Maths - Hyperbolic Geometry

In Rienmannian geometry space can curve at different places (see manifolds) here we look at geometries where the curve of space is constant.

Eulidean Geometry

flat space

euclidean triangle

Hyperbolic Geometry

space curves outward

hyperbolic triangle

Spherical Geometry and Elliptic Geometry

space curves inward

spherical tringle

Terminology

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:

noneucliean 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
and Bolyai
   

Spherical Geometry

Elliptic Geometry

0 space curves inward so all lines meet      

 

Models

Examples are:

Projections

Stereographic Projection

Transforms

Invariant

Examples are:

 

Parallel Postulate

If we take away the parallel postulate from Euclidean Space.

This leads to:

Possible mappings between Euclidean Space and Hyperbolic Space:

properties:

  projective conformal (Poincaré disc )
straight lines - geodesics   segments of circles
angle   preserved
internal angles of triangle   π-(α+β+γ)=CΔ
distance between two points   ln
     
     
     
     
     

 


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Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

flag flag flag flag flag flag Geometric Algebra for Computer Science: An Object-oriented Approach to Geometry. This book stresses the Geometry in Geometric Algebra, although it is still very mathematically orientated. Programmers using this book will need to have a lot of mathematical knowledge. Its good to have a Geometric Algebra book aimed at computer scientists rather than physicists. There is more information about this book here.

 

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