Re: [boost] Interest in dimension/unit-checking library?

From: Deane Yang (deane_yang_at_[hidden])
Date: 2005-10-12 09:21:55

Jeff Garland wrote:
> On Mon, 10 Oct 2005 23:22:24 -0400, Deane Yang wrote
>> Jeff Garland wrote:
>>> Wow, that's amazing.  Replace quantity_vector with time_duration and
>>> quantity_point with time_point and you have the core concepts behind
>>> date-time.  And the 32 versus zero -- seems an awful lot like an epoch_time
>>> adjustment.  So I suppose I'm saying I believe there is something fundamental
>>> here :-)
>>>   
>> Yes, there is.
>>
>> If you dig back in the archives, I tried to explain it all before, 
>> but obviously not very well.

 > (stuff deleted)
> 
> Sorry if I missed or didn't respond to your comments the first time.

Definitely understandable.

>  However,
> I now have reasons to be more interested in the theoretical foundations. 
> Specifically, the LWG committee is rightly an intense group that needs to
> ensure the quality and correctness of things that get added to the standard --
> having additional theoretical foundations for the date-time concepts can't
> hurt.  Any chance you can point me to some sort of references (preferably
> readable to folks that have only taken ~15 college math courses ;-)
> 

Here's a brief summary of the idea (assuming that you learned about 
vector spaces (http://en.wikipedia.org/wiki/Vector_space)  in linear 
algebra) (In anything you read about vector and affine spaces, if they 
say "field F", just assume that F is the set of real numbers):

A "dimension" or, as Matt calls it, a "point quantity" or, as you call 
it the set of "time_points", without choosing any particular unit, 
corresponds to the mathematical concept of an "affine line" or an 
"1-dimensional affine space" A (see 
http://en.wikipedia.org/wiki/Affine_space).

Associated with any "point quantity" or "dimension" or the set of 
"time_points" is the set of all possible differences between two points. 
Matt calls this a "vector quantity"; you call it the set of 
"time_durations"; mathematicians call it the (1-dimensional) vector 
space (http://en.wikipedia.org/wiki/Vector_space) associated with the 
affine space.

Assigning (relative) units to a vector quantity corresponds to fixing a 
nontrivial real-valued linear function f: R -> V on the vector space V. 
Think of V as a straight line with the "zero vector" marked and the map 
f as like putting a ruler on top of the straight line, assigning a real 
number to each point on the line. Notice, however, that you have to line 
up the ruler so that the 0 on the ruler is placed against the 
zero-vector. So your only flexibility in what you do is stretching or 
shrinking the ruler.

Assigning (absolute) units to a point quantity corresponds to fixing a 
non-trivial affine map 
(http://en.wikipedia.org/wiki/Affine_transformation) f: R -> A, where A 
is the set of all possible values of the point quantity. Think of A as a 
  competely blank straight line and the map f as putting a ruler on top 
of the line. This time, since the line is blank, you can slide the ruler 
along the line and put the 0 on the ruler against any point on A. So you 
have more flexibility with your units; you can stretch, shrink, or slide 
the ruler.

Unit casting corresponds to the situation where you have two different 
rulers you want to use, and you need to convert a number on one ruler 
into the corresponding number on the other. For vector quantities, you 
just need to multiply the number by a fixed proportionality constant. 
For point quantities (temperature is the best example here), you might 
also need to do an addition or subtraction, too.

You can always turn any vector space V into an affine space, by simply 
"forgetting" where the zero vector is. You can always turn any affine 
space into a vector space V by arbitrarily choosing a point in A and 
declaring it to be the zero vector.