Transwedge products

The transwedge product is a generalization of the exterior product and interior product that also includes a transitional sequence of liminal products between exterior and interior.

The transwedge product of order $$k$$ is written with a double upward wedge having either a subscript or an underscript indicating the order. It is defined as

$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf{\underline c} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf{\tilde c^{\unicode{x2605}}})}$$.

The set $$\mathcal B_k$$ is the set of all basis elements having grade $$k$$. When $$k = 0$$, this set is simply $$\{\mathbf 1\}$$, and the transwedge product reduces to the wedge (exterior) product. That is,

$$\displaystyle\mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b$$.

When $$k = \operatorname{gr}(\mathbf a)$$, the transwedge product reduces to the right contraction $$\mathbf b \vee \mathbf{\tilde a^{\unicode{x2605}}}$$, which is an interior product. If $$k > \operatorname{gr}(\mathbf a)$$, then $$\mathbf a \mathbin{\unicode{x2A53}_k} \mathbf b = 0$$.

For operands $$\mathbf a$$ and $$\mathbf b$$ having grades $$g$$ and $$h$$, the transwedge product of order $$k$$ generates a result having grade $$g + h - 2k$$, assuming it’s nonzero.

The sum of all possible transwedge products yields the geometric product. That is,

$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \sum_{k=0}^n{\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b}$$,

where $$n$$ is the dimension of the algebra, which is 4 in the 3D rigid geometric algebra.