In Rienmannian geometry space can curve at different places (see manifolds) here we look at geometries where the curve of space is constant.
space curves outward
Spherical Geometry and Elliptic Geometry
space curves inward
Here we look at the terminology such as geometries, spaces, models, projections and transforms. Its quite difficult when we start dealing with non-Euclidean geometries because we use similar terminology that we are used to in conventional Euclidean space but the terms can have slightly different properties. For example, the concept of a 'line' can look different in different geometries:
Geometries and Spaces
Rienmannian geometry defines spaces generally in terms of manifolds, here we are interested in homogeneous, isotropic spaces which have no preferred points or directions, examples are:
|parallel postulate - number of unique parallel lines through point||space curves||Inventors||Point at Infinity||Distance Measure|
|Eulidean Geometry||1||none||Euclid||where parallel lines meet||√(x² + y²)|
|Hyperbolic Geometry||∞||space curves outward so lines dont meet||Lobachevskii
|0||space curves inward so all lines meet|
- Möbius Transform
- Lorentz Transform
If we take away the parallel postulate from Euclidean Space.
This leads to:
- parallel postulate is false
- the angles of a triangle do not add to π
- for a shape of a given size, there does not in general, exist a similar shape of a larger size.
Possible mappings between Euclidean Space and Hyperbolic Space:
|projective||conformal (Poincaré disc )|
|straight lines - geodesics||segments of circles|
|internal angles of triangle||π-(α+β+γ)=CΔ|
|distance between two points||ln|