Subject: Re: Coordinate systems

Different Coordinate Systems

Subject:      Re: Coordinate systems

Author:       John Conway <conway@math.Princeton.EDU>

Date:         Sun, 8 Feb 1998 10:59:07 -0500 (EST)

On 8 Feb 1998, Clifford J. Nelson wrote:

>  I have spent about fifty dollars a month for about twenty years on math

> books and some time in libraries and I have only run across two coordinate

> systems for the uncurved plane: the perpendicular XY system and polar

> coordinates. I discovered what I call the Synergetics or simplex

> coordinates for the plane in 1994, but I can't find any books about them.

>  Could you tell me the names of the coordinate systems for the uncurved

> plane and maybe steer me to some books? Thank you.

>       Cliff Nelson

   It's a bit hard to answer this, precisely because switching to a new

coordinate system is a pretty trivial business; people just say something

like "use the following as coordinates" rather than "use So-and-so

coordinates".

   First, there are various systems of trilinear coordinates for the

plane (becoming (n+1)-linear in n-dimensions), which were used

particularly in projective geometry and so are often called "projective

coordinates".  These were popularized by Mobius in the middle of the

last century in his little book "The barycentric calculus", and

one particular variety of them is called "barycentric" coordinates.

They are still very much used in the geometry of a triangle, and

since they shade off into a variety of other systems, I'll describe

them first.

    Mobius' idea was to use (x,y,z) for the center of gravity you get

by putting masses x,y,z at the vertices A,B,C of a fixed triangle.

Obviously you get the same CG if you use masses kx,ky,kz, so that

the coordinates (kx,ky,kz) (k not 0) represent the same point as (x,y,z).

This is the "projective" property, so that barycentric coordinates

are a particular case of projective coordinates.  We call the

coordinates normalized if x+y+z = 1.  There are some triples

(x,y,z) that you can't normalize, because x+y+z = 0 : then you can think

of (x,y,z) as representing a vector, and the set of all (kx,ky,kz)

as either a "direction" or a "point at infinity".

    Suppose you take Euclidean 3-dimensional coordinates.  Then the

condition x+y+z = 1 determines a plane, and so in this case the normalized

barycentrics can be thought of as using 3 Euclidean coordinates in this

plane.  As k varies, the points (kx,ky,kz) represent all the points of

a line through the origin, so making them represent the same point

is really centrally projecting the rest of 3-space onto this plane

 from the origin.  But we could use other projections - for instance

orthogonal projection, under which (x+k,y+k,z+k) would represent

the same point as (x,y,z).  Now you could normalize instead by

taking x+y+z = 0 if you like.  Sometimes the name "simplicial"

coordinates has been used for this, so that using n+1 coordinates

for an n-space with the condition that their sum is zero would

be "normalized simplicial coordinates", from which you get the

unnormalized ones by letting (x+k,y+k,...) represent the same

point as (x,y,...).

    This is the same as projecting in the direction of the vector

(1,1,1,...) - in generalized simplicial coordinates you'd project

in the direction of some other vector.  You can combine the two

types of projection by using n+2 coordinates for an n-space, with

the understanding that   (Kx+k,Ky+k,...)  should represent the

same point as (x,y,...).  The name "pentahedral coordinates" is

used for this in the case n = 3.  Of course pentahedral coordinates

would be the natural choice to use for a problem that involved 5

particular planes.

    The above are the most common systems of "linear" coordinates,

that adjective meaning that lines, planes, etc., are determined by

linear equations.  People studying such subjects as potential theory,

fluid dynamics and the like use all sorts of non-linear coordinate

systems determined by the particular shapes that concern them.  So

for instance you'd use spherical polar coordinates (r,theta,phi) for

a problem involving spheres, cylindrical polars (r,theta,z) for one

involving cylinders, ellipsoidal coordinates (often called "confocal"

coordinates" for one involving ellipsoids, and so on.  Confocal

coordinates are so called because their level surfaces are a confocal

system of quadrics (or conics in 2 dimensions, where they are also

called "elliptic coordinates").

     John Conway

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