The multivector method .is_versor() purports to follow the test on p. 533 of Leo Dorst's Geometric Algebra for Computer Science. But:

1. Dorst's test itself, as it appears in GACS, is incorrect, as shown by a simple counterexample.
2. The algorithm used in .is_versor() does not faithfully implement Dorst's test.
3. The algorithm used in .is_versor() has a simple counterexample.

I have devised four conditions satisfied by versors which, if satisfied, also imply that a multivector is a versor. I am currently checking with Dorst as to his opinion, but I'm pretty confident in my reasoning.

Let V be a multivector, and let V.rev() and V.g_invol() be its reverse and grade involute. Let x be a generic 1-vector. Then V will be a versor if and only if it meets each of the following conditions:

1. V * V.rev() and V.rev() * V are both scalars.
2. V * V.rev() is not zero.
3. V.g_invol() * V.rev() is a scalar.
4. V.g_invol() * x * V.rev() is a vector.

The counterexamples, if anyone's interested, are these:

• To Dorst's test as it appears in GACS: Let V = 1 + I where I = e0 e1 e2 e3 is the unit pseudoscalar of the spacetime algebra G(1,3). Then V meets the conditions described by GACS, but the geometric product V (reverse of V) is not a scalar, showing that V cannot be a versor.
• To the algorithm currently used in .is_versor(): Let V = e1 + e2 e3 in the algebra G(3,0) of Euclidean 3-space. Then V passes the test in .is_versor(), but V cannot be a versor as it is a mixture of odd and even grades.

I will post further after hearing back from Dorst. In the meanwhile, I have a proposed rewrite of the .is_versor() method that I am testing. Once my testing is satisfied, I'll post my rewrite here.

Greg Grunberg
[email protected]
2025 March 04