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.
Transwedge Product
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 $$\mathbf a \mathbin{\unicode{x2A53}_k} \mathbf b = 0$$.
An equivalent definition for the transwedge product is given by
- $$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf{\tilde c_{\unicode{x2605}}} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf{\overline c})}$$,
where the right contraction on the right side of the summand is now a left contraction on the left side of the summand.
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.
The geometric product of each pair of the 16 basis elements in the 3D rigid algebra is given by exactly one of the transwedge products. These are shown in the following table, which color codes the transwedge products of order 0, 1, 2, and 3. The transwedge product of order 4 is always zero in this algebra due to the degenerate metric.
For vectors $$\mathbf a$$ and $$\mathbf b$$, we have
- $$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf b$$,
where the dot product on the right is the result of the right contraction $$\mathbf b \vee \mathbf a^{\unicode{x2605}}$$, and we have dropped the reverse operation because $$\mathbf a$$ is a vector.
For bivectors $$\mathbf a$$ and $$\mathbf b$$, we have
- $$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{2}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf b$$,
where the dot product is again attributable to the right contraction $$\mathbf b \vee \mathbf{\tilde a}^{\unicode{x2605}}$$. The middle term given by the transwedge product of order 1 generates the grade-2 part of the geometric product.
Transwedge Antiproduct
The transwedge antiproduct of order $$k$$ is the dual of the transwedge product of order $$k$$. It is written with a double downward wedge having either a subscript or an underscript indicating the order. It is defined as
- $$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_{n - k}}{(\mathbf{\underline c} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf{\smash{\mathbf{\underset{\Large\unicode{x7E}}{c}}}^{\unicode{x2606}}})}$$,
where the wedge and antiwedge products have traded places, the bulk dual has become the weight dual, and the reverse has become the antireverse.
The sum of all possible transwedge antiproducts yields the geometric antiproduct. That is,
- $$\displaystyle\mathbf a \mathbin{\unicode{x27C7}} \mathbf b = \sum_{k=0}^n{\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b}$$,
The components of the geometric antiproduct are shown in the following table, which color codes the transwedge antiproducts of order 0, 1, 2, and 3.
