71–80 of 225 posts
Earlier quoted context omitted.
> all angles are without a unit. Dimensionless, sure, but what do you mean here? Radians and degrees are units, are they not?
In a very awkward way: rad is m/m, which is 1...
I think I cautiously agree with this notion to some extent, but IMHO the real answer is that it's application-dependent, and if you're writing a low-level trig library and you have to pick one or the other, it really isn't clear to me that turns should win over radians. I expect many systems that use trigonometry would sometimes use small-angle approximations either for efficiency or to bootstrap to the general case.…
I don't have a super-wide gamut of experience here and numerical analysis isn't my specialty, but nearly all trig implementations I've looked into (in both software and hardware) make heavy use of lookup tables and other shortcuts. I've never seen a Taylor series used in a general implementation - not saying it doesn't exist anywhere, but in most cases that I'm familiar with you could support turns just as easily wit…
Tend to use radians because we're moving from written proofs or simulations into embedded code in such systems. The code needs to read and work the same as those.
Earlier quoted context omitted.
You generally can’t apply functions to dimensional units. The only thing units can do is be multiplied or divided together. So I can multiply a mass by a distance or divide a distance by a speed, and I can multiply the result by a scalar; but I can’t take the sine of a distance or the logarithm of a time or exponentiate a mass. Those are things I can only do to scalars. ‘But wait!’ You may cry: ‘the formula for a tra…
I basically agree, at least for standard functions like sin, cos, tan, exp etc. It is even possible to see mistakes in equations just by checking that all the units to standard functions cancel out making the arguments dimensionless. On the other hand I am still unhappy with calling the ratio of two quantities, that happen to have the same units, "dimensionless". This way, you could create any two "dimensionless" qua…
Edit: Apparently "same dimension" doesn't imply "same unit".
Same advantages as here but multiplied times 400...
I like to store angles as turns in my own code, because (as noted) it makes quarter-turns computable without rounding. OTOH if you need, say, twelfths of a turn, you might want to just store angles as degrees since that’s already common. Michael Spivak, in Calculus (3rd ed p. 301) considers the unit choice to be a property of the function and initially defines sin° and sinʳ (before settling on sin meaning sinʳ) and c…
You generally can’t apply functions to dimensional units. The only thing units can do is be multiplied or divided together. So I can multiply a mass by a distance or divide a distance by a speed, and I can multiply the result by a scalar; but I can’t take the sine of a distance or the logarithm of a time or exponentiate a mass. Those are things I can only do to scalars. ‘But wait!’ You may cry: ‘the formula for a tra…
Perhaps not in mathematics, but in programming that's clearly possible. I guess programming is more general than mathematics.
In the old days of making 8 bit video games we used BRADs of 0-255 - worked well and the wrap was easy.
I miss strict, deterministic unsigned addition overflow. In many modern languages, all kinds of verbose hoops are required to get this behavior and there's a chance it will generate terrible machine code.
sin(1*pi) = 1.2246e-16
That should put to bed that whole τ crap. "But the symbol τ is used for other things!" Yeah, yeah, yeah, just write turn. Even better, because it's more international and has more precedent, write rev for revolution.