[boost] Coordinates as abstractions [was: Preview 3 of the Geometry library]

Subject: [boost] Coordinates as abstractions [was: Preview 3 of the Geometry library]
From: John Phillips (phillips_at_[hidden])
Date: 2008-10-22 11:02:18

   After a little more thought, I have a different way to talk about 
coordinate systems that may make more sense in a computer science setting.

   Philosophical arguments aside, let us assume that some piece of space 
is the real object you are trying to use and describe in your program. 
In that case, as with any real thing brought into a computer program, 
you produce some abstraction and work with the abstraction as an 
approximate replacement for the real thing.

   The quality of that abstraction is determined by a few factors. Does 
it capture the important facets of the problem at hand? Does it provide 
an interface that allows for a clear description and solution of the 
problem? Can it be implemented in a way that translates from the 
interface to the hardware in a reasonably efficient and accurate way? 
Does it allow the user to solve the problem in terms of already familiar 
  concepts? (There are some others, but this is enough of a list for the 
moment.)

   Cartesian coordinates are one example of such an abstraction. They 
are very useful for a number of real problems and they are the simplest 
to visualize, so they are often the first version anyone learns. By 
analogy, a simple array is typically the first data structure anyone 
learns when programming. It is easy to visualize, and easy to implement 
so it is a good first example. It is also useful for many problems so 
the time spent learning it is valuable.

   In the context of this analogy, other coordinate systems (such as 
spherical, or cylindrical, or confluent hypergeometric) are equivalent 
to other data structures (such as lists, queues or trees). They are a 
different abstraction of the same underlying real object (the piece of 
space, or the collection of data). They are a good choice exact in those 
cases where they make problems easier to think about and to solve. Just 
as anyone who does anything with a computer must know how to use at 
least one data structure, experts know the proper situations in which to 
apply many different data structures, and there are specialty data 
structure that are only known to the small communities that need them 
regularly; the same can be said for coordinate systems.

   Everyone who has ever made a graph knows at least one (usually, 
Cartesian), experts know many (most physicists are comfortable with at 
least 3 and could apply dozens at need), and some specialized 
communities have systems they use frequently that the rest of us have 
never spent time thinking about.

   The quality of this abstraction is measured the same way as any 
other, and by that measure of quality it is true that Cartesian is just 
not always the best choice.

   I don't know if this helps, but I'm trying to move my reasons out of 
specialized math and science considerations and into the language more 
common in good design.

                        John