"""Laguerre geometry module.
Contains model classes and functions for conversion between the models.
"""
import numpy as np
from ddg.geometry.geometry_model_templates import CayleyKleinGeometry
from ddg.geometry.quadrics import Quadric
from ddg.geometry.subspaces import Point, normal_with_level
__all__ = [
"ProjectiveModel",
"EuclideanModel",
"euclidean_to_projective",
"projective_to_euclidean",
]
[docs]class ProjectiveModel(CayleyKleinGeometry):
"""Projective / Blaschke cylinder model of Laguerre geometry.
.. rubric:: Model space
The model space is the quadric with matrix `diag([1,...,1, 0, -1])` without
the point at infinity [0,...,0, 1 ,0].
.. rubric:: Representation of objects
Points in this cylinder represent hyperplanes in R^n: In affine
coordinates, they have a "direction" v and a "height" h. The point (v, h)
corresponds to the hyperplane ``<x, v> = h.``
Parameters
----------
dimension : int
Attributes
----------
dimension : int
"""
@property
def absolute(self):
"""The absolute quadric with matrix ``diag([1,...,1, 0, -1])``.
Returns
-------
Quadric
"""
diag = np.ones(self.dimension + 2)
diag[-2] = 0
diag[-1] = -1
return Quadric(np.diag(diag))
def __contains__(self, point):
return point in self.absolute
[docs]class EuclideanModel:
"""Euclidean model of Laguerre geometry.
.. rubric:: Model space
The model space is R^n.
.. rubric:: Representation of objects
Objects are represented by their Euclidean equivalents, mostly subspaces.
"""
[docs]def euclidean_to_projective(object_):
"""Convert from Euclidean model to Projective model / Blaschke cylinder.
Parameters
----------
object_ : Subspace of dimension n-1
Returns
-------
Point
"""
return Point([*normal_with_level(object_), 1])
[docs]def projective_to_euclidean(object_):
"""Convert from Projective model / Blaschke cylinder to Euclidean model.
Parameters
----------
object_ : Point
Returns
-------
Subspace
"""
p = object_.point[:-1]
return Point(p).dualize()