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Different Classes of Geometries
Posted:
Apr 5, 2010 1:24 PM



Begin forwarded message:
> From: Walter Whiteley <whiteley@mathstat.yorku.ca> > Date: April 5, 2010 1:14:58 PM GMT04:00 > To: Tim <timlee@yahoo.com> > Cc: geometryresearch@support1.mathforum.org, approve@support1.mathforum.org > Subject: Re: Different Classes of Geometries > > Tim > Yes there are a variety of 'geometries' and different ways to build > a network of connections among them: > (a) Felix Klein worked out a general definition of geometry in his > Erlanger Program. > In this, a geometry is defined by its group of transformations. > Closest to this spirit is the classification from Euclidean (all > isometries), adding similarity maps, then shearing (affine geometry) > to projective geometry (maps preserving straight lines and > intersections). The more transformations, the fewer the > properties, and the simpler the geometry. You can go one more step > and add arbitrary invertible continuous maps and start to do topology. > > Any of these geometries  for example projective geometry  the > methods used could be analytic (using formulae, variables over some > field), mostly rational functions); or synthetic (using > constructions of intersecting lines, joining points etc.); or > axiomatic. The first fundamental theorem of projective geometry > states that with some basic axioms, the constructions of points and > lines actually can replicate the underlying field  so that the > analytic geometry and the axiomatic / synthetic geometry talk about > the same properties  but with very different complexity. > > (b) Euclidean, spherical, hyperbolic, even Minkowskian geometries > can be seen as overlaying a metric (a way of measuring lengths and > angles) onto the shared projective geometry. This is valuable as > an overview, since it helps figure out when properties are shared in > all the geometries, and when they are different. So the conditions > for triangles to be 'congruent' (overlayed by distance preserving > maps or isometries) are different: AAA does not work in Euclidean > Geometry but does work in hyperbolic and spherical geometry. Or the > sum of the angles in a triangle is constant in Euclidean Geometry, > but not in spherical or hyperbolic geometry. > > (c) Algebraic Geometry is another tool  using results and methods > in algebra to solve geometric problems. (Geometric algebra uses > geometric results and methods from Geometry to solve problems in > Algebra!) > > Differential Geometry is another world, studying smooth objects > (with continuous tangents etc.) rather than discrete geometry  > finite lists of points, line, planes, .. . Again the objects, the > questions, and the methods shift. > > I work in discrete applied geometry: objects in robotics, in > mechanical linkages, in rigid frameworks, in protein structures, > etc. One of the key questions when starting to work on an applied > 'geometric' problem is to figure out which transformations don't > impact possible solutions / relevant properties. This is key to > picking out the appropriate geometry. > > If the properties actually are unchanged by projective > transformations, then working in Euclidean Geometry will add layers > of complicated discussion which hide the simplicity of the possible > answers. This has happened in areas of spline approximations to > surfaces etc., where Euclidean methods missed the underlying > projective transformations  as well as the fact that it could work > in hyperbolic geometry. On the other hand, if it really does > change when moving to projective methods when it is different in the > sphere and the euclidean geometry, will prevent any valid solution > (I have seen that happen as well). > > In the end, one is trying to build up a network / concept map of > connections and results, of methods and options  then picking out > where to start and what to probe according to the problems one is > trying to solve, what one is trying to model, or generalize, or ..... > > Walter Whiteley > York University > > > > On 5Apr10, at 11:52 AM, Tim wrote: > >> Hi, >> >> I learn that Geometry has several categories/subfields from >> Wikipedia. But I am still not clear about the standards according >> to which they are classified. >> >> 1. It seems Euclidean Geometry, Affine Geometry and Projective >> Geometry are classified according some rule, while Hyperbolic >> Geometry, Elliptic Geometry and Riemann Geometry according to >> another, and Axiomatic, Analytic, Algebraic and Differential >> Geometry perhaps according to a different one? What rules are they? >> >> 2. Are Affine Geometry, Projective Geometry, Hyperbolic Geometry, >> Elliptic Geometry and Riemann Geometry all NonEuclidean Geometry? >> What are their common characteristics that make them NOnEuclidean >> Geometry? >> >> Really appreciate if someone could clarify these questions for me >> and also hope you can provide more insights into the subfields of >> Geometry not necessarily the specific questions I asked. >> >> Thanks and regards!



