Interior products

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.

Contractions

The bulk contraction between elements $$\mathbf a$$ and $$\mathbf b$$ is defined as

$$\mathrm{bulk\ contraction}(\mathbf a, \mathbf b) = \mathbf a \vee \mathbf b^\unicode["segoe ui symbol"]{x2605}$$ .

The weight contraction between elements $$\mathbf a$$ and $$\mathbf b$$ is defined as

$$\mathrm{weight\ contraction}(\mathbf a, \mathbf b) = \mathbf a \vee \mathbf b^\unicode["segoe ui symbol"]{x2606}$$ .

The contractions have the effect of subtracting grades so that

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

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

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

and

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

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 $$\mathbf G$$.

Expansions

The bulk expansion between elements $$\mathbf a$$ and $$\mathbf b$$ is defined as

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

The weight expansion between elements $$\mathbf a$$ and $$\mathbf b$$ is defined as

$$\mathrm{weight\ expansion}(\mathbf a, \mathbf b) = \mathbf a \wedge \mathbf b^\unicode["segoe ui symbol"]{x2606}$$ .

The expansions have the effect of subtracting antigrades so that

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

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

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

and

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

The weight expansion satisfies the identity

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

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 $$\mathbb G$$.

In the Book

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

See Also

• Projections