4 ms·

> The concept is that you can think of a line as a circle with infinite radius That's a different thing though. "Thinking of something as" is not the same as "

> The concept is that you can think of a line as a circle with infinite radius That's a different thing though. "Thinking of something as" is not the same as "is" in terms of types. The OP is right that in terms of programming ergonomics, it would be better to have distinct types that share the same operations, so you can _treat_ them in similar ways.

jiggawatts 6y ago
You can have mathematically meaningful sub-types of a GA multi-vector that also map nicely to some basic shapes or transformations. Generally speaking, a multi-vector can be broken down by which "grades" it contains. There are subspaces like "all even grades" that are closures under all basic operations. If I remember correctly, the "even subset" of a 3D GA multi-vector is isomorphic to the Quarternions, which is why they turn up so often in 3D graphics libraries. But think about how messy this is! These 3D libraries are blending together subsets of vaguely related algebras that are not "compatible" from a type theory perspective and so they're forced to convert back-and-forth. It's like mixing Unicode UTF-8 with Latin-1 byte strings. If you squint, it looks like one is the subset of the other... but not really. With a pure GA-base library, the "rotation interpolation" functions would use the "even multivector subset" type, and be totally compatible with the rest of the library. It's the same type definitions, just with different parameters. This would be like using UTF-8 Unicode throughout the program, but having the ability to define a string-based type that only allows the "Basic Multilingual Plane".
- valenterry 6y ago
  > It's the same type definitions, just with different parameters The wording is important here. These "type definitions" you are talking about are not types. We usually call them "type constructors". So we mean the same thing, but please don't call multivector out to be a type without specifying the "grade" (then it becomes a type).

jiggawatts 6y ago

You can have mathematically meaningful sub-types of a GA multi-vector that also map nicely to some basic shapes or transformations. Generally speaking, a multi-vector can be broken down by which "grades" it contains. There are subspaces like "all even grades" that are closures under all basic operations. If I remember correctly, the "even subset" of a 3D GA multi-vector is isomorphic to the Quarternions, which is why they turn up so often in 3D graphics libraries. But think about how messy this is! These 3D libraries are blending together subsets of vaguely related algebras that are not "compatible" from a type theory perspective and so they're forced to convert back-and-forth. It's like mixing Unicode UTF-8 with Latin-1 byte strings. If you squint, it looks like one is the subset of the other... but not really. With a pure GA-base library, the "rotation interpolation" functions would use the "even multivector subset" type, and be totally compatible with the rest of the library. It's the same type definitions, just with different parameters. This would be like using UTF-8 Unicode throughout the program, but having the ability to define a string-based type that only allows the "Basic Multilingual Plane".

valenterry 6y ago
> It's the same type definitions, just with different parameters The wording is important here. These "type definitions" you are talking about are not types. We usually call them "type constructors". So we mean the same thing, but please don't call multivector out to be a type without specifying the "grade" (then it becomes a type).