ddg.geometry.projections module

Module containing projections, e.g. central and stereographic.

A word on projection functions: Central projection is usually defined as a map between two k-planes S1, S2 in RPⁿ via another, (n-k-1)-dim. subspace C disjoint from S1, S2. The projection maps x in S1 to meet(S2, join(x, S1)) and this is a well-defined bijective map and a projective transformation. In this module, we take a more general approach: If objects Y and C are given, we map any object X to meet(Y, join(X, C)).

ddg.geometry.projections.central_project_subspace_onto_subspace(subspace, target_subspace, center)

Parameters:

subspaceSubspace

If possible, the basis of the resulting subspace will consist of the projections of the basis points of subspace. For this to work, target_subspace.dimension+center.dimension must be equal to ambient_dimension-1 and they must be disjoint. Otherwise, this function simply returns meet(target_subspace, join(subspace, center)).

target_subspaceSubspace

centerSubspace

Returns:

Subspace

ddg.geometry.projections.central_project_quadric_onto_subspace(quadric, subspace, center)

Parameters:

quadricQuadric

target_subspaceSubspace

centerSubspace

Returns:

Quadric

Raises:

NotImplementedError

If quadric is not contained in a subspace.

ddg.geometry.projections.central_project_subspace_onto_quadric(subspace, quadric, center)

Parameters:

subspaceSubspace

Subspace to be projected.

quadricQuadric

Quadric to project onto.

centerSubspace

Projection center.

Returns:

Subspace

ddg.geometry.projections.central_project(object_, target, center)

Central projection dispatcher.

ddg.geometry.projections.central_project_subspace_onto_subspace_contour(subspace, target_subspace, center)

Central project subspace onto subspace (contour mode).

If image is a proper subspace of target_subspace, returns the regular central projection. If image is all of target_subspace, returns the empty subspace.

Parameters:

subspace, target_subspace, centerSubspace

Returns:

Subspace

ddg.geometry.projections.central_project_quadric_onto_subspace_contour(quadric, subspace, center)

Central project quadric onto subspace (contour version).

This function also supports quadrics in the whole space.

Parameters:

quadricQuadric

subspaceSubspace

centerPoint

Returns:

Quadric

Raises:

NotImplementedError

If quadric is in the whole space and center is not a point.

ddg.geometry.projections.central_project_subspace_onto_quadric_contour(subspace, quadric, center)

Central project subspace onto quadric (contour mode).

If image is a proper subset of quadric, returns image. If image is all of quadric, returns an empty quadric.

Parameters:

subspaceSubspace

quadricQuadric

centerSubspace

Returns:

Quadric

ddg.geometry.projections.central_project_contour(object_, target, center)

Central projection dispatcher (contour mode).

This returns the boundary of the shadow instead of the shadow itself, so to speak.

ddg.geometry.projections.central_project_quadric_onto_subspace_complex(quadric, subspace, center)

Central project quadric onto subspace (complex version).

If we also consider complex intersections of lines through center with quadric, the projection image will always be all of subspace (or the intersection of subspace with quadric.subspace).

Parameters:

quadricQuadric

subspaceSubspace

centerSubspace

Returns:

Subspace

ddg.geometry.projections.central_project_complex(object_, target, center)

Central projection dispatcher (complex mode).

ddg.geometry.projections.stereographic_project(object_)

ddg.geometry.projections.stereographic_project(subspace: Subspace, hyperplane=None, projection_point=None)

ddg.geometry.projections.stereographic_project(quadric: Quadric, hyperplane=None, projection_point=None)

Stereographic projection (Quadric to subspace) dispatcher.

Accepts different keyword arguments based on type. See the type-specific functions for details.

ddg.geometry.projections.stereographic_project_subspace(subspace, hyperplane=None, projection_point=None)

Parameters:

subspaceSubspace

Subspace to be projected.

If this is the projection point itself, returns inf.

hyperplaneSubspace or None (default=None)

Hyperplane not containing projection_point. Default is the plane spanned by the homogeneous vectors e_1,…,e_n-1, e_{n+1}.

projection_pointPoint or None (default=None)

Point to project from. Default is the “north pole” [e_n+e_{n+1}].

Returns:

Subspace or float

ddg.geometry.projections.stereographic_project_quadric(quadric, hyperplane=None, projection_point=None)

Parameters:

quadricQuadric

Quadric to be projected.

hyperplaneSubspace or None (default=None)

Hyperplane not containing projection_point. Default is the plane spanned by the homogeneous vectors e_1,…,e_n-1, e_{n+1}.

projection_pointPoint or None (default=None)

Point to project from. Default is the “north pole” [e_n+e_{n+1}].

Returns:

Quadric

ddg.geometry.projections.inverse_stereographic_project(object_)

ddg.geometry.projections.inverse_stereographic_project(point: Point, quadric=None, projection_point=None)

ddg.geometry.projections.inverse_stereographic_project(subspace: Subspace, quadric=None, projection_point=None)

ddg.geometry.projections.inverse_stereographic_project(sphere: SphereLike, quadric=None, projection_point=None)

Inverse stereographic projection (Subspace to Quadric) dispatcher.

Accepts different keyword arguments based on type. See the type-specific functions for details.

ddg.geometry.projections.inverse_stereographic_project_point(point, quadric=None, projection_point=None)

Parameters:

pointPoint

Point to project.

quadricQuadric or None (default=None)

Quadric to project onto. Default is the Möbius quadric.

projection_pointPoint

Center of projection. Default is the “north pole” [e_n+e_{n+1}].

Returns:

Point

Raises:

ValueError

If projection_point does not lie on quadric.

Notes

This is separate from inverse_stereographic_project_subspace because in this case there is an explicit formula and because of some special cases.

ddg.geometry.projections.inverse_stereographic_project_subspace(subspace, quadric=None, projection_point=None)

Parameters:

subspaceSubspace

Subspace to project.

quadricQuadric or None (default=None)

Quadric to project onto. Default is the Möbius quadric.

projection_pointPoint

Center of projection. Default is the “north pole” [e_n+e_{n+1}].

Returns:

Quadric

Raises:

ValueError

If projection_point does not lie on quadric.

ddg.geometry.projections.inverse_stereographic_project_sphere(sphere, quadric=None, projection_point=None)

Stereographically project a sphere contained in a subspace to a quadric.

Works in all cases in which planar sections of quadric are stereographically projected to spheres (in the sense of the Euclidean metric in affine coordinates) in sphere.subspace.

Parameters:

sphereEuclidean sphere

quadricQuadric or None (default=None)

projection_pointPoint or None (default=None)

Center of projection. Default is the “north pole” [e_n+e_{n+1}].

Returns:

Quadric

Raises:

NotImplementedError

• If the intersection of sphere.subspace and the tangent plane at projection_point is not at infinity.

• If the induced metric is distorted, i.e. planar sections of the quadric would not map to spheres in sphere.subspace.

• If the intersection of the line through projection_point and polarize(sphere.subspace) with sphere.subspace is at infinity. This point would normally act as a sort of origin.

ValueError

If projection_point does not lie on quadric.

Notes

This function uses the euclidean distance on a quadric of signature (n+1, 1) given by:

d([x], [y])^2 = -1/2 * <x,y>/(<x,p><y,p>)

for a point [p] on the quadric with a fixed representative vector. We use that formula to find all points a constant distance from a point. This set will be a planar section of the quadric.

This metric corresponds to the Euclidean metric in affine coordinates after stereographic projection, but only to certain subspaces via certain points. We have to determine whether we have such a case and then find the right representative vector p to get the right scaling.

See supplemental material for details.

ddg.geometry.projections.lift_sphere_to_quadric(sphere, quadric, projection_point)

Find the lift of a sphere to a quadric via a point.

This is the inverse of the following map: Map a planar section of quadric to quadric.polarize(projection_point) via projection_point, giving a Cayley-Klein sphere.

Because the inverse map (the projection) is a double cover, this will always return two quadrics.

Parameters:

sphereCayleyKleinSphere

quadricQuadric

Quadric to lift to.

projection_pointPoint

A point not on the quadric.

Returns:

Quadric, Quadric

Raises:

ValueError

• If projection_point is on the quadric

• If center of sphere does not lie in quadric.polarize(projection_point).

• If no preimage exists. This is the case if the center of the sphere does not lie in the projection image of the quadric.

Notes

Taken from Non-Euclidean Laguerre Geometry and Incircular Nets, page 43.

ddg.geometry.projections.north_pole(n)

“North pole” in n-dimensional space.

Point(en + en+1)