Interior products

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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.

Interior products are also known as contraction products.

Left Interior Product

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 following Cayley table shows the left interior products between all pairs of basis elements in the 4D projective geometric algebra $$\mathcal G_{3,0,1}$$.

a   b $$\mathbf 1$$ $$\mathbf e_1$$ $$\mathbf e_2$$ $$\mathbf e_3$$ $$\mathbf e_4$$ $$\mathbf e_{23}$$ $$\mathbf e_{31}$$ $$\mathbf e_{12}$$ $$\mathbf e_{43}$$ $$\mathbf e_{42}$$ $$\mathbf e_{41}$$ $$\mathbf e_{321}$$ $$\mathbf e_{124}$$ $$\mathbf e_{314}$$ $$\mathbf e_{234}$$ $$\large\unicode{x1D7D9}$$
$$\mathbf 1$$ $$\mathbf 1$$ $$\mathbf e_1$$ $$\mathbf e_2$$ $$\mathbf e_3$$ $$\mathbf e_4$$ $$\mathbf e_{23}$$ $$\mathbf e_{31}$$ $$\mathbf e_{12}$$ $$\mathbf e_{43}$$ $$\mathbf e_{42}$$ $$\mathbf e_{41}$$ $$\mathbf e_{321}$$ $$\mathbf e_{124}$$ $$\mathbf e_{314}$$ $$\mathbf e_{234}$$ $$0$$
$$\mathbf e_1$$ $$0$$ $$\mathbf 1$$ $$0$$ $$0$$ $$0$$ $$0$$ $$\mathbf e_3$$ $$-\mathbf e_2$$ $$0$$ $$0$$ $$\mathbf e_4$$ $$-\mathbf e_{23}$$ $$-\mathbf e_{42}$$ $$\mathbf e_{43}$$ $$0$$ $$-\mathbf e_{234}$$
$$\mathbf e_2$$ $$0$$ $$0$$ $$\mathbf 1$$ $$0$$ $$0$$ $$-\mathbf e_3$$ $$0$$ $$\mathbf e_1$$ $$0$$ $$\mathbf e_4$$ $$0$$ $$-\mathbf e_{31}$$ $$\mathbf e_{41}$$ $$0$$ $$-\mathbf e_{43}$$ $$-\mathbf e_{314}$$
$$\mathbf e_3$$ $$0$$ $$0$$ $$0$$ $$\mathbf 1$$ $$0$$ $$\mathbf e_2$$ $$-\mathbf e_1$$ $$0$$ $$\mathbf e_4$$ $$0$$ $$0$$ $$-\mathbf e_{12}$$ $$0$$ $$-\mathbf e_{41}$$ $$\mathbf e_{42}$$ $$-\mathbf e_{124}$$
$$\mathbf e_4$$ $$0$$ $$0$$ $$0$$ $$0$$ $$\mathbf 1$$ $$0$$ $$0$$ $$0$$ $$-\mathbf e_3$$ $$-\mathbf e_2$$ $$-\mathbf e_1$$ $$0$$ $$\mathbf e_{12}$$ $$\mathbf e_{31}$$ $$\mathbf e_{23}$$ $$-\mathbf e_{321}$$
$$\mathbf e_{23}$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$\mathbf 1$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$-\mathbf e_1$$ $$0$$ $$0$$ $$\mathbf e_4$$ $$-\mathbf e_{41}$$
$$\mathbf e_{31}$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$\mathbf 1$$ $$0$$ $$0$$ $$0$$ $$0$$ $$-\mathbf e_2$$ $$0$$ $$\mathbf e_4$$ $$0$$ $$-\mathbf e_{42}$$
$$\mathbf e_{12}$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$\mathbf 1$$ $$0$$ $$0$$ $$0$$ $$-\mathbf e_3$$ $$\mathbf e_4$$ $$0$$ $$0$$ $$-\mathbf e_{43}$$
$$\mathbf e_{43}$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$\mathbf 1$$ $$0$$ $$0$$ $$0$$ $$0$$ $$\mathbf e_1$$ $$-\mathbf e_2$$ $$-\mathbf e_{12}$$
$$\mathbf e_{42}$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$\mathbf 1$$ $$0$$ $$0$$ $$-\mathbf e_1$$ $$0$$ $$\mathbf e_3$$ $$-\mathbf e_{31}$$
$$\mathbf e_{41}$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$\mathbf 1$$ $$0$$ $$\mathbf e_2$$ $$-\mathbf e_3$$ $$0$$ $$-\mathbf e_{23}$$
$$\mathbf e_{321}$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$\mathbf 1$$ $$0$$ $$0$$ $$0$$ $$\mathbf e_4$$
$$\mathbf e_{124}$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$\mathbf 1$$ $$0$$ $$0$$ $$\mathbf e_3$$
$$\mathbf e_{314}$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$\mathbf 1$$ $$0$$ $$\mathbf e_2$$
$$\mathbf e_{234}$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$\mathbf 1$$ $$\mathbf e_1$$
$$\large\unicode{x1D7D9}$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$\mathbf 1$$

Right Interior Product

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 right interior products between all pairs of basis elements in the 4D projective geometric algebra $$\mathcal G_{3,0,1}$$.

a   b $$\mathbf 1$$ $$\mathbf e_1$$ $$\mathbf e_2$$ $$\mathbf e_3$$ $$\mathbf e_4$$ $$\mathbf e_{23}$$ $$\mathbf e_{31}$$ $$\mathbf e_{12}$$ $$\mathbf e_{43}$$ $$\mathbf e_{42}$$ $$\mathbf e_{41}$$ $$\mathbf e_{321}$$ $$\mathbf e_{124}$$ $$\mathbf e_{314}$$ $$\mathbf e_{234}$$ $$\large\unicode{x1D7D9}$$
$$\mathbf 1$$ $$\mathbf 1$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$
$$\mathbf e_1$$ $$\mathbf e_1$$ $$\mathbf 1$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$
$$\mathbf e_2$$ $$\mathbf e_2$$ $$0$$ $$\mathbf 1$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$
$$\mathbf e_3$$ $$\mathbf e_3$$ $$0$$ $$0$$ $$\mathbf 1$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$
$$\mathbf e_4$$ $$\mathbf e_4$$ $$0$$ $$0$$ $$0$$ $$\mathbf 1$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$
$$\mathbf e_{23}$$ $$\mathbf e_{23}$$ $$0$$ $$\mathbf e_3$$ $$-\mathbf e_2$$ $$0$$ $$\mathbf 1$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$
$$\mathbf e_{31}$$ $$\mathbf e_{31}$$ $$-\mathbf e_3$$ $$0$$ $$\mathbf e_1$$ $$0$$ $$0$$ $$\mathbf 1$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$
$$\mathbf e_{12}$$ $$\mathbf e_{12}$$ $$\mathbf e_2$$ $$-\mathbf e_1$$ $$0$$ $$0$$ $$0$$ $$0$$ $$\mathbf 1$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$
$$\mathbf e_{43}$$ $$\mathbf e_{43}$$ $$0$$ $$0$$ $$-\mathbf e_4$$ $$\mathbf e_3$$ $$0$$ $$0$$ $$0$$ $$\mathbf 1$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$
$$\mathbf e_{42}$$ $$\mathbf e_{42}$$ $$0$$ $$-\mathbf e_4$$ $$0$$ $$\mathbf e_2$$ $$0$$ $$0$$ $$0$$ $$0$$ $$\mathbf 1$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$
$$\mathbf e_{41}$$ $$\mathbf e_{41}$$ $$-\mathbf e_4$$ $$0$$ $$0$$ $$\mathbf e_1$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$ $$\mathbf 1$$ $$0$$ $$0$$ $$0$$ $$0$$ $$0$$
$$\mathbf e_{321}$$ $$\mathbf e_{321}$$ $$-\mathbf e_{23}$$ $$-\mathbf e_{31}$$ $$-\mathbf e_{12}$$ $$0$$ $$-\mathbf e_1$$ $$-\mathbf e_2$$ $$-\mathbf e_3$$ $$0$$ $$0$$ $$0$$ $$\mathbf 1$$ $$0$$ $$0$$ $$0$$ $$0$$
$$\mathbf e_{124}$$ $$\mathbf e_{124}$$ $$-\mathbf e_{42}$$ $$\mathbf e_{41}$$ $$0$$ $$\mathbf e_{12}$$ $$0$$ $$0$$ $$\mathbf e_4$$ $$0$$ $$-\mathbf e_1$$ $$\mathbf e_2$$ $$0$$ $$\mathbf 1$$ $$0$$ $$0$$ $$0$$
$$\mathbf e_{314}$$ $$\mathbf e_{314}$$ $$\mathbf e_{43}$$ $$0$$ $$-\mathbf e_{41}$$ $$\mathbf e_{31}$$ $$0$$ $$\mathbf e_4$$ $$0$$ $$\mathbf e_1$$ $$0$$ $$-\mathbf e_3$$ $$0$$ $$0$$ $$\mathbf 1$$ $$0$$ $$0$$
$$\mathbf e_{234}$$ $$\mathbf e_{234}$$ $$0$$ $$-\mathbf e_{43}$$ $$\mathbf e_{42}$$ $$\mathbf e_{23}$$ $$\mathbf e_4$$ $$0$$ $$0$$ $$-\mathbf e_2$$ $$\mathbf e_3$$ $$0$$ $$0$$ $$0$$ $$0$$ $$\mathbf 1$$ $$0$$
$$\large\unicode{x1D7D9}$$ $$0$$ $$\mathbf e_{234}$$ $$\mathbf e_{314}$$ $$\mathbf e_{124}$$ $$\mathbf e_{321}$$ $$-\mathbf e_{41}$$ $$-\mathbf e_{42}$$ $$-\mathbf e_{43}$$ $$-\mathbf e_{12}$$ $$-\mathbf e_{31}$$ $$-\mathbf e_{23}$$ $$-\mathbf e_4$$ $$-\mathbf e_3$$ $$-\mathbf e_2$$ $$-\mathbf e_1$$ $$\mathbf 1$$

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.

$$\mathbf a \mathbin{\unicode{x22A8}} \mathbf b = \overline{\underline{\mathbf a} \mathbin{\unicode{x22A2}} \underline{\mathbf b}} = \mathbf a \wedge \overline{\mathbf b}$$
$$\mathbf a \mathbin{\unicode{x2AE4}} \mathbf b = \overline{\underline{\mathbf a} \mathbin{\unicode{x22A3}} \underline{\mathbf b}} = \underline{\mathbf a} \wedge \mathbf b$$

See Also