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
 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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