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Different Classes of Geometries
Posted:
Apr 5, 2010 1:24 PM
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Begin forwarded message:
> From: Walter Whiteley <whiteley@mathstat.yorku.ca>
> Date: April 5, 2010 1:14:58 PM GMT-04:00
> To: Tim <timlee@yahoo.com>
> Cc: geometry-research@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 5-Apr-10, 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 Non-Euclidean Geometry?
>> What are their common characteristics that make them NOn-Euclidean
>> 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!
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