I recently read John Vince's Geometric Algebra for Computer Graphics, which seemed excellent – written, the author explains, because he realized you had to virtually be a mathematician to understand the books in the field, and he wanted to write something much more accessible. It sounded great until Chapter 11, Conformal Geometry..
Various arguments have been proposed for adopting this model such as, conformal space:
• is homogeneous and consequently removes the origin as being a special point
• supports points and lines at infinity
• provides a single geometric mechanism for representing lines, circles, planes and spheres, which introduces a rare quality of elegance to problem solving
• enjoys all the normal features of Euclidean space.
A model of space that possesses such a range of positive features sounds too good to be true! But remember, in mathematics there is no such thing as a free lunch. So what price must we pay for this model?
Then Vince explains that the conformal model, "an alternative space to Euclidean space for solving 3D geometric problems", is covered by a patent..of Hestenes, US Patent 6,853,964 System for encoding and manipulating models of objects. Vince says "Licences to employ this system for commercial purposes are available through the patent holders, although there are no restrictions for academic research and educational use. All of the above may be a small price to pay for the benefits associated with the conformal model, and I will do my best to explain the model in this chapter."
Uh what?! So it actually costs money to use... That rather put me off Hestenes' noble mission. It seems...bizarre that what seems a whole subfield of mathematics is patented, rather different to patenting a particular algorithm or technique. Very strange, after reading so much of Hestenes' wondering why the superior GA isn't used everywhere. (Is it to make the world a better place..or to make Hestenes rich?..) I stopped reading at that point.
As far as using Geometric Algebra in a Euclidean setting goes, most of my programming is in 2D, but when I do 3D next I will consider using it, it seems very cool. Most of the features/differences/advantages over the usual vector/matrix geometry only appear in 3 or more dimensions. Vince's book is great at explaining it, highly recommended.
It seems the patent has been discontinued: https://patents.google.com/patent/US6853964B1/en
Oh thank you. Hmm "Status Expired - Fee Related .. 2022-07-13 Adjusted expiration" .. is that like when your power gets cut off from not paying the bill? p.s. Vince's book was published in 2008.
Is this... legal? How on earth was a patent like that approved, and surely it wouldn't hold up to scrutiny, right?
Most people familiar with the US patent office will agree that it's totally broken. The reviewers are not scientists, and aren't required to understand what's being proposed in a patent. They simply perform a search for prior art.
As a result, all kinds of pseudoscience has gotten "patented", for example this patent for detecting "tachyonized" materials (you know, materials that have been infused with tachyons, the hypothetical particles that travel faster than light):
https://patents.google.com/patent/US7760357B2/en
It looks like the patent was approved in between the CAFC decision of State St. (subject matter is patentable if it produces a "useful, concrete, and tangible result") and SCOTUS's Bilski decision (State St. is on crack and not good case law). Bilski and later Alice have made it rather clear that these kinds of abstract patents aren't going to fly, not that it stops patent attorneys from pretending otherwise.
So this patent very likely wouldn't hold up in court. It was approved in a time when the USPTO more or less gave up on adjudicating patentability (thanks to the State St. decision).