ddg.geometry.geometry_model_templates module

Abstract base classes for geometries.

This module defines abstract base classes for different types of geometries. Among other things, these can be used to pre-define metrics and absolute quadrics and they can be given to spheres.

class ddg.geometry.geometry_model_templates.MetricGeometry(dimension)

Bases: ABC

Geometry that just has a dimension and defines a metric.

Parameters:

dimensionint

Notes

This class supports the == operator. Two metric geometries are equal if and only if they have the same type and dimension.

Attributes:

dimensionint

abstract d(v, w) → float

Metric function

Parameters:

v, wPoint or array_like of shape (n,)

Returns:

float

class ddg.geometry.geometry_model_templates.CayleyKleinGeometry(dimension)

Bases: CayleyKleinSphereFactory

Cayley-Klein geometry abstract base class.

Parameters:

dimensionint

Dimension of the geometry. This is the actual “intuitive” dimension of the geometry, not the ambient dimension of the model.

Notes

This class supports __eq__, i.e. ==. Two quadric geometries are equal if and only if they have the same type, dimension and quadric.

Attributes:

dimensionint

absolute: Quadric

The absolute quadric.

property ambient_dimension

inner_product(v, w)

Alias for self.absolute.inner_product.

cayley_klein_distance(v, w)

Alias for self.absolute.cayley_klein_distance.

cayley_klein_sphere(center, radius, subspace=None, atol=None, rtol=None)

Create a Cayley-Klein sphere.

Parameters:

centerddg.geometry.Point or numpy.ndarray of shape (n+1,)

radiusfloat

Cayley-Klein radius

subspaceddg.geometry.Subspace or list of numpy.ndarray of shape (k,)

(default=None)

Returns:

ddg.geometry.spheres.CayleyKleinSphere

generalized_cayley_klein_sphere(center, radius, subspace=None, atol=None, rtol=None)

Create a generalized Cayley-Klein sphere.

Parameters:

centerddg.geometry.Point or numpy.ndarray of shape (n+1,)

radiusfloat

Generalized radius

subspaceddg.geometry.Subspace or list of numpy.ndarray of shape (k,)

(default=None)

Returns:

ddg.geometry.spheres.GeneralizedCayleyKleinSphere

class ddg.geometry.geometry_model_templates.MetricCayleyKleinGeometry(dimension)

Bases: CayleyKleinGeometry, MetricGeometry, SphereFactory

Metric Cayley-Klein geometry.

A metric Cayley-Klein geometry is a Geometry with a metric that is computed from a Cayley-Klein “distance” induced by a quadric.

Parameters:

dimensionint

Notes

This class supports __eq__, i.e. ==. Two metric Cayley-Klein geometries are equal if and only if they have the same type, dimension and quadric.

Attributes:

dimensionint

abstract metric_to_cayley_klein_distance(d) → float

Return Metric distance converted to Cayley-Klein “distance”.

Parameters:

dfloat

Returns:

Kfloat

abstract cayley_klein_distance_to_metric(d) → float

Convert Cayley-Klein “distance” to metric.

Parameters:

Kfloat

Returns:

dfloat

d(v, w) → float

Cayley-Klein metric

Parameters:

v, wnumpy.ndarray or ddg.geometry.Point

Homogeneous coordinate vectors or Point instances. Both points must be either inside or outside the absolute quadric.

Returns:

float

sphere(center, radius, subspace=None, atol=None, rtol=None)

Create a sphere with metric radius.

Parameters:

centerddg.geometry.Point or numpy.ndarray of shape (n+1,)

radiusfloat

Radius in terms of the Cayley-Klein Metric self.d.

subspaceddg.geometry.Subspace or array_like of shape (m,n) (default=None)

Returns:

ddg.geometry.spheres.MetricCayleyKleinSphere

property ambient_dimension

cayley_klein_distance(v, w)

Alias for self.absolute.cayley_klein_distance.

cayley_klein_sphere(center, radius, subspace=None, atol=None, rtol=None)

Create a Cayley-Klein sphere.

Parameters:

centerddg.geometry.Point or numpy.ndarray of shape (n+1,)

radiusfloat

Cayley-Klein radius

subspaceddg.geometry.Subspace or list of numpy.ndarray of shape (k,)

(default=None)

Returns:

ddg.geometry.spheres.CayleyKleinSphere

generalized_cayley_klein_sphere(center, radius, subspace=None, atol=None, rtol=None)

Create a generalized Cayley-Klein sphere.

Parameters:

centerddg.geometry.Point or numpy.ndarray of shape (n+1,)

radiusfloat

Generalized radius

subspaceddg.geometry.Subspace or list of numpy.ndarray of shape (k,)

(default=None)

Returns:

ddg.geometry.spheres.GeneralizedCayleyKleinSphere

inner_product(v, w)

Alias for self.absolute.inner_product.

absolute: Quadric

The absolute quadric.