Different Classes of Geometries

[Tim]

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!

[Walter Whiteley]

Begin forwarded message:

From: Walter Whiteley <whit...@mathstat.yorku.ca>

Date: April 5, 2010 1:14:58 PM GMT-04:00

To: Tim <tim...@yahoo.com>

Cc: geometry...@support1.mathforum.org, app...@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