Normal

Hyperbolic Geometry in Solid Geometry

Lobatschewsky-Bolyai Geometry in Euclidean Geometry can be extend into by solid geometry by:

Space = Region of points inside a sphere.

Point = Point inside of a Sphere

Strait Line = Chord in the Sphere (i.e. Volume Cap is Pi*h^2*(3r-h)/3 and the spherical cap is 2*Pi*r*h)

Plane = Points of a plane section which are inside of the Sphe

Displacements = Projective transformation in the space (i.e. dot products, cross product and linear mapping), which change the region of the point inside the sphere itself 

Free Groups definitions

Torus Mapping

Non-Linear Optimization

Simplex-Method

Projection of Volumes

Angles and the Plane

Linear Function

Orthogonal

Arc-Wise connected open subsets

Gauss Curvature Definition