisfying $$1 \le a_i \le n$$. The right complement of $$\mathbf u$$ is given by

	:$$\overline{\mathbf u} = \operatorname{sgn}(a_1, a_2, \dots, a_k) \left(\prod_{i=1}^k (-1)^{a_i-k}\right) \left(\bigwedge_{j=1}^n \varphi(j)\right)$$ ,		:$$\overline{\mathbf u} = m \operatorname{sgn}(a_1, a_2, \dots, a_k) \left(\prod_{i=1}^k (-1)^{a_i-k}\right) \left(\bigwedge_{j=1}^n \varphi(j)\right)$$ ,

	where $$\operatorname{sgn}$$ is the signature of the permutation $$(a_1, a_2, \dots, a_k)$$, and $$\varphi(j)$$ is defined as		where $$\operatorname{sgn}$$ is the signature of the permutation $$(a_1, a_2, \dots, a_k)$$, and $$\varphi(j)$$ is defined as

Eric Lengyel: /* Explicit Formula */

2024-08-25T06:33:10Z

Explicit Formula

← Older revision		Revision as of 06:33, 25 August 2024
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	where each $$a_i$$ is a unique index satisfying $$1 \le a_i \le n$$. The right complement of $$\mathbf u$$ is given by		where each $$a_i$$ is a unique index satisfying $$1 \le a_i \le n$$. The right complement of $$\mathbf u$$ is given by

	:$$\overline{\mathbf u} = \operatorname{sgn}(a_1, a_2, \dots, a_k) \left(\prod_{i=1}^k (-1)^{a_i-1}\right) \left(\bigwedge_{j=1}^n \varphi(j)\right)$$ ,		:$$\overline{\mathbf u} = \operatorname{sgn}(a_1, a_2, \dots, a_k) \left(\prod_{i=1}^k (-1)^{a_i-k}\right) \left(\bigwedge_{j=1}^n \varphi(j)\right)$$ ,

	where $$\operatorname{sgn}$$ is the signature of the permutation $$(a_1, a_2, \dots, a_k)$$, and $$\varphi(j)$$ is defined as		where $$\operatorname{sgn}$$ is the signature of the permutation $$(a_1, a_2, \dots, a_k)$$, and $$\varphi(j)$$ is defined as

Eric Lengyel at 06:30, 25 August 2024

2024-08-25T06:30:10Z

← Older revision		Revision as of 06:30, 25 August 2024
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	[[Image:Complements.svg\|720px]]		[[Image:Complements.svg\|720px]]

			=== Explicit Formula ===

			Let $$\mathbf u$$ be a basis element with grade $$k$$. Then $$\mathbf u$$ can be expressed as

			:$$\mathbf u = \mathbf e_{a_1} \wedge \mathbf e_{a_2} \cdots \mathbf e_{a_k}$$ ,

			where each $$a_i$$ is a unique index satisfying $$1 \le a_i \le n$$. The right complement of $$\mathbf u$$ is given by

			:$$\overline{\mathbf u} = \operatorname{sgn}(a_1, a_2, \dots, a_k) \left(\prod_{i=1}^k (-1)^{a_i-1}\right) \left(\bigwedge_{j=1}^n \varphi(j)\right)$$ ,

			where $$\operatorname{sgn}$$ is the signature of the permutation $$(a_1, a_2, \dots, a_k)$$, and $$\varphi(j)$$ is defined as

			:$$\varphi(j) = \begin{cases}\mathbf e_j, & \text{if } j \notin \{a_1, a_2, \dots, a_k\}; \\ \mathbf 1, & \text{otherwise.}\end{cases}$$

	== In the Book ==		== In the Book ==

Eric Lengyel at 23:31, 13 April 2024

2024-04-13T23:31:28Z

← Older revision		Revision as of 23:31, 13 April 2024
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	[[Image:Complements.svg\|720px]]		[[Image:Complements.svg\|720px]]

			== In the Book ==

			* Complements are introduced in Section 2.2.

	== See Also ==		== See Also ==
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	* [[Reverses]]		* [[Reverses]]
	* [[Duality]]		* [[Duality]]

	~~== In the Book ==~~

	* Complements are introduced in Section 2.2.

Eric Lengyel at 05:06, 13 April 2024

2024-04-13T05:06:42Z

← Older revision		Revision as of 05:06, 13 April 2024
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	* [[Reverses]]		* [[Reverses]]
	* [[Duality]]		* [[Duality]]

			== In the Book ==

			* Complements are introduced in Section 2.2.

Eric Lengyel at 20:39, 12 April 2024

2024-04-12T20:39:44Z

← Older revision		Revision as of 20:39, 12 April 2024
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	This shows that the left and right complements of an element $$\mathbf u$$ are always the same if either its [[grade]] $$\operatorname{gr}(\mathbf u)$$ or its [[antigrade]] $$\operatorname{ag}(\mathbf u)$$ 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 $$\overline{\underline{\mathbf u}} = \mathbf u$$.		This shows that the left and right complements of an element $$\mathbf u$$ are always the same if either its [[grade]] $$\operatorname{gr}(\mathbf u)$$ or its [[antigrade]] $$\operatorname{ag}(\mathbf u)$$ 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 $$\overline{\underline{\mathbf u}} = \mathbf u$$.

			Taking the right or left complement twice causes the sign to change according to the formula

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

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

Eric Lengyel: /* See Also */

2024-04-12T06:26:50Z

Eric Lengyel at 08:01, 22 January 2024

2024-01-22T08:01:26Z

← Older revision		Revision as of 08:01, 22 January 2024
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	The following table lists the left and right complements for all of the basis elements in the 4D geometric algebra $$\mathcal G_{3,0,1}$$.		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~~]]		[[Image:Complements.svg\|720px]]

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

Eric Lengyel at 07:02, 23 October 2023

2023-10-23T07:02:30Z

← Older revision		Revision as of 07:02, 23 October 2023
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	''Complements'' are unary operations in geometric algebra that perform a specific type of dualization.		''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		Every basis element $$\mathbf u$$ has a ''right complement'', which we denote by $$\overline{\mathbf u}$$, that satisfies the equation

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

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

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

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

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

	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 $$\overline{\underline{\mathbf x}} = \mathbf x$$.		This shows that the left and right complements of an element $$\mathbf u$$ are always the same if either its [[grade]] $$\operatorname{gr}(\mathbf u)$$ or its [[antigrade]] $$\operatorname{ag}(\mathbf u)$$ 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 $$\overline{\underline{\mathbf u}} = \mathbf u$$.

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

	:$$\mathbf x \vee \overline{\mathbf x} = \mathbf 1$$		:$$\mathbf u \vee \overline{\mathbf u} = \mathbf 1$$
	:$$\underline{\mathbf x} \vee\mathbf x = \mathbf 1$$ .		:$$\underline{\mathbf u} \vee\mathbf u = \mathbf 1$$ .

	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,		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,

Eric Lengyel at 06:07, 23 October 2023

2023-10-23T06:07:35Z

← Older revision		Revision as of 06:07, 23 October 2023
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	[[Image:Complements.svg\|900px]]		[[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 ==		== See Also ==

← Older revision		Revision as of 06:26, 12 April 2024
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	== See Also ==		== See Also ==

	* [[~~Duality~~]]		* [[Duals]]
	* [[Grade and antigrade]]		* [[Grade and antigrade]]
	* [[Bulk and weight]]		* [[Bulk and weight]]
	* [[Reverses]]		* [[Reverses]]
			* [[Duality]]

← Older revision		Revision as of 06:07, 23 October 2023
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	[[Image:Complements.svg\|900px]]		[[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 ==		== See Also ==