Interior products - Revision history

Eric Lengyel: /* Expansions */

2026-01-01T21:57:03Z

Expansions

← Older revision		Revision as of 21:57, 1 January 2026
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	The expansions have the effect of subtracting antigrades so that		The expansions have the effect of subtracting antigrades so that

	:$$\operatorname{ag}(\mathbf a \~~vee~~ \mathbf b^\unicode["segoe ui symbol"]{x2605}) = \operatorname{ag}(\mathbf a \~~vee~~ \mathbf b^\unicode["segoe ui symbol"]{x2606}) = \operatorname{ag}(\mathbf a) - \operatorname{ag}(\mathbf b)$$ .		:$$\operatorname{ag}(\mathbf a \wedge \mathbf b^\unicode["segoe ui symbol"]{x2605}) = \operatorname{ag}(\mathbf a \wedge \mathbf b^\unicode["segoe ui symbol"]{x2606}) = \operatorname{ag}(\mathbf a) - \operatorname{ag}(\mathbf b)$$ .

	In the case that the grades (and thus antigrades) of $$\mathbf a$$ and $$\mathbf b$$ are the same, the bulk and weight expansions are equal to dot products written as		In the case that the grades (and thus antigrades) of $$\mathbf a$$ and $$\mathbf b$$ are the same, the bulk and weight expansions are equal to dot products written as

Eric Lengyel: /* Expansions */

2024-08-22T03:13:29Z

Expansions

← Older revision		Revision as of 03:13, 22 August 2024
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	The weight expansion satisfies the identity		The weight expansion satisfies the identity

	:$$\mathbf a \wedge \mathbf b^\unicode["segoe ui symbol"]{x2606} = \langle\smash{\mathbf{\underset{\Large\unicode{x7E}}{b}}} \mathbin{\unicode{x27C7}} \mathbf a\rangle_{~~n - [~~\operatorname{ag}(\mathbf a) - \operatorname{ag}(\mathbf b)]}$$ .		:$$\mathbf a \wedge \mathbf b^\unicode["segoe ui symbol"]{x2606} = \langle\!\langle\smash{\mathbf{\underset{\Large\unicode{x7E}}{b}}} \mathbin{\unicode{x27C7}} \mathbf a\rangle\!\rangle_{\operatorname{ag}(\mathbf a) - \operatorname{ag}(\mathbf b)}$$ .

	The following Cayley table shows the bulk and weight expansion products between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The values of the bulk expansion are highlighted in green, and the values of the weight expansion are highlighted in purple. The darker purple cells along the diagonal correspond to the nonzero entries of the antimetric $$\mathbb G$$.		The following Cayley table shows the bulk and weight expansion products between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The values of the bulk expansion are highlighted in green, and the values of the weight expansion are highlighted in purple. The darker purple cells along the diagonal correspond to the nonzero entries of the antimetric $$\mathbb G$$.
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	[[Image:Expansions.svg\|720px]]		[[Image:Expansions.svg\|720px]]


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

Eric Lengyel at 01:22, 22 August 2024

2024-08-22T01:22:14Z

← Older revision		Revision as of 01:22, 22 August 2024
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	The bulk contraction satisfies the identity		The bulk contraction satisfies the identity

	:$$\mathbf a \vee \mathbf b^\unicode["segoe ui symbol"]{x2605} = \langle\mathbf{\tilde b} \mathbin{\unicode{x27D1}} \mathbf a\rangle_{\operatorname{gr}(\mathbf a) - \operatorname{gr}(\mathbf b)}$$		:$$\mathbf a \vee \mathbf b^\unicode["segoe ui symbol"]{x2605} = \langle\mathbf{\tilde b} \mathbin{\unicode{x27D1}} \mathbf a\rangle_{\operatorname{gr}(\mathbf a) - \operatorname{gr}(\mathbf b)}$$ .

	The following Cayley table shows the bulk and weight contraction products between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The values of the bulk contraction are highlighted in green, and the values of the weight contraction are highlighted in purple. The darker green cells along the diagonal correspond to the nonzero entries of the metric $$\mathbf G$$.		The following Cayley table shows the bulk and weight contraction products between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The values of the bulk contraction are highlighted in green, and the values of the weight contraction are highlighted in purple. The darker green cells along the diagonal correspond to the nonzero entries of the metric $$\mathbf G$$.
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	The expansions have the effect of subtracting antigrades so that		The expansions have the effect of subtracting antigrades so that

	:$$\operatorname{ag}(\mathbf a \vee \mathbf b^\unicode["segoe ui symbol"]{x2605}) = \operatorname{ag}(\mathbf a \vee \mathbf b^\unicode["segoe ui symbol"]{x2606}) = \operatorname{ag}(\mathbf a) - \operatorname{gr}(\mathbf b)$$ .		:$$\operatorname{ag}(\mathbf a \vee \mathbf b^\unicode["segoe ui symbol"]{x2605}) = \operatorname{ag}(\mathbf a \vee \mathbf b^\unicode["segoe ui symbol"]{x2606}) = \operatorname{ag}(\mathbf a) - \operatorname{ag}(\mathbf b)$$ .

	In the case that the grades (and thus antigrades) of $$\mathbf a$$ and $$\mathbf b$$ are the same, the bulk and weight expansions are equal to dot products written as		In the case that the grades (and thus antigrades) of $$\mathbf a$$ and $$\mathbf b$$ are the same, the bulk and weight expansions are equal to dot products written as
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	:$$\mathbf a \wedge \mathbf b^\unicode["segoe ui symbol"]{x2605} = (\mathbf a \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf b) \wedge \mathbf {\large\unicode{x1D7D9}}$$ .		:$$\mathbf a \wedge \mathbf b^\unicode["segoe ui symbol"]{x2605} = (\mathbf a \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf b) \wedge \mathbf {\large\unicode{x1D7D9}}$$ .

			The weight expansion satisfies the identity

			:$$\mathbf a \wedge \mathbf b^\unicode["segoe ui symbol"]{x2606} = \langle\smash{\mathbf{\underset{\Large\unicode{x7E}}{b}}} \mathbin{\unicode{x27C7}} \mathbf a\rangle_{n - [\operatorname{ag}(\mathbf a) - \operatorname{ag}(\mathbf b)]}$$ .

	The following Cayley table shows the bulk and weight expansion products between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The values of the bulk expansion are highlighted in green, and the values of the weight expansion are highlighted in purple. The darker purple cells along the diagonal correspond to the nonzero entries of the antimetric $$\mathbb G$$.		The following Cayley table shows the bulk and weight expansion products between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The values of the bulk expansion are highlighted in green, and the values of the weight expansion are highlighted in purple. The darker purple cells along the diagonal correspond to the nonzero entries of the antimetric $$\mathbb G$$.

Eric Lengyel at 01:14, 22 August 2024

2024-08-22T01:14:49Z

← Older revision		Revision as of 01:14, 22 August 2024
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	:$$\mathbf a \vee \mathbf b^\unicode["segoe ui symbol"]{x2606} = (\mathbf a \mathbin{\unicode["segoe ui symbol"]{x2218}} \mathbf b) \vee \mathbf 1$$ .		:$$\mathbf a \vee \mathbf b^\unicode["segoe ui symbol"]{x2606} = (\mathbf a \mathbin{\unicode["segoe ui symbol"]{x2218}} \mathbf b) \vee \mathbf 1$$ .

			The bulk contraction satisfies the identity

			:$$\mathbf a \vee \mathbf b^\unicode["segoe ui symbol"]{x2605} = \langle\mathbf{\tilde b} \mathbin{\unicode{x27D1}} \mathbf a\rangle_{\operatorname{gr}(\mathbf a) - \operatorname{gr}(\mathbf b)}$$

	The following Cayley table shows the bulk and weight contraction products between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The values of the bulk contraction are highlighted in green, and the values of the weight contraction are highlighted in purple. The darker green cells along the diagonal correspond to the nonzero entries of the metric $$\mathbf G$$.		The following Cayley table shows the bulk and weight contraction products between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The values of the bulk contraction are highlighted in green, and the values of the weight contraction are highlighted in purple. The darker green cells along the diagonal correspond to the nonzero entries of the metric $$\mathbf G$$.

Eric Lengyel at 01:29, 8 July 2024

2024-07-08T01:29:33Z

← Older revision		Revision as of 01:29, 8 July 2024
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	In the case that the grades of $$\mathbf a$$ and $$\mathbf b$$ are the same, the bulk and weight contractions are equal to dot products written as		In the case that the grades of $$\mathbf a$$ and $$\mathbf b$$ are the same, the bulk and weight contractions are equal to dot products written as

	:$$\mathbf a \vee \mathbf b^\unicode["segoe ui symbol"]{x2605} = \mathbf a \mathbin{\unicode{~~x25CF~~}} \mathbf b$$		:$$\mathbf a \vee \mathbf b^\unicode["segoe ui symbol"]{x2605} = \mathbf a \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf b$$

	and		and

	:$$\mathbf a \vee \mathbf b^\unicode["segoe ui symbol"]{x2606} = (\mathbf a \mathbin{\unicode{~~x25CB~~}} \mathbf b) \vee \mathbf 1$$ .		:$$\mathbf a \vee \mathbf b^\unicode["segoe ui symbol"]{x2606} = (\mathbf a \mathbin{\unicode["segoe ui symbol"]{x2218}} \mathbf b) \vee \mathbf 1$$ .

	The following Cayley table shows the bulk and weight contraction products between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The values of the bulk contraction are highlighted in green, and the values of the weight contraction are highlighted in purple. The darker green cells along the diagonal correspond to the nonzero entries of the metric $$\mathbf G$$.		The following Cayley table shows the bulk and weight contraction products between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The values of the bulk contraction are highlighted in green, and the values of the weight contraction are highlighted in purple. The darker green cells along the diagonal correspond to the nonzero entries of the metric $$\mathbf G$$.
Line 44:		Line 44:
	In the case that the grades (and thus antigrades) of $$\mathbf a$$ and $$\mathbf b$$ are the same, the bulk and weight expansions are equal to dot products written as		In the case that the grades (and thus antigrades) of $$\mathbf a$$ and $$\mathbf b$$ are the same, the bulk and weight expansions are equal to dot products written as

	:$$\mathbf a \wedge \mathbf b^\unicode["segoe ui symbol"]{x2606} = \mathbf a \mathbin{\unicode{~~x25CB~~}} \mathbf b$$		:$$\mathbf a \wedge \mathbf b^\unicode["segoe ui symbol"]{x2606} = \mathbf a \mathbin{\unicode["segoe ui symbol"]{x2218}} \mathbf b$$

	and		and

	:$$\mathbf a \wedge \mathbf b^\unicode["segoe ui symbol"]{x2605} = (\mathbf a \mathbin{\unicode{~~x25CF~~}} \mathbf b) \wedge \mathbf {\large\unicode{x1D7D9}}$$ .		:$$\mathbf a \wedge \mathbf b^\unicode["segoe ui symbol"]{x2605} = (\mathbf a \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf b) \wedge \mathbf {\large\unicode{x1D7D9}}$$ .

	The following Cayley table shows the bulk and weight expansion products between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The values of the bulk expansion are highlighted in green, and the values of the weight expansion are highlighted in purple. The darker purple cells along the diagonal correspond to the nonzero entries of the antimetric $$\mathbb G$$.		The following Cayley table shows the bulk and weight expansion products between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The values of the bulk expansion are highlighted in green, and the values of the weight expansion are highlighted in purple. The darker purple cells along the diagonal correspond to the nonzero entries of the antimetric $$\mathbb G$$.

Eric Lengyel at 23:27, 13 April 2024

2024-04-13T23:27:50Z

← Older revision		Revision as of 23:27, 13 April 2024
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	[[Image:Expansions.svg\|720px]]		[[Image:Expansions.svg\|720px]]


			== In the Book ==

			* Interior products, contractions, and expansions are discussed in Section 2.13.

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

	* [[Projections]]		* [[Projections]]

Eric Lengyel at 07:56, 22 January 2024

2024-01-22T07:56:36Z

← Older revision		Revision as of 07:56, 22 January 2024
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	The ~~left and right~~ ''interior products'' are special products in geometric algebra that are useful for ~~performing~~ [[projections]]. ~~These products cancel common factors in their operands and thus reduce grade. Depending on the choice of dualization function, there~~ are ~~several possible interior products. We define~~ the ~~interior products in terms of~~ the ~~left and right~~ [[~~complements~~]].		The ''interior products'' are special products in geometric algebra that are useful for a number of operations including [[projections]]. They are defined as the [[wedge product]] or [[antiwedge product]] between one object and either the [[bulk dual]] or [[weight dual]] of another object.

	~~Interior products are also known as contraction products.~~		== Contractions ==

	~~== Left~~ and ~~Right Interior Products ==~~		The ''bulk contraction'' between elements $$\mathbf a$$ and $$\mathbf b$$ is defined as

	~~The ''left interior product'' between elements~~ $$\mathbf a~~$$ and $$~~\mathbf b~~$$ is written as $$~~\mathbf a \~~mathbin{~~\unicode{~~x22A3~~}~~} \mathbf b~~$$ ~~and defined as~~		:$$\mathrm{bulk\ contraction}(\mathbf a, \mathbf b) = \mathbf a \vee \mathbf b^\unicode["segoe ui symbol"]{x2605}$$ .

	:$$\mathbf a ~~\mathbin{\unicode{x22A3}} \mathbf b = \underline{\mathbf a} \vee~~ \mathbf b$$ .		The ''weight contraction'' between elements $$\mathbf a$$ and $$\mathbf b$$ is defined as

	~~The ''right interior product'' between elements~~ $$\mathbf a~~$$ and $$~~\mathbf b~~$$ is written as $$~~\mathbf a \~~mathbin{~~\unicode{~~x22A2~~}~~} \mathbf b~~$$ ~~and defined as~~		:$$\mathrm{weight\ contraction}(\mathbf a, \mathbf b) = \mathbf a \vee \mathbf b^\unicode["segoe ui symbol"]{x2606}$$ .

	~~:$$\mathbf a \mathbin{\unicode{x22A2}} \mathbf b = \mathbf a \vee \overline{\mathbf b}$$ .~~		The contractions have the effect of subtracting grades so that

	~~The left and right interior products satisfy the relationship~~		:$$\operatorname{gr}(\mathbf a \vee \mathbf b^\unicode["segoe ui symbol"]{x2605}) = \operatorname{gr}(\mathbf a \vee \mathbf b^\unicode["segoe ui symbol"]{x2606}) = \operatorname{gr}(\mathbf a) - \operatorname{gr}(\mathbf b)$$ .

	:$$\mathbf a ~~\mathbin{\unicode{x22A3}} \mathbf b = (-1)^{\operatorname{gr}(\mathbf a)\left[\operatorname{gr}(\mathbf a) + \operatorname{gr}(\mathbf b)\right]}~~\mathbf b ~~\mathbin{\unicode{x22A2}} \mathbf a~~$$ .		In the case that the grades of $$\mathbf a$$ and $$\mathbf b$$ are the same, the bulk and weight contractions are equal to dot products written as

	~~The following Cayley table shows the interior products between all pairs of basis elements in the 4D rigid geometric algebra~~ $$\~~mathcal G_~~{~~3,0,1~~}$$. Cells colored blue belong only to the left interior product and are zero for the right interior product. Cells colored red belong only to the right interior product and are zero for the left interior product. Cells colored purple along the diagonal belong to both the left and right interior products.		:$$\mathbf a \vee \mathbf b^\unicode["segoe ui symbol"]{x2605} = \mathbf a \mathbin{\unicode{x25CF}} \mathbf b$$

			and

	[~~[Image:InteriorProduct~~.~~svg\|720px]]~~		:$$\mathbf a \vee \mathbf b^\unicode["segoe ui symbol"]{x2606} = (\mathbf a \mathbin{\unicode{x25CB}} \mathbf b) \vee \mathbf 1$$ .

	~~== Left~~ and ~~Right Interior Antiproducts ==~~		The following Cayley table shows the bulk and weight contraction products between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The values of the bulk contraction are highlighted in green, and the values of the weight contraction are highlighted in purple. The darker green cells along the diagonal correspond to the nonzero entries of the metric $$\mathbf G$$.

	Like all operations in geometric algebra, the interior products have duals, which we call ''interior antiproducts''. The right and left interior antiproducts between elements $$\mathbf a$$ and $$\mathbf b$$ are written as $$\mathbf a \mathbin{\unicode{x22A8}} \mathbf b$$ and $$\mathbf a \mathbin{\unicode{x2AE4}} \mathbf b$$, respectively, and they are defined as follows.

	:~~$$\mathbf a \mathbin{\unicode{x22A8}} \mathbf b = \overline{\underline{\mathbf a} \mathbin{\unicode{x22A2}} \underline{\mathbf b}} = \mathbf a \wedge \overline{\mathbf b}$$~~		[[Image:Constractions.svg\|720px]]

	~~:$$\mathbf a \mathbin{\unicode{x2AE4}} \mathbf b~~ = ~~\overline{\underline{\mathbf a} \mathbin{\unicode{x22A3}} \underline{\mathbf b}}~~ = ~~\underline{\mathbf a} \wedge \mathbf b$$~~		== Expansions ==

	The ~~interior antiproducts are also related to the opposite interior products through the following equalities.~~		The ''bulk expansion'' between elements $$\mathbf a$$ and $$\mathbf b$$ is defined as

	:$$\~~mathbf a \mathbin~~{\~~unicode{x22A8}~~} \mathbf b = ~~\overline{~~\mathbf a \~~mathbin{~~\unicode{~~x22A3}} \mathbf b~~}$$		:$$\mathrm{bulk\ expansion}(\mathbf a, \mathbf b) = \mathbf a \wedge \mathbf b^\unicode["segoe ui symbol"]{x2605}$$ .

	:$$\mathbf a ~~\mathbin{\unicode{x2AE4}} \mathbf b = \underline{\mathbf a \mathbin{\unicode{x22A2}}~~ \mathbf b}$$		The ''weight expansion'' between elements $$\mathbf a$$ and $$\mathbf b$$ is defined as

	~~The following Cayley table shows the interior antiproducts between all pairs of basis elements in the 4D rigid geometric algebra~~ $$\~~mathcal G_~~{3,~~0,1~~}$$. Cells colored blue belong only to the left interior antiproduct and are zero for the right interior antiproduct. Cells colored red belong only to the right interior antiproduct and are zero for the left interior antiproduct. Cells colored purple along the diagonal belong to both the left and right interior antiproducts.		:$$\mathrm{weight\ expansion}(\mathbf a, \mathbf b) = \mathbf a \wedge \mathbf b^\unicode["segoe ui symbol"]{x2606}$$ .

			The expansions have the effect of subtracting antigrades so that

	[[Image:~~InteriorAntiproduct~~.svg\|720px]]		:$$\operatorname{ag}(\mathbf a \vee \mathbf b^\unicode["segoe ui symbol"]{x2605}) = \operatorname{ag}(\mathbf a \vee \mathbf b^\unicode["segoe ui symbol"]{x2606}) = \operatorname{ag}(\mathbf a) - \operatorname{gr}(\mathbf b)$$ .

			In the case that the grades (and thus antigrades) of $$\mathbf a$$ and $$\mathbf b$$ are the same, the bulk and weight expansions are equal to dot products written as

			:$$\mathbf a \wedge \mathbf b^\unicode["segoe ui symbol"]{x2606} = \mathbf a \mathbin{\unicode{x25CB}} \mathbf b$$

			and

			:$$\mathbf a \wedge \mathbf b^\unicode["segoe ui symbol"]{x2605} = (\mathbf a \mathbin{\unicode{x25CF}} \mathbf b) \wedge \mathbf {\large\unicode{x1D7D9}}$$ .

			The following Cayley table shows the bulk and weight expansion products between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The values of the bulk expansion are highlighted in green, and the values of the weight expansion are highlighted in purple. The darker purple cells along the diagonal correspond to the nonzero entries of the antimetric $$\mathbb G$$.


			[[Image:Expansions.svg\|720px]]

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

	* [[Projections]]		* [[Projections]]

Eric Lengyel: Created page with "The left and right ''interior products'' are special products in geometric algebra that are useful for performing projections. These products cancel common factors in their operands and thus reduce grade. Depending on the choice of dualization function, there are several possible interior products. We define the interior products in terms of the left and right complements. Interior products are also known as contraction products. == Left and Right Interior Prod..."

2023-07-15T06:27:41Z

Created page with "The left and right ''interior products'' are special products in geometric algebra that are useful for performing projections. These products cancel common factors in their operands and thus reduce grade. Depending on the choice of dualization function, there are several possible interior products. We define the interior products in terms of the left and right complements. Interior products are also known as contraction products. == Left and Right Interior Prod..."

New page

The left and right ''interior products'' are special products in geometric algebra that are useful for performing [[projections]]. These products cancel common factors in their operands and thus reduce grade. Depending on the choice of dualization function, there are several possible interior products. We define the interior products in terms of the left and right [[complements]].

Interior products are also known as contraction products.

== Left and Right Interior Products ==

The ''left interior product'' between elements $$\mathbf a$$ and $$\mathbf b$$ is written as $$\mathbf a \mathbin{\unicode{x22A3}} \mathbf b$$ and defined as

:$$\mathbf a \mathbin{\unicode{x22A3}} \mathbf b = \underline{\mathbf a} \vee \mathbf b$$ .

The ''right interior product'' between elements $$\mathbf a$$ and $$\mathbf b$$ is written as $$\mathbf a \mathbin{\unicode{x22A2}} \mathbf b$$ and defined as

:$$\mathbf a \mathbin{\unicode{x22A2}} \mathbf b = \mathbf a \vee \overline{\mathbf b}$$ .

The left and right interior products satisfy the relationship

:$$\mathbf a \mathbin{\unicode{x22A3}} \mathbf b = (-1)^{\operatorname{gr}(\mathbf a)\left[\operatorname{gr}(\mathbf a) + \operatorname{gr}(\mathbf b)\right]}\mathbf b \mathbin{\unicode{x22A2}} \mathbf a$$ .

The following Cayley table shows the interior products between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. Cells colored blue belong only to the left interior product and are zero for the right interior product. Cells colored red belong only to the right interior product and are zero for the left interior product. Cells colored purple along the diagonal belong to both the left and right interior products.

[[Image:InteriorProduct.svg|720px]]

== Left and Right Interior Antiproducts ==

Like all operations in geometric algebra, the interior products have duals, which we call ''interior antiproducts''. The right and left interior antiproducts between elements $$\mathbf a$$ and $$\mathbf b$$ are written as $$\mathbf a \mathbin{\unicode{x22A8}} \mathbf b$$ and $$\mathbf a \mathbin{\unicode{x2AE4}} \mathbf b$$, respectively, and they are defined as follows.

:$$\mathbf a \mathbin{\unicode{x22A8}} \mathbf b = \overline{\underline{\mathbf a} \mathbin{\unicode{x22A2}} \underline{\mathbf b}} = \mathbf a \wedge \overline{\mathbf b}$$

:$$\mathbf a \mathbin{\unicode{x2AE4}} \mathbf b = \overline{\underline{\mathbf a} \mathbin{\unicode{x22A3}} \underline{\mathbf b}} = \underline{\mathbf a} \wedge \mathbf b$$

The interior antiproducts are also related to the opposite interior products through the following equalities.

:$$\mathbf a \mathbin{\unicode{x22A8}} \mathbf b = \overline{\mathbf a \mathbin{\unicode{x22A3}} \mathbf b}$$

:$$\mathbf a \mathbin{\unicode{x2AE4}} \mathbf b = \underline{\mathbf a \mathbin{\unicode{x22A2}} \mathbf b}$$

The following Cayley table shows the interior antiproducts between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. Cells colored blue belong only to the left interior antiproduct and are zero for the right interior antiproduct. Cells colored red belong only to the right interior antiproduct and are zero for the left interior antiproduct. Cells colored purple along the diagonal belong to both the left and right interior antiproducts.

[[Image:InteriorAntiproduct.svg|720px]]

== See Also ==

* [[Projections]]