4 ms·

This makes a lot of sense when you are expecting to take derivatives, integrals, or use complex numbers. In cases outside of that, radians lose their advantage

by rocqua 4y ago

This makes a lot of sense when you are expecting to take derivatives, integrals, or use complex numbers. In cases outside of that, radians lose their advantage over turns.

kragen 4y ago
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. Doing trigonometry in Galois fields sounds super cool though and I hope I understand it one day. 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-raytracer http://canonical.org/~kragen/sw/aspmisc/my-very-first-raytra... and https://gitlab.com/kragen/bubbleos/-/blob/master/yeso/sdf.lua 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! ______ * His main concern is that irrational numbers don't, in some sense, really exist, so they're a bad basis for trigonometry. As I understand it, not only is Platonism now a minority among foundations-of-mathematics types, but even Platonists generally believe that irrational numbers are just as real as rational ones, so as I understand it, Wildberger's viewpoint is held by quite a small minority. That doesn't, of course, imply anything about whether it's correct.
- kazinator 4y ago
  So if I wanted to, say, calculate the height of a pole from the length of its shadow, I should use Wildberger's rational trig, because I don't need derivatives, integrals, or complex numbers? :)
  - kragen 4y ago
    You can do that without derivatives, integrals, complex numbers, sines, cosines, tangents, exponentials, square roots, or even addition and subtraction. If the pole is plumb, erect a pole smaller than you with a plumb bob, measure its length a and its shadow's length b, and then calculate the height of the original pole from its shadow's length c as ac/b. If you have a slide rule, you can do this in a single motion: align c on the C scale over b on D and read off the answer on C above a on the D scale.
  - lupire 4y ago
    Sure. No smiley face needed, unless geometry makes you smile.
- rocqua 4y ago
  I never saw rational trigonometry, but I imagine it is mathematically overkill in another sense. When calling `sin` or `cos` in code you get back a rational approximation to the presumably irrational answer. From what I quickly gleamed, using spreads would be really annoying to represent rotations, because you lose 'aditivity'. Two subsequent rotations with spreads a and b do not have spread `a` and spread `b`. Lots of code using trig is about rotations, so losing that feature would probably not be the nicest.
  - kragen 4y ago
    In order to reason about the quality of the rational approximation, you need to reason about the irrational number it's an approximation to, for example as a Dedekind cut. Wildberger, rightly or wrongly, doesn't trust reasoning about that sort of thing; he wants to ground everything in the rationals. Maybe he's worried about finding some equivalent of Russell's barber paradox in the irrationals. Also, of course, irrational numbers don't allow you to extend trigonometric theorems to Galois fields, complex numbers, and so on, which to my mind is a much more interesting direction. I don't know how much you lose if you use Wildberger's construction into a division algebra like the quaternions. I don't actually know how you compute the "angle sum" in terms of Wildberger's spreads, but I'm pretty confident that there's a way to compute it, and it's pretty simple. In the unit-vector representation I described, angle-sum is not just simple addition, but it's really not that bad: (a + bi)(c + di) = (ac - bd) + (ad + bc)i. That's four real multiplications, an addition, and a subtraction, and the result is exact if computed in bignums or bignum rationals. This is usually cheaper than computing sine and cosine, especially if you can use SIMD or vectors, and of course if you want to rotate some points around a center, you end up having to multiply by the sin and cos in exactly the same way anyway. (Maybe in strength-reduced fashion if you're texture-mapping or something, but that applies just as well to representing the angles as sin and cos.)

by rocqua 4y ago

This makes a lot of sense when you are expecting to take derivatives, integrals, or use complex numbers. In cases outside of that, radians lose their advantage over turns.

kragen 4y ago

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. Doing trigonometry in Galois fields sounds super cool though and I hope I understand it one day. 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-raytracer http://canonical.org/~kragen/sw/aspmisc/my-very-first-raytra... and https://gitlab.com/kragen/bubbleos/-/blob/master/yeso/sdf.lua 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! ______ * His main concern is that irrational numbers don't, in some sense, really exist, so they're a bad basis for trigonometry. As I understand it, not only is Platonism now a minority among foundations-of-mathematics types, but even Platonists generally believe that irrational numbers are just as real as rational ones, so as I understand it, Wildberger's viewpoint is held by quite a small minority. That doesn't, of course, imply anything about whether it's correct.

kazinator 4y ago
So if I wanted to, say, calculate the height of a pole from the length of its shadow, I should use Wildberger's rational trig, because I don't need derivatives, integrals, or complex numbers? :)
- kragen 4y ago
  You can do that without derivatives, integrals, complex numbers, sines, cosines, tangents, exponentials, square roots, or even addition and subtraction. If the pole is plumb, erect a pole smaller than you with a plumb bob, measure its length a and its shadow's length b, and then calculate the height of the original pole from its shadow's length c as ac/b. If you have a slide rule, you can do this in a single motion: align c on the C scale over b on D and read off the answer on C above a on the D scale.
- lupire 4y ago
  Sure. No smiley face needed, unless geometry makes you smile.
rocqua 4y ago
I never saw rational trigonometry, but I imagine it is mathematically overkill in another sense. When calling `sin` or `cos` in code you get back a rational approximation to the presumably irrational answer. From what I quickly gleamed, using spreads would be really annoying to represent rotations, because you lose 'aditivity'. Two subsequent rotations with spreads a and b do not have spread `a` and spread `b`. Lots of code using trig is about rotations, so losing that feature would probably not be the nicest.
- kragen 4y ago
  In order to reason about the quality of the rational approximation, you need to reason about the irrational number it's an approximation to, for example as a Dedekind cut. Wildberger, rightly or wrongly, doesn't trust reasoning about that sort of thing; he wants to ground everything in the rationals. Maybe he's worried about finding some equivalent of Russell's barber paradox in the irrationals. Also, of course, irrational numbers don't allow you to extend trigonometric theorems to Galois fields, complex numbers, and so on, which to my mind is a much more interesting direction. I don't know how much you lose if you use Wildberger's construction into a division algebra like the quaternions. I don't actually know how you compute the "angle sum" in terms of Wildberger's spreads, but I'm pretty confident that there's a way to compute it, and it's pretty simple. In the unit-vector representation I described, angle-sum is not just simple addition, but it's really not that bad: (a + bi)(c + di) = (ac - bd) + (ad + bc)i. That's four real multiplications, an addition, and a subtraction, and the result is exact if computed in bignums or bignum rationals. This is usually cheaper than computing sine and cosine, especially if you can use SIMD or vectors, and of course if you want to rotate some points around a center, you end up having to multiply by the sin and cos in exactly the same way anyway. (Maybe in strength-reduced fashion if you're texture-mapping or something, but that applies just as well to representing the angles as sin and cos.)