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I have heard that 2-dimensional Hyperbolic Geometry can be reduced to 3-dimensional Euclidean Geometry, with certain restrictions. And also that Euclidean Geometry is a special case of Affine Geometry.

As such, some geometries can be modelled, or interpreted as special cases of other geometries. What I would like to know is, among the geometries below which is the most general case?

Euclidean Geometry Affine Geometry Hyperbolic Geometry Elliptic Geometry Inversive Geometry Projective Geometry

And also, is there any other geometry which includes those as well?

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    $\begingroup$ You need to explain what you mean by 'reduced to'. $\endgroup$
    – guest
    Oct 1, 2015 at 22:17
  • $\begingroup$ By a X-geometry being "reduced to" an Y-geometry, I mean that, for some m, n-dimensional X-geometry can be modelled as m-dimensional Y-geometry. $\endgroup$
    – user207032
    Oct 1, 2015 at 22:51
  • $\begingroup$ I say 2-dimensional Hyperbolic Geometry can be reduced to 3-dimensional Euclidean Geometry because it can be modelled as a surface there $\endgroup$
    – user207032
    Oct 1, 2015 at 22:55

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None of those geometries is most general overall: several of them have contradictory axioms.

There are a couple of relationships though. Affine geometry includes Playfair's axiom and requires that transformations preserve lines and parallels, and Euclidean geometry (whose transformations preserve distance) has a stricter set of transformations which are all affine.

Now, projective geometry can be viewed as a completion of affine geometry. In a projective space, you can delete a hyperplane and restrict to the projective transformations that map that hyperplane into itself, and the result is an affine space with affine transformations. So, thus could be considered an argument for viewing projective geometry as very general.

Elliptic geometry is often viewed as a type of projective geometry, although there is good reason to think of geometry in the surface of a sphere as elliptic too.

Inversive geometry and hyperbolic geometry can't be slotted into anything discussed so far.

There is a geometry absent from your list that sits above Euclidean and hyperbolic geometry: the so-called absolute geometry (or neutral geometry) which is agnostic about parallelism.

As for what you've heard about "reducing the hyperbolic plane to 3d Euclidean space": this is not really making one geometry special case of the other, but rather building a model of one out of the other.

There are indeed ways to model hyperbolic geometry in $\Bbb R^2$ and $\Bbb R^3$, but synthetically their axioms are inconsistent with each other.

If you are only concerned about which geometric constructs allow modeling of the others, then projective geometry would be the best bet.

On one hand it is a complete version of affine and Euclidean geometry which is very close to our intuitions. But it admits models of both hyperbolic and elliptic geometries as subsets of projective spaces.

Invertible geometry is really the odd man out: its transformations do not even preserve lines.

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