Introduction to geometry

The projective approach

Our library takes an approach to geometry following Felix Klein’s Erlangen program.[1] In the following paragraphs, we will attempt to introduce this approach to those unfamiliar with it:

We consider all of our geometric objects (everything that is defined in the ddg.geometry package) as subsets of a projective space \(\RP^n\) or in some special cases of \(\CP^n\). All other geometries, including Euclidean, hyperbolic and spherical geometry, are realized as subgeometries of projective geometry. This means that the model space for them is a subset of \(\RP^n\) and they define additional structure like angles and distances. For example, Euclidean geometry is \(\R^n \subset \RP^n\) (identified with the points “not at infinity”, see the Conventions section below) together with the usual Euclidean metric and angles.

More abstractly, a geometry can be thought of as a set \(X\) called the model space together with a group \(G\) of transformations acting on the space, which preserves the relevant structure like distances and angles. Euclidean geometry could be defined in this way as \(\R^n\) together with the group of isometries \(x \mapsto Ax + b\) with \(A \in \operatorname{O}(n)\) and \(b \in \R^n\). A subgeometry is then a subset \(Y \subseteq X\) together with a subgroup \(H < G\). It turns out that projective geometry, i.e. \(\RP^n\) together with the group of projective transformations \(\operatorname{PGL}(n+1,\R)\), contains many other familiar geometries as subgeometries. This is the motivation for our approach.

Conventions

In the geometry package, all points are assumed to be given in homogeneous coordinates, unless stated otherwise. Exceptions are functions like subspace_from_affine_points().

When switching from homogeneous coordinates to affine coordinates, we follow the convention of homogenizing and dehomogenizing with respect to the last component:

>>> import numpy as np
>>> from ddg.math.projective import homogenize, dehomogenize
>>> print(homogenize(np.array([1.0, 2.0, 3.0])))
[1. 2. 3. 1.]
>>> print(dehomogenize(np.array([1.0, 2.0, 3.0, 1.0])))
[1. 2. 3.]
>>> print(dehomogenize(np.array([2.0, 4.0, 6.0, 2.0])))
[1. 2. 3.]

So “points at infinity” for us are points whose last component is 0.

To see what an object would look like in a different affine chart, use the method change_affine_picture() of ddg.abc.LinearTransformable objects to transform the object with an appropriate projective transformation before visualizing it in the usual way.

Footnotes