# Interior products

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 projective 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.

## 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 projective 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.