Avoiding Trigonometry (2013)

My take as a graphics programmer is that angles are perfectly fine as inputs. Bring 'em! And we'll use the trig to turn those into matrices/quaternions/whatever to do the linear algebra. Not a problem.

I'm a trig-avoider too, but see it more as about not wiggling back and forth. You don't want to be computing angle -> linear algebra -> angle -> linear algebra... (I.e., once you've computed derived values from angles, you can usually stay in the derived values realm.)

Pro-tip I once learned from Eric Haines (https://erich.realtimerendering.com/) at a conference: angles should be represented in degrees until you have to convert them to radians to do the trig. That way, user-friendly angles like 90, 45, 30, 60, 180 are all exact and you can add and subtract and multiply them without floating-point drift. I.e., 90.0f is exactly representable in FP32, pi/2 is not. 1000 full revolutions of 360.0f degrees is exact, 1000 full revolutions of float(2*pi) is not.

Almost every critique of the axiom of infinity is philosophical. I don't think you can just say "the axiom is sound, so what's your point". And you don't even get to claim that because of Godel's incompleteness theorem.

The axioms were not handed to us from above. They were a product of a thought process anchored to intuition about the real world. The outcomes of that process can be argued about. This includes the belief that the outcomes are wrong even if we can't point to any obvious paradox.

You're right that quaternions don't work for those. Vec3 is the move IMO. Direction is axis; len is magnitude.

He's still computing cross(z, d) and dot(z, d) separately. that looks like a code smell to me. with quaternions this would be easier: just calculate the quotient between z and d and take the square root (which means adding 1 and renormalising). the square root is necessary if one is dealing with vectors, which live in a kind of square-y space. finding the rotation between two spinors is even simpler: it's just the quotient of the the spinors as quaternions. unfortunately hamilton's view that quaternions are the quotient of vectors has never been quite abandoned. it's much more natural to think of them as quotients of spinors.

I had to check the precise definition for trigonometry while writing my comment, found it interesting so I added a reference.

As with many subject that we learn early in school, it's often interesting revisiting them as adult to perceive additional layer of depth by casting a new look.

With trigonometry we tend to associate it with circle. But fundamentally it's the study of tri-angles.

What is interesting is that the whole theory is "relative". I would reference the wikipedia page for angle but it may make me look like an LLM. The triangle doesn't have positions and orientation baked-in, what matters is the length of the sides and the angle between them.

The theory by definition becomes translation and rotation invariant. And from this symmetry emerge the concept of rotations.

What is also interesting about the concept of angle is that it is a scalar whereas the original objects like lines live in an higher dimension. To avoid losing information you therefore need multiple of these scalars to fully describe the scene.

But there is a degree of redundancy because the angles of a triangle sums to pi. And from this degree of freedom results multiple paths to do the computations. But with this liberty comes the risks of not making progress and going in circles. Also it's harder to see if two points coming from different paths are the same or not, and that's why you have "identities".

Often for doing the computation it's useful to break the symmetry, by picking a center, even though all points could be centers, (but you pick one and that has made all the difference).

Similar situation arise in Elliptic Curve Cryptography, where all points could have the same role, but you pick one as your generator. Also in physics the concept of gauge invariance.