home | flowmeter articles | temperature study | circular geometry | duonyms | news | osborne says... | philosophy articles | services | about the founder | photos | jesse's homepage | contact us
Different Coordinate SystemsSubject: 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
©1999-2000
Flow Research
27 Water Street
Wakefield, MA 01880
781-224-7550
781-224-7552 (fax)
email: info@flowresearch.com