Get the latest tech news
The Transwedge Product
Eric Lengyel • May 23, 2025 Introductory texts on geometric algebra often begin by showing how the geometric product is a combination of the wedge product and the dot product, giving us the formula[1] \(\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \cdot \mathbf b\).(1) However, the above formula holds only for vectors \(\mathbf a\) and \(\mathbf b\). When \(\mathbf a\) and \(\mathbf b\) are allowed to assume values of higher grade, the geometric product generally yields more terms, especially in higher dimensions.
This table includes almost all of the ways that we can combine two objects that could each be a point, line, or plane to create a new related geometry, but there is one operation that’s conspicuously absent. The new circle also needs to be adjusted to satisfy the geometric constraint, but in CGA, that involves dividing by the norm of a trivector in a six-dimensional base ring. It’s usually convenient to work in a basis for which the metric is not diagonal in CGA, but this can make it difficult to implement a geometric product for general calculations.
Or read this on Hacker News