equivalence in projective geometry calculator

Resultsīefore we delve into the theory, here is a summary of some of the useful results, so that we know where we are heading. Two of these models, the hemisphere model and the stereographic model are described below.

There are different ways to deal with this redundant information regarding the length of the vector in the projective plane. In the projective domain the length of the vector is less important because the projection onto the euclidean domain depends on the direction of the vector and not its length. We can combine quantities to give 'meet' and 'join'.which go through the origin, projective geometry provides a way to represent quantities offset from the origin. Geometric algebra provides a powerful way to represent lines, planes, etc.This can greatly simplify the representation of objects moving in space. Representing isometries (translations and rotations) as a single multiplication operation.

Projecting a 3D scene onto a 2D screen (used by openGL).As a representation of perspective drawing.This can be important, for instance, any two lines will meet at a point and we don't have to make an exception for parallel lines. As a geometry in n+1 dimensions with with the additional dimension added to allow us to represent points at infinity without the need to use the ∞ symbol.That is it has the same symmetries as the individual geometries with additional symmetries. As a more general case of Elliptical, Euclidean and Hyperbolic geometries which can represent the properties of all of these.There are different ways to think about projective geometry:

Equivalence in projective geometry calculator