[a_e_k]

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.

[the__alchemist]

Hah. I think we're and the author of both articles on the same page about this. (I had to review my implementations to be sure). I'm a fan of all angles are radians for consistency, and it's more intuitive to me. I.e. a full rot is τ. 1/2 rot is 1/2 τ etc. Pi is standard but makes me do extra mental math, and degrees has the risk of mixing up units, and doesn't have that neat rotation mapping.

Very good tip about the degrees mapping neatly to fp... I had not considered that in my reasoning.

[adrian_b]

If you want consistency, you should measure all angles in cycles, not in radians.

Degrees are better than radians, but usually they lead to more complications than using consistently only cycles as the unit of measure for angles (i.e. to plenty of unnecessary multiplications or divisions, the only advantage of degrees of being able to express exactly the angle of 30 degrees and its multiples is not worth in comparison with the disadvantages).

The use of radians introduces additional rounding errors that can be great at each trigonometric function evaluation, and it also wastes time. When the angles are measured in cycles, the reduction of the input range for the function arguments is done exactly and very fast (by just taking the fractional part), unlike with the case when angles are measured in radians.

The use of radians is useful only for certain problems that are solved symbolically with pen on paper, because the use of radians removes the proportionality constant from the integration and derivation formulae for trigonometric function. However this is a mistake, because those formulae are applied seldom, while the use of radians does not eliminate the proportionality constant (2*Pi), but it moves the constant into each function evaluation, with much worse overhead.

Because of this, even in the 19th century, when the use of radians became widespread for symbolic computations, whenever they did numeric computations, not symbolic, the same authors used sexagesimal degrees, not radians.

The use of radians with digital computers has always been a mistake, caused by people who have been taught in school to use radians, because there they were doing mostly symbolic computations, not numeric, and they have passed this habit to computer programs, without ever questioning whether this is the appropriate method for numeric computations.

[the__alchemist]

Holy shit I have to try this.

[adrian_b]

Unfortunately, IEEE Std 754 contains a huge mistake, which has been followed by some standard libraries for programming languages.

As an alternative to the trigonometric functions with arguments measured in radians, it recommends a set of functions with arguments measured in half-cycles: sinPi, cosPi, atanPi, atan2Pi and so on.

I do not who is guilty for this, because I have never ever encountered a case when you want to measure angles in half-cycles. There are cases when it would be more convenient to measure angles in right angles (i.e. quarters of a cycle), but half-cycles are always worse than both cycles and right angles. An example where measuring angles in cycles is optimal is when you deal with Fourier series or Fourier transforms. When the unit is the cycle that deletes a proportionality constant from the Fourier formulae, and that constant is always present when any other unit is used, e.g. the radian, the degree or the half-cycle.

Due to this mistake in the standard, it is more likely to find a standard library that includes these functions with angles measured in half-cycles than a library with the corresponding functions for cycles. Half-cycles are still better than radians, by producing more accurate results and being faster, but it may be hard to avoid some scalings by two. However, usually it is not necessary to do a scaling at every invocation, but there are chances that the scalings can be moved outside of loops.

Such functions written for angles measured in half-cycles can be easily modified to work with arguments measured in cycles, but if one does not want to touch a standard library, they may be used as they are.

When one uses consistently the cycle as the unit of angle, which is consistent with measuring frequencies in Hertz, i.e. cycle per second, instead of measuring them in radian per second, one must pay attention to the fact that a lot of formulae from most handbooks of physics are incorrect. Despite the claim that those formulae are written in a form that is independent of the system of units, this claim is false because many formulae are written in a form that is valid only when the unit of angle is the radian.

For example, all formulae for quantities related to rotation movements, as they are written in modern handbooks contain the "radius". This use of the "radius" creates a wrong mental model of the rotational quantities, both for students and also even for many experienced physicists.

In reality, in all those formulae, e.g. in the definition of the angular momentum, in order to obtain the correct formulae one must replace the "radius" with the inverse of the curvature of the trajectory of the movement. Thus the angular momentum is not the product of the linear momentum by the radius, but it is the ratio between the linear momentum and the curvature.

Then one must use the correct definition for the curvature. Most handbooks define the curvature as the inverse of the radius. This is a wrong definition, which is based on the non-explicit assumption that angles are measured in radians.

The correct definition of the curvature is as the ratio between rotation angle and length, for the movement, i.e. more precisely it is the derivative of the rotation angle as a function of the length of the curve on which something moves. When angles are measured in radians, the curvature is the inverse of the radius. When angles are measured in cycles, the curvature is the inverse of the perimeter. Thus with angles measured in cycles the angular momentum is defined as the product between the linear momentum and perimeter. Similarly for the other rotational quantities, like angular velocity and acceleration, moment of inertia and so on.

[the__alchemist]

Great's great info on curvature! Also don't see the use of the half cycle, although obviously see use of cycle. (As was thinking in terms of them anyway, just with a tau term to convert to radians.)

[donkeybeer]

Then why restrict to arbitrary degrees, might as well use 1/2**31sts of the circle. Or a larger higly composite number if you want your 3's and 5's.