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> When you approximate functions by polynomials, including the trigonometric functions, the Taylor series are never used, because they are inefficient (too much computation for a given error). Other kinds of polynomials are used for function approximations.
Can you point me to some implementation of sin that’s not actually using Taylor expansion in some form? Because most that I am aware of do in fact use Taylor series. See glibc for example:
https://github.com/bminor/glibc/blob/release/2.34/master/sys...
This might have something to do with the Taylor’s theorem. You know, that the Taylor’s polynomial of the order n is the only polynomial of order n that satisfies |f(x)-T(x)|/(x-a)^(n+1) -> 0 as x -> a. In other words, the Taylor polynomial of order n is the unique polynomial approximation to f around a.
If you're not using derivatives, integrals, or complex numbers, maybe you'd be better off using Wildberger's "rational trigonometry" with quadrances and spreads instead of angles? I haven't actually tried it myself. Wildberger's motivation is a sort of ultra-strict Platonism mixed with the desire to extend analytic geometry to fields other than the real numbers, though, so it wouldn't be surprising if it wasn't actually a simpler way to write Asteroids.
Alternatively you can just directly represent angles as unit vectors in the desired direction, which is pretty much the same as using complex numbers. Angle addition is complex multiplication, angle bisection is complex square root, and computing the sine and cosine is simplicity itself. (This takes twice as much space. If you choose to store only the real part of the complex number, you can only represent angles up to half a turn, same as in Wildberger's approach, you lose some precision near the limits, and the other operations require some extra computations.) I have tried this, for example in http://canonical.org/~kragen/sw/aspmisc/my-very-first-raytra... and https://gitlab.com/kragen/bubbleos/-/blob/master/yeso/sdf.lu..., and in the cases I've tried it, it works great.
I'm interested to hear other people's experiences on this count!