`Different Coordinate Systems`
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`Subject:      Re: Coordinate systems`
`Author:       John Conway <conway@math.Princeton.EDU>`
`Date:         Sun, 8 Feb 1998 10:59:07 -0500 (EST)`
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`On 8 Feb 1998, Clifford J. Nelson wrote:`
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`>  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`

http://forum.swarthmore.edu/epigone/geometry-research/sarswimpbli/Pine.3.07.9802081007.H2555-d100000@broccoli.princeton.edu