ddg.geometry.abc module

class ddg.geometry.abc.ProjectiveObject

Bases: ABC

ABC for geometric objects that are subsets of a Projective space RP^n.

ambient_dimension: int

abstract at_infinity()

Whether the object is contained in the hyperplane at infinity.

Returns:

bool

class ddg.geometry.abc.Embeddable

Bases: ProjectiveObject

ABC for objects that can be embedded in subspaces in a higher-dimensional projective space.

abstract embed(subspace=None)

The embedded object.

We can embed a subset X of n-dimensional projective space into m-dimensional space (where m >= n) by choosing an n-dimensional subspace S in RP^m together with a basis B. We then interpret the original homogeneous coordinates as coordinates w.r.t. this basis B.

More concretely: If M is the matrix of S, we define

X.embed(S) := { [M @ x] | [x] in X }.

X.embed(S) must be the same kind of object as X, for example if X is a euclidean sphere, the basis M must be chosen so that X.embed(S) is also a euclidean sphere.

Parameters:

subspaceddg.geometry.Subspace (default=None)

Subspace in m-dimensional ambient space whose dimension is the ambient dimension of self (where m >= n). The homogeneous coordinates of points are used as cordinates w.r.t. the given basis of subspace.

The default is the coordinate hyperplane {x_n+1 = 0} in (n+1)-dimensional space, where n is the ambient dimension of self. The basis is the standard basis with the second to last basis vector missing.

Returns:

type(self)

Object in m-dimensional ambient space contained in subspace.

Raises:

ValueError

If embedding w.r.t. the given basis would result in an object of a different type.

abstract unembed(subspace=None)

Inverse of embed.

Parameters:

subspaceddg.geometry.Subspace (default=None)

k-dim. subspace to treat as the new ambient space. coordinates of points will be computed w.r.t. the given basis of subspace.

The default is the coordinate hyperplane {x_n = 0} which will fail if the object is not contained in this subspace.

Returns:

type(self)

Object in k-dim. ambient space.

Raises:

ValueError

• If object is not contained in subspace.

• If coordinate computation would result in an object of a different type.

abstract at_infinity()

Whether the object is contained in the hyperplane at infinity.

Returns:

bool

ambient_dimension: int

class ddg.geometry.abc.QuadricConvertible

Bases: ABC

ABC for objects that can be converted to quadrics.

abstract quadric()

Return object as quadric.

Returns:

ddg.geometry.Quadric

class ddg.geometry.abc.SphereFactory

Bases: ABC

Factory class for general spheres.

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

General sphere factory method.

Parameters:

centerddg.geometry.Point

radiusfloat

subspaceddg.geometry.Subspace (default=None)

atol, rtolfloat (default=None)

Returns:

ddg.geometry.spheres.SphereLike

class ddg.geometry.abc.CayleyKleinSphereFactory

Bases: ABC

Factory class for (generalized) Cayley-Klein spheres but not Metric Cayley-Klein spheres, use SphereFactory for those.

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

Cayley-Klein sphere factory method.

Parameters:

centerddg.geometry.Point

radiusfloat

subspaceddg.geometry.Subspace (default=None)

atol, rtolfloat (default=None)

Returns:

ddg.geometry.spheres.CayleyKleinSphere

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

Generalized Cayley-Klein sphere factory method.

Parameters:

centerddg.geometry.Point

radiusfloat

subspaceddg.geometry.Subspace (default=None)

atol, rtolfloat (default=None)

Returns:

ddg.geometry.spheres.GeneralizedCayleyKleinSphere