Re: [boost] Re: new compile-time dimensional analysis and unitlibrary

From: Matthias Schabel (boost_at_[hidden])
Date: 2003-12-11 00:22:57

> I guess my main thesis is that an angle is not just something that you
> feed into a trig function.  It is common to manipulate angles as just
> the numbers they are.  If you want to provide an implementation of the
> trig library that accepts degrees and gradians, that's fine with me.
> But don't make radians into a "unit".

I'd recommend you check into the IBWM (International Bureau for Weights  
and Measures, home of the SI unit system) web site for clarification.   
In particular :

http://www1.bipm.org/en/si/derived_units/2-2-3.html
http://www1.bipm.org/en/si/si_brochure/chapter2/2-2/2-2-2.html#table3
http://www1.bipm.org/en/si/si_brochure/chapter2/2-2/2-2-2.html#table4

To quote the notes below Table 3 :

"a) The radian and steradian may be used with advantage in expressions  
for derived units to distinguish between quantities of different nature  
but the same dimension. Some examples of their use in forming derived  
units are given in Table 4.
(b) In practice, the symbols rad and sr are used where appropriate, but  
the derived unit "1" is generally omitted in combination with a  
numerical value.
(c) In photometry, the name steradian and the symbol sr are usually  
retained in expressions for units."

It should be clear that radian (length/length) and steradian  
(area/area) are fundamentally different quantities, even though they  
are technically both dimensionless numbers.

> Radians are the ratio of one length to another.  Degrees are a unit
> that can be converted to a raw number by multiplying by pi/180.  Much
> like you can convert a speed into a raw number by dividing by the
> speed of light or the speed of sound.

No.  Degrees and radians are two different ways of defining the same  
dimensionless quantity.  For more clarification, see

http://en2.wikipedia.org/wiki/Right_angle

 From the "Measuring angles" section :

"In order to measure an angle, a circle centered at the vertex is  
drawn. The radian measure of the angle is the length of the arc cut out  
by the angle, divided by the circle's radius. The degree measure of the  
angle is the length of the arc, divided by the circumference of the  
circle, and multiplied by 360. "

and

"Mathematicians generally prefer angle measurements in radians because  
this removes the arbitrariness of the number 360 in the degree system  
and because the trigonometric functions can be developed into  
particularly simple Taylor series if their arguments are specified in  
radians. The SI system of units uses radians as the (derived) unit for  
angles."

or, more simply,

http://mathforum.org/library/drmath/view/52546.html

Matthias

------------------------------------------------------------------------ 
---------------------------
Matthias Schabel, Ph.D.                           	
Utah Center for Advanced Imaging Research
729 Arapeen Drive
Salt Lake City, UT  84108
801-587-9413 (work)
801-585-3592 (fax)
801-706-5760 (cell)
801-484-0811 (home)
mschabel at ucair med utah edu