ddg.geometry.geometries module

class ddg.geometry.geometries.Geometry(name=None, dimension=None, dimension_offset=None)

Bases: object

Basic geometry class.

Predefined geometries and geometries that are named upon initialization can be retrieved with get_geometry.

Parameters:

namestr (default=None)

Name of the geometry. Note that geometries can only be retrieved with get_geometry, if its name is not None.

dimensionint (default=None)

Dimension of the geometry. This is the ambient dimension of the space the model for the geometry lives in.

dimension_offsetint (default=None)

Offset to subtract from dimension to get the ‘interpreted’ dimension. For example, Möbius geometry has an offset of 1: The model space of n-dimensional Möbius geometry is an n-dimensional quadric surface in (n+1)-dimensional space.

See also

get_geometry

Attributes:

namestr

Name of the geometry.

dimensionint

Dimension of the geometry.

dimension_offsetint

name_dimensionint

Equal to dimension - dimension_offset.

Methods

set_alias(name[, overwrite])

Set alias for geometry.

dimension_offset = 0

Dimension offset, see above.

This will be overwritten if a dimension_offset is passed to a geometry upon creation.

set_alias(name, overwrite=False)

Set alias for geometry.

A geometry can also be retrieved by its alias using get_geometry.

Parameters:

namestr

New alias for the geometry.

overwritebool (default=False)

If set to True, the internal name of the geometry will be overwritten.

class ddg.geometry.geometries.ProjectiveGeometry(name='projective', dimension=None, dimension_offset=None, is_dual=False)

Bases: Geometry

Projective geometry class.

Describes an n-dimensional projective space (or its dual space).

Parameters:

namestr (default=None)

Name of the geometry. Note that geometries can only be retrieved with get_geometry, if its name is not None.

dimensionint (default=None)

Dimension of the projective space.

is_dualbool (default=False)

Indicates whether the geometry describes the primal or dual projective space.

See also

get_geometry

Attributes:

namestr

Name of the geometry.

dimensionint

Dimension of the projective space.

is_dualbool

True if geometry is dual, else False.

dualProjectiveGeometry

The dual geometry.

Methods

set_alias(name[, overwrite])

Set alias for geometry.

dualize()

set_alias(name, overwrite=False)

Set alias for geometry.

A geometry can also be retrieved by its alias using get_geometry.

Parameters:

namestr

New alias for the geometry.

overwritebool (default=False)

If set to True, the internal name of the geometry will be overwritten.

dimension_offset = 0

Dimension offset, see above.

This will be overwritten if a dimension_offset is passed to a geometry upon creation.

class ddg.geometry.geometries.CayleyKleinGeometry(quadric, name=None, dimension=None, is_dual=False)

Bases: ProjectiveGeometry

Cayley-Klein geometry class.

Describes an n-dimensional Cayley-Klein space using an absolute quadric, or a (pseudo-)conformal geometry on that quadric.

Parameters:

namestr (default=None)

Name of the geometry. Note that geometries can only be retrieved with get_geometry, if its name is not None.

dimensionint (default=None)

Dimension of the projective space. This is the ambient dimension of the absolute quadric.

is_dualbool (default=False)

Indicates whether the geometry describes the primal or dual projective space.

quadricddg.Quadric, numpy.array, callable

Absolute quadric of the geometry. When given as an array it is taken as the symmetric matrix of the absolute quadric. When given as a callable inner product, the parameter dimension becomes mandatory.

See also

get_geometry

Attributes:

namestr

Name of the geometry.

dimensionint

Dimension of the projective space. This is the ambient dimension of the absolute quadric.

is_dualbool

True if geometry is dual, else False

dualCayleyKleinGeometry

The dual geometry.

Methods

set_alias(name[, overwrite])	Set alias for geometry.
inner_product(v, w)	Inner product induced by the absolute quadric.
d(v, w)	Distance function.

inner_product(v, w)

Inner product induced by the absolute quadric.

See also

ddg.geometry.quadrics.Quadric.inner_product

cayley_klein_distance(v, w)

Cayley-Klein “distance” induced by the absolute quadric.

See also

ddg.geometry.quadrics.Quadric.cayley_klein_distance

metric_to_cayley_klein_distance(d)

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

Parameters:

dfloat

Returns:

Kfloat

cayley_klein_distance_to_metric(K)

Return Cayley-Klein distance converted to actual metric distance.

Parameters:

Kfloat

Returns:

dfloat

d(v, w)

Distance function.

The default implementation attempts to use cayley_klein_distance_to_metric (an abstract method) to convert the Cayley-Klein distance, wich is always defined in the same way, to a metric distance.

Parameters:

v, wnumpy.ndarray or Point

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

Returns:

float

dualize()

dimension_offset = 0

Dimension offset, see above.

This will be overwritten if a dimension_offset is passed to a geometry upon creation.

set_alias(name, overwrite=False)

Set alias for geometry.

A geometry can also be retrieved by its alias using get_geometry.

Parameters:

namestr

New alias for the geometry.

overwritebool (default=False)

If set to True, the internal name of the geometry will be overwritten.

class ddg.geometry.geometries.EllipticGeometry(dimension, is_dual=False)

Bases: CayleyKleinGeometry

Elliptic geometry class.

Parameters:

dimensionint

Dimension of the geometry.

is_dualbool (default=False)

Whether geometry is dual or not.

See also

Geometry, ProjectiveGeometry, CayleyKleinGeometry

Attributes:

dimensionint

Dimension of the geometry. This is the dimension of the model itself.

is_dualbool

True if geometry is dual, else False

dualEllipticGeometry

The dual geometry.

Methods

set_alias(name[, overwrite])	Set alias for geometry.
inner_product(v, w)	Inner product induced by the absolute quadric.

metric_to_cayley_klein_distance(d)

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

K = cos(d) ** 2

Parameters:

dfloat

Returns:

Kfloat

cayley_klein_distance_to_metric(K)

Return Cayley-Klein distance converted to metric distance.

d is given by the equation

K = cos(d) ** 2

Parameters:

dfloat

Returns:

Kfloat

dualize()

cayley_klein_distance(v, w)

Cayley-Klein “distance” induced by the absolute quadric.

See also

ddg.geometry.quadrics.Quadric.cayley_klein_distance

d(v, w)

Distance function.

The default implementation attempts to use cayley_klein_distance_to_metric (an abstract method) to convert the Cayley-Klein distance, wich is always defined in the same way, to a metric distance.

Parameters:

v, wnumpy.ndarray or Point

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

Returns:

float

dimension_offset = 0

Dimension offset, see above.

This will be overwritten if a dimension_offset is passed to a geometry upon creation.

inner_product(v, w)

Inner product induced by the absolute quadric.

See also

ddg.geometry.quadrics.Quadric.inner_product

set_alias(name, overwrite=False)

Set alias for geometry.

A geometry can also be retrieved by its alias using get_geometry.

Parameters:

namestr

New alias for the geometry.

overwritebool (default=False)

If set to True, the internal name of the geometry will be overwritten.

class ddg.geometry.geometries.EuclideanGeometry(dimension, is_dual=False, affine_component=-1)

Bases: CayleyKleinGeometry

Euclidean geometry class.

Parameters:

dimensionint

Dimension of the geometry.

is_dualbool (default=False)

Whether geometry is dual or not.

See also

Geometry, ProjectiveGeometry, CayleyKleinGeometry

Attributes:

namestr

Name of the geometry.

dimensionint

Dimension of the geometry. This is the dimension of the model itself.

is_dualbool

True if geometry is dual, else False

dualEuclideanGeometry

The dual geometry.

Methods

set_alias(name[, overwrite])	Set alias for geometry.
inner_product(v, w)	Inner product induced by the absolute quadric.

dualize()

d(x, y)

Euclidean distance.

Dehomogenizes by self.affine_component and computes the norm of the difference of x and y.

Note that the affine_component attribute of Point objects is ignored.

Parameters:

x, yPoint or array_like of shape (dimension+1,)

Returns:

float

cayley_klein_distance(v, w)

Cayley-Klein “distance” induced by the absolute quadric.

See also

ddg.geometry.quadrics.Quadric.cayley_klein_distance

cayley_klein_distance_to_metric(K)

Return Cayley-Klein distance converted to actual metric distance.

Parameters:

Kfloat

Returns:

dfloat

dimension_offset = 0

Dimension offset, see above.

This will be overwritten if a dimension_offset is passed to a geometry upon creation.

inner_product(v, w)

Inner product induced by the absolute quadric.

See also

ddg.geometry.quadrics.Quadric.inner_product

metric_to_cayley_klein_distance(d)

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

Parameters:

dfloat

Returns:

Kfloat

set_alias(name, overwrite=False)

Set alias for geometry.

A geometry can also be retrieved by its alias using get_geometry.

Parameters:

namestr

New alias for the geometry.

overwritebool (default=False)

If set to True, the internal name of the geometry will be overwritten.

class ddg.geometry.geometries.HyperbolicGeometry(dimension, is_dual=False)

Bases: CayleyKleinGeometry

Hyperbolic geometry class.

Parameters:

dimensionint

Dimension of the geometry.

is_dualbool (default=False)

Whether geometry is dual or not.

See also

Geometry, ProjectiveGeometry, CayleyKleinGeometry

Attributes:

namestr

Name of the geometry.

dimensionint

Dimension of the geometry. This is the dimension of the model itself.

is_dualbool

True if geometry is dual, else False

dualHyperbolicGeometry

The dual geometry.

Methods

set_alias(name[, overwrite])	Set alias for geometry.
inner_product(v, w)	Inner product induced by the absolute quadric.

metric_to_cayley_klein_distance(d)

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

K = cosh(d) ** 2

Parameters:

dfloat

Returns:

Kfloat

cayley_klein_distance_to_metric(K)

Return Cayley-Klein distance converted to metric distance.

d is given by the equation.

K = cosh(d) ** 2

Parameters:

dfloat

Returns:

Kfloat

dualize()

cayley_klein_distance(v, w)

Cayley-Klein “distance” induced by the absolute quadric.

See also

ddg.geometry.quadrics.Quadric.cayley_klein_distance

d(v, w)

Distance function.

The default implementation attempts to use cayley_klein_distance_to_metric (an abstract method) to convert the Cayley-Klein distance, wich is always defined in the same way, to a metric distance.

Parameters:

v, wnumpy.ndarray or Point

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

Returns:

float

dimension_offset = 0

Dimension offset, see above.

This will be overwritten if a dimension_offset is passed to a geometry upon creation.

inner_product(v, w)

Inner product induced by the absolute quadric.

See also

ddg.geometry.quadrics.Quadric.inner_product

set_alias(name, overwrite=False)

Set alias for geometry.

A geometry can also be retrieved by its alias using get_geometry.

Parameters:

namestr

New alias for the geometry.

overwritebool (default=False)

If set to True, the internal name of the geometry will be overwritten.

class ddg.geometry.geometries.LaguerreGeometry(dimension, is_dual=False)

Bases: CayleyKleinGeometry

Laguerre geometry class.

Parameters:

dimensionint

Dimension of the geometry.

is_dualbool (default=False)

Whether geometry is dual or not.

See also

Geometry, ProjectiveGeometry, CayleyKleinGeometry

Attributes:

namestr

Name of the geometry.

dimensionint

Dimension of the geometry. This is the dimension of the model itself.

is_dualbool

True if geometry is dual, else False

dualLaguerreGeometry

The dual geometry.

Methods

set_alias(name[, overwrite])	Set alias for geometry.
inner_product(v, w)	Inner product induced by the absolute quadric.

dimension_offset = 1

Dimension offset, see above.

This will be overwritten if a dimension_offset is passed to a geometry upon creation.

dualize()

cayley_klein_distance(v, w)

Cayley-Klein “distance” induced by the absolute quadric.

See also

ddg.geometry.quadrics.Quadric.cayley_klein_distance

cayley_klein_distance_to_metric(K)

Return Cayley-Klein distance converted to actual metric distance.

Parameters:

Kfloat

Returns:

dfloat

d(v, w)

Distance function.

The default implementation attempts to use cayley_klein_distance_to_metric (an abstract method) to convert the Cayley-Klein distance, wich is always defined in the same way, to a metric distance.

Parameters:

v, wnumpy.ndarray or Point

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

Returns:

float

inner_product(v, w)

Inner product induced by the absolute quadric.

See also

ddg.geometry.quadrics.Quadric.inner_product

metric_to_cayley_klein_distance(d)

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

Parameters:

dfloat

Returns:

Kfloat

set_alias(name, overwrite=False)

Set alias for geometry.

A geometry can also be retrieved by its alias using get_geometry.

Parameters:

namestr

New alias for the geometry.

overwritebool (default=False)

If set to True, the internal name of the geometry will be overwritten.

class ddg.geometry.geometries.LieGeometry(dimension, is_dual=False)

Bases: CayleyKleinGeometry

Elliptic geometry class.

Parameters:

dimensionint

Dimension of the geometry.

is_dualbool (default=False)

Whether geometry is dual or not.

See also

Geometry, ProjectiveGeometry, CayleyKleinGeometry

Attributes:

namestr

Name of the geometry.

dimensionint

Dimension of the geometry. This is the dimension of the model itself.

is_dualbool

True if geometry is dual, else False

dualLieGeometry

The dual geometry.

Methods

set_alias(name[, overwrite])	Set alias for geometry.
inner_product(v, w)	Inner product induced by the absolute quadric.

dimension_offset = 2

Dimension offset, see above.

This will be overwritten if a dimension_offset is passed to a geometry upon creation.

dualize()

cayley_klein_distance(v, w)

Cayley-Klein “distance” induced by the absolute quadric.

See also

ddg.geometry.quadrics.Quadric.cayley_klein_distance

cayley_klein_distance_to_metric(K)

Return Cayley-Klein distance converted to actual metric distance.

Parameters:

Kfloat

Returns:

dfloat

d(v, w)

Distance function.

The default implementation attempts to use cayley_klein_distance_to_metric (an abstract method) to convert the Cayley-Klein distance, wich is always defined in the same way, to a metric distance.

Parameters:

v, wnumpy.ndarray or Point

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

Returns:

float

inner_product(v, w)

Inner product induced by the absolute quadric.

See also

ddg.geometry.quadrics.Quadric.inner_product

metric_to_cayley_klein_distance(d)

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

Parameters:

dfloat

Returns:

Kfloat

set_alias(name, overwrite=False)

Set alias for geometry.

A geometry can also be retrieved by its alias using get_geometry.

Parameters:

namestr

New alias for the geometry.

overwritebool (default=False)

If set to True, the internal name of the geometry will be overwritten.

class ddg.geometry.geometries.MoebiusGeometry(dimension, is_dual=False)

Bases: CayleyKleinGeometry

Moebius geometry class.

Parameters:

dimensionint

Dimension of the geometry.

is_dualbool (default=False)

Whether geometry is dual or not.

See also

Geometry, ProjectiveGeometry, CayleyKleinGeometry

Attributes:

namestr

Name of the geometry.

dimensionint

Dimension of the geometry. This is the dimension of the model itself.

is_dualbool

True if geometry is dual, else False

dualMoebiusGeometry

The dual geometry.

Methods

set_alias(name[, overwrite])	Set alias for geometry.
inner_product(v, w)	Inner product induced by the absolute quadric.

dimension_offset = 1

Dimension offset, see above.

This will be overwritten if a dimension_offset is passed to a geometry upon creation.

dualize()

cayley_klein_distance(v, w)

Cayley-Klein “distance” induced by the absolute quadric.

See also

ddg.geometry.quadrics.Quadric.cayley_klein_distance

cayley_klein_distance_to_metric(K)

Return Cayley-Klein distance converted to actual metric distance.

Parameters:

Kfloat

Returns:

dfloat

d(v, w)

Distance function.

The default implementation attempts to use cayley_klein_distance_to_metric (an abstract method) to convert the Cayley-Klein distance, wich is always defined in the same way, to a metric distance.

Parameters:

v, wnumpy.ndarray or Point

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

Returns:

float

inner_product(v, w)

Inner product induced by the absolute quadric.

See also

ddg.geometry.quadrics.Quadric.inner_product

metric_to_cayley_klein_distance(d)

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

Parameters:

dfloat

Returns:

Kfloat

set_alias(name, overwrite=False)

Set alias for geometry.

A geometry can also be retrieved by its alias using get_geometry.

Parameters:

namestr

New alias for the geometry.

overwritebool (default=False)

If set to True, the internal name of the geometry will be overwritten.

ddg.geometry.geometries.get_geometry(name, dimension=None)

Retrieve a geometry.

Geometries can only be retrieved, if they are named upon initialization, or are one of the predefined geometries (elliptic, hyperbolic, laguerre, moebius, euclicean and lie).

Parameters:

namestr

Name of the geometry

dimensionint (default=None)

Dimension of the geometry. This should be the model dimension.

Returns:

ddg.Geometry

Raises:

KeyError

when the geometry could not be found nor created.