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Avoiding Trigonometry (2013)

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Re: Avoiding Trigonometry (2013)

#2

Post by srean »

This has been some sort of a mix of peeve and a moment of enlightenment of mine when I understood this.

I wholeheartedly agree with the point being made in the post. I had commented about this in the recent asin() post but deleted thinking it might not be of general interest.

If you care about angles and rotations in the plane, it is often profitable to represent an angle not by a scalar such as a degree or a radian but as a tuple

    (cos \theta, sin \theta)
or as a complex number.

This way one can often avoid calls to expensive trigonometric functions. One may need calls to square roots and general polynomial root finding.

In Python you can represent an angle as a unit complex numbers and the runtime will do the computations for you.

For example, if you needed the angular bisector of an angle subtended at the origin (you can translate the vertex there and later undo the translation), the bisector is just the geometric mean of the arms of the angle

   sqrt(z1 * z2)
Along with stereographic transform and its inverse you can do a lot.

This is directly related to the field of algebraic numbers.

With complex numbers you get translations, scaled rotations and reflections. Sufficient for Euclidean geometry.

Re: Avoiding Trigonometry (2013)

#3

Post by chriswarbo »

Norman Wildberger takes this to the extreme with Rational Trigonometry https://en.wikipedia.org/wiki/Divine_Proportions:_Rational_T...

It eschews angles entirely, sticking to ratios. It avoids square roots by sticking to "quadrances" (squared distance; i.e. pythagoras/euclidean-distance without taking square roots).

I highly recommend Wildberger's extensive Youtube channels too https://www.youtube.com/@njwildberger and https://www.youtube.com/@WildEggmathematicscourses

He's quite contrarian, so I'd take his informal statements with a pinch of salt (e.g. that there's no such thing as Real numbers; the underlying argument is reasonable, but the grand statements lose all that nuance); but he ends up approaching many subjects from an interesting perspective, and presents lots of nice connections e.g. between projective geometry, linear algebra, etc.

Re: Avoiding Trigonometry (2013)

#4

Post by srean »

chriswarbo wrote: post #3

Norman Wildberger takes this to the extreme with Rational Trigonometry https://en.wikipedia.org/wiki/Divine_Proportions:_Rational_T... It eschews angles entirely, sticking to ratios. It avoids square roots by sticking to "quadrances" (squared distance; i.e. pythagoras/euclidean-distance without taking square roots). I highly recommend Wildberger's extensive Youtube channels too https://www.youtube.com/@njwildberger a…

He maybe considered contrarian but his math is sound.

Re: Avoiding Trigonometry (2013)

#6

Post by srean »

storus wrote: post #5

In principle, wouldn't a change of basis be all that is needed?

It's a little more than change of basis, although change of basis is an important part of it. It converts many apparently trigonometric operations into algebraic ones, root finding included.

There are certain drawbacks. If the solution involves non-algebraic numbers there is no getting away from the transcendental numbers (that ultimately get approximated by algebraic numbers).

Re: Avoiding Trigonometry (2013)

#9

Post by GistNoesis »

I think it boils down to the alternate view of rotations as two successive reflections.

You can then use householder matrix to avoid trigonometry.

These geometric math tricks are sometimes useful for efficient computations.

For example you can improve Vector-Quantization Variational AutoEncoder (VQ-VAE) using a rotation trick, and compute it efficiently without trigonometry using Householder matrix to find the optimal rotation which map one vector to the other. See section 4.2 of [1]

The question why would someone avoid trigonometry instead of looking toward it is another one. Trigonometry [2] is related to the study of the triangles and connect it naturally to the notion of rotation.

Rotations [3] are a very rich concept related to exponentiation (Multiplication is repeated addition, Exponentiation is repeated multiplication).

As doing things repeatedly tend to diverge, rotations are self stabilizing, which makes them good candidates as building blocks for the universe [4].

Because those operations are non commutative, tremendous complexity emerge just from the order in which the simple operations are repeated, yet it's stable by construction [5][6]

[0]https://en.wikipedia.org/wiki/Householder_transformation

[1]https://arxiv.org/abs/2410.06424

[2]https://en.wikipedia.org/wiki/Trigonometry

[3]https://en.wikipedia.org/wiki/Matrix_exponential

[4]https://en.wikipedia.org/wiki/Exponential_map_(Lie_theory)

[5]https://en.wikipedia.org/wiki/Geometric_algebra

[6]https://en.wikipedia.org/wiki/Clifford_algebra

Re: Avoiding Trigonometry (2013)

#10

Post by djmips »

xeonmc wrote: post #7

This is just https://en.wikipedia.org/wiki/Rodrigues%27_rotation_formula

This is avoiding an common but unnecesary round trip. When your inputs are vectors, angles are an unnecessary intermediate representation. You can substitute the geometric meaning of dot and cross product directly into the Rodrigues matrix and get by with less operations overall. It's more elegant, uses less instructions.
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