Re: [boost] Controversy over angles in radians

From: Matthias Schabel (boost_at_[hidden])
Date: 2003-12-18 11:42:42

> Although radians clearly are units, an angle given in radians is also
> a pure unitless ratio (the ratio of the circular arc cut out by the  
> angle over
> the radius of the circle) and, as has been observed by

If I choose to use cycles as a unit of angular measure, they too are  
pure
_dimensionless_ ratio : the ratio of the circular arc length subtended  
by
the specified angle to the circumference of the circle.

> others, therefore can be used in two ways
> that quantities given in units normally cannot be:
> 1) You can multiply a length by a
> n angle in radians and get a length.

This holds true for any dimensionless quantity, technically speaking.   
The
question here is whether the dimensionlessness of a quantity should  
supersede
the units associated with that quantity.  That is, radians (and degrees  
and all other
units of angular measure) are dimensionless quantities having distinct  
units.  If
I specify to an engineer that the angle between two planes should be  
0.25 (a pure
number), you can be sure that she will come back to me with the  
question : 0.25
what?  So it is clear that the angular quantities are in fact not pure  
numbers - the
number itself is not sufficient to fully specify the quantity.

> 2) You can feed an angle in radians into a pure mathematical
> nonlinear function.
> In both cases, the angle in radians is being treated as a pure
> unitless real number expressing a ratio, rather than a quantity
> with units.

Again, this is just a consequence of the fact that the Taylor series  
expansions of
commonly used functions are particularly simple when expressed in  
radian measure.

> But on further thought I now feel differently. You CAN define radians
> as a unit and, contrary to some of the objections, it IS useful to do  
> so. For example, suppose I want to write some generic numerical code  
> involving angles, but without being unit-specific. Two simple examples
> are:
>
> // Function for computing the length of a circular arc
> template <class length_units, class angle_units>
> length_units arc_length(length_units radius, angle_units angle);
>
> // Sine function
> template <class angle_units>
> double sine(angle_units angle);
>
> I would want this to work with either degrees or radians,
> and I would want the unit library to sort this out automatically for  
> me.
> I think this is important in many scientific and engineering  
> applications.

Thank you - this is precisely what I've been trying to get at.

> It appears to me that any code involving angles can be divided cleanly  
> into two pieces, one where angles are viewed as quanitities
> defined with respect to a unit and one where angles are viewed as pure
> ratios. Roughly speaking, the former is usually the library interface,  
> and the latter is the implementation. I would use the units library  
> for
> the former and represent angles as doubles (and in radians) for the  
> latter.

Absolutely - I've never proposed changing the underlying  
implementations of trig functions (or other functions) - using radians  
is obviously the best implementation.  But, as you've observed, from a  
practical standpoint being able to trap degree/radian mismatch errors  
is a desirable goal (unless your philosophical attachment to a  
particular dogma trumps your pragmatic interest in producing clean,  
error-inhibiting code).

> Finally, if all of the physicists agree that angle is not a physical  
> dimension, then it certainly should not appear in a dimensions  
> library.
> Just in the units library.

As it stands now, radians do not appear in the dimensional analysis  
code, only in the implementation of the SI unit system.

------------------------------------------------------------------------ 
---------------------------
Matthias Schabel, Ph.D.                           	
Utah Center for Advanced Imaging Research
729 Arapeen Drive
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801-587-9413 (work)
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mschabel at ucair med utah edu