4 ms·

y’’ = -y Radians are God’s chosen angular unit. If you want to do mathematics, you have to use radians.

gugagore 4y ago
Because I didn't see it mentioned yet, this is the same feature that makes the small-angle approximation of sin elegant: sin(x) ≈ x .
- tsimionescu 4y ago
  With a sin designed for turns, that is just sin(x) ~= 2pi * x or tau * x. Slightly uglier, but still quite simple.
anonymoushn 4y ago
video games don't usually need to take unitless derivatives of sines.
- 7steps2much 4y ago
  True, but if you are talking about video games in particular you could just build a sin/cos function with different parameters into the game engine. have a `sin(x)` where the unit of x is radiants and a `sin_turn(x)` where x is expressed in turns. Video games especially are a great situation to do it like this because they often use a framework (game engine) that was specifically created for this purpose.
  - anonymoushn 4y ago
    Yes, this is the topic of the fine article and the point that is being made. Ideally both of the multiplies (one in the user code to multiply by pi and another in the engine code to divide by pi) can be omitted.
bmacho 4y ago
What subfield of mathematics? Surely if you do differential equations, your trigonometric functions will eat radians. But e.g. for geometry, where you do basic arithmetic operations on angles, turns are a bit more convenient than radians. (Radians are not that inconvenient if you denote 6.28 somehow, but then again, why not just use turns.)
- zarzavat 4y ago
  The situation is similar to that of the logarithm and exponential functions[0]. There’s a log2 function and a log10 function and they are both useful. But when we talk about the log() function there can be no doubt that it is to base e. If you want to define a sinT() function that works in turns then that’s totally fine. But the sin() function is defined as taking an argument scaled in radians, because it is mathematically natural. Mathematically speaking, all trig functions are in radians. When you write sin(90°) the degree symbol ° is a conversion factor. I blame calculators for confusing high schoolers into believing that there is a separate set of functions that work in degrees. [0] unsurprisingly because Euler’s formula equates the trigonometric and exponential functions.
  - bmacho 4y ago
    The sine function, defined as x-x^3/3!+... doesn't take argument scaled in radians. It takes real numbers. It has nothing to do with radians, really. Or even with angles. The other sine function, defined using right triangles takes argument in Angles, also has nothing to do with the measuring unit. (Also I don't know who told you that log() denotes log_e. Maybe in your narrow environment, but definitely not universal between fields and languages. Personally I prefer ln, ld and lb for natural, decimal and binary logarithms.)
    - lkuty 4y ago
      "Also I don't know who told you that log() denotes log_e. Maybe in your narrow environment," e.g. https://reference.wolfram.com/language/ref/Log.html https://reference.wolfram.com/language/ref/Log.html But I must confess that we had ln() in university courses and by default log used base 10. Now I use ln and a base for the log as a subscript like log_10, log_2, etc.
sph 4y ago
> If you want to do mathematics I'm a software engineer, not a mathematician. Context matters.

Because I didn't see it mentioned yet, this is the same feature that makes the small-angle approximation of sin elegant: sin(x) ≈ x .

tsimionescu 4y ago
With a sin designed for turns, that is just sin(x) ~= 2pi * x or tau * x. Slightly uglier, but still quite simple.

anonymoushn 4y ago

video games don't usually need to take unitless derivatives of sines.

7steps2much 4y ago
True, but if you are talking about video games in particular you could just build a sin/cos function with different parameters into the game engine. have a `sin(x)` where the unit of x is radiants and a `sin_turn(x)` where x is expressed in turns. Video games especially are a great situation to do it like this because they often use a framework (game engine) that was specifically created for this purpose.
- anonymoushn 4y ago
  Yes, this is the topic of the fine article and the point that is being made. Ideally both of the multiplies (one in the user code to multiply by pi and another in the engine code to divide by pi) can be omitted.

bmacho 4y ago

What subfield of mathematics? Surely if you do differential equations, your trigonometric functions will eat radians. But e.g. for geometry, where you do basic arithmetic operations on angles, turns are a bit more convenient than radians. (Radians are not that inconvenient if you denote 6.28 somehow, but then again, why not just use turns.)

zarzavat 4y ago
The situation is similar to that of the logarithm and exponential functions[0]. There’s a log2 function and a log10 function and they are both useful. But when we talk about the log() function there can be no doubt that it is to base e. If you want to define a sinT() function that works in turns then that’s totally fine. But the sin() function is defined as taking an argument scaled in radians, because it is mathematically natural. Mathematically speaking, all trig functions are in radians. When you write sin(90°) the degree symbol ° is a conversion factor. I blame calculators for confusing high schoolers into believing that there is a separate set of functions that work in degrees. [0] unsurprisingly because Euler’s formula equates the trigonometric and exponential functions.
- bmacho 4y ago
  The sine function, defined as x-x^3/3!+... doesn't take argument scaled in radians. It takes real numbers. It has nothing to do with radians, really. Or even with angles. The other sine function, defined using right triangles takes argument in Angles, also has nothing to do with the measuring unit. (Also I don't know who told you that log() denotes log_e. Maybe in your narrow environment, but definitely not universal between fields and languages. Personally I prefer ln, ld and lb for natural, decimal and binary logarithms.)
  - lkuty 4y ago
    "Also I don't know who told you that log() denotes log_e. Maybe in your narrow environment," e.g. https://reference.wolfram.com/language/ref/Log.html https://reference.wolfram.com/language/ref/Log.html But I must confess that we had ln() in university courses and by default log used base 10. Now I use ln and a base for the log as a subscript like log_10, log_2, etc.

sph 4y ago

> If you want to do mathematics I'm a software engineer, not a mathematician. Context matters.