ddg.geometry.quadrics module

Quadrics

class ddg.geometry.quadrics.Quadric(Q, subspace=None, atol=None, rtol=None)[source]

Bases: LinearTransformable, NonExact

Quadric in a projective space.

Parameters:

Qnp.ndarray: Symmetric matrix representing the quadratic form in homogeneous coordinates. The matrix is interpreted as the Gram matrix with respect to the given basis of the containing subspace.
subspaceddg.Subspace or iterable of numpy.ndarray (default=None): Subspace containing the quadric. Can be given as a list of homogeneous coordinate vectors. If None is given, uses whole space with standard basis.

Raises:

ValueError

If Q is not symmetric
If the dimension of the subspace does not match the shape of Q.

property ambient_dimension

dimension(real=True)[source]

The (real) dimension of the quadric.

More precisely, the maximum of the dimensions of the manifolds contained in the set of real points in the quadric.

Ignores affine coordinates! I.e. we don’t distinguish between the quadric lying at infinity or not.

Returns:

int

Notes

This function was fully intuited. There is no guarantee that this gives correct results.

at_infinity(affine_component=-1)[source]

Whether or not quadric is at infinity.

Parameters:

affine_componentint (default=-1)

Returns:

bool

coordinates(v)[source]

Get coordinates of vectors in terms of self.subspace’s basis.

Parameters:

vSubspace or numpy.ndarray: Subspace contained in self.subspace or Matrix representing such a subspace (or Point).

Returns:

numpy.ndarray of shape (self.subspace.dimension + 1,)

inner_product(v, w)[source]

Inner product defined by / defining the quadric.

Parameters:

v, wnumpy.ndarray of shape (ambient_dimension + 1,): Two vectors that lie in the containing subspace.

Returns:

float: a.T @ matrix @ b, where a and b are the coordinates of v and w in terms of the given basis of the containing subspace.

cayley_klein_distance(v, w)[source]

Cayley-Klein “distance”.

This is defined as

b(v, w) ** 2 / (b(v, v) * b(w, w))

Where b is the inner product induced by the absolute. None of the points can lie on the absolute.

Parameters:

v, wnumpy.ndarray or Point: Homogeneous coordinate vectors or Point instances

Returns:

float

Raises:

ValueError: If either of the points is contained in the absolute quadric.

conjugate(S1, S2)[source]

Conjugacy of two subspaces.

Whether two subspaces are conjugate, i.e. if approximately inner_product(v, w) == 0 for all v in S1 and w in S2.

Parameters:

S1, S2ddg.geometry.subspaces.Subspace or numpy.ndarray: Subspaces contained in self.subspace. Arrays will be interpreted as matrices whose columns are points in homogeneous coordinates. In particular, 1D arrays are simply points in homogeneous coordinates.

Returns:

bool

polarize(obj)[source]

Alias for polarize(obj, self).

See also

polarize

signature(subspace=None, affine=False, affine_component=-1)[source]: Alias for signature().

normalize(affine=False, affine_component=-1)[source]

Return an equal quadric expressed in coordinates that normalize it.

Returns an equal quadric Q_normalized such that

Q_normalized.matrix == sgn.matrix

where sgn = self.signature(affine, affine_component). See ddg.math.symmetric_matrices.Signature and ddg.math.symmetric_matrices.AffineSignature for more information.

If affine=True, Q_normalized.subspace.points will be in the form [d1,…,dn, p], where n is the dimension of the subspace, d1,…,dn are directions at infinity and p is a point in the subspace not at infinity (according to affine_component). This means that the returned AffineSignature instance will always have affine_component -1.

Parameters:

affinebool (default=False)
affine_componentint (default=-1)

Returns:

Q_normalizedQuadric

property rank

Rank of self.matrix.

Returns:

int

property corank

Corank of self.matrix.

Returns:

int

property singular_subspace

Singular subspace.

Subspace containing all singular points of the quadric (points in the kernel of self.matrix).

Returns:

ddg.geometry.subspaces.Subspace of dimension corank - 1 and the
same ambient dimension as the quadric.

property non_degenerate_subspace

Non-degenerate subspace.

Complementary subspace of the kernel of self.matrix.

Returns:

ddg.geometry.subspaces.Subspace: Has dimension rank - 1 and the same ambient dimension as the quadric.

property is_degenerate

Whether the quadric is degenerate.

Returns:

bool

dualize()[source]

dual_transformation(F)[source]

Dualizes a transformation of the ambient space.

Parameters:

Fnumpy.ndarray of shape (ambient_dimension+1, ambient_dimension+1): An invertible matrix.

Returns:

F_dualnumpy.ndarray of shape (ambient_dimension+1,ambient_dimension+1): This is an invertible matrix that approximately satisfies F_dual @ B_dual == F @ B @ np.linalg.inv((F @ B).T @ (F @ B)), where B_dual is the subspace of the dual quadric. That is, it maps the old dual basis to the new one.

push_transformation(F)[source]

Transform the quadric by a transformation of the ambient space.

Parameters:

Fnumpy.ndarray of shape (ambient_dimension+1, ambient_dimension+1): Invertible matrix.

pop_transformation()[source]

Pop transformation from the trafo-stack.

Returns:

function

property atol

atol_default = 1e-07

Absolute tolerance default.

Used for all purposes if atol is None.

classmethod nonexact_function(f)

Decorator to use global tolerance defaults in functions.

If None is given to the wrapped function as the value for atol or rtol, it will be replaced by the global default.

Parameters:

fcallable: Function that takes atol and rtol arguments.

Returns:

callable

classmethod reset_tol_defaults()

property rtol

rtol_default = 0.0

Relative tolerance default.

Used for all purposes if rtol is None.

transform(f)

Wrapper for push_transformation.

See also

push_transformation

property transformation

Product of all matrices on the stack.

Returns:

numpy.ndarray

class ddg.geometry.quadrics.Pencil(Q1, Q2, **kwargs)[source]

Bases: LinearTransformable

Pencil of quadrics in a projective space.

Parameters:

Q1, Q2Quadric or numpy.ndarray: Quadrics spanning the pencil. If given as arrays, they will be created. If one is given as a quadric and the other as an array, the newly created one will have matching affine component to the given one, unless overridden by kwargs.
**kwargsdict: Keyword arguments to be passed to Quadric during creation of Q1 and Q2, if they are given as arrays.

Raises:

ValueError

If the geometries of the quadrics don’t match
If the quadrics are not in the same subspace

See also

ddg.geometry.quadrics.Quadric

property ambient_dimension

push_transformation(f)[source]

Add a transformation to the trafo-stack.

Parameters:

fnumpy.ndarray

pop_transformation()[source]

Pop transformation from the trafo-stack.

Returns:

function

quadric(u)[source]

Return a quadric in the pencil.

The pencil is parametrized as [u1*Q1 + u2*Q2]. We identify [1, a] <-> a and [0, 1] <-> inf.

Parameters:

uarray_like (float, float) or float including inf.: A finite float will be interpreted as (1, u) and inf will be interpreted as (0, 1).

Returns:

Quadric

roots()[source]

Find values for which quadrics in the pencil are degenerate.

This function calculates the roots of the polynomial det(Q1 + lambda*Q2) and their multiplicity.

Returns:

dict {complex or inf: int}: Dictionary whose keys are the roots as complex numbers (or infinity) and whose values are their multiplicity.

degenerate_quadrics(real=True)[source]

Calculate degenerate quadrics in the pencil.

Parameters:

realbool (default=True): If True, only use real roots.

Returns:

list of Quadric

transform(f)

Wrapper for push_transformation.

See also

push_transformation

property transformation

Product of all matrices on the stack.

Returns:

numpy.ndarray

ddg.geometry.quadrics.touching_cone(p, quadric, in_subspace=False)[source]

Calculate touching cone.

The touching cone of a quadric Q and a point P is the set of lines through P that are tangent to Q.

Note that by default, this function looks at tangency in the ambient space as opposed to within quadric.subspace. This means that if the quadric is contained in a proper subspace, the function returns the join of p and quadric.

Parameters:

pnumpy.ndarray of shape (quadric.ambient_dimension+1,) or Point: Point in homogeneous coordinates. Must lie in quadric.subspace if in_subspace is True.
quadricQuadric
in_subspacebool (default=False)

Returns:

Quadric or Subspace

Raises:

ValueError: If in_subspace is True and p is not contained in quadric.subspace.

ddg.geometry.quadrics.cayley_klein_sphere(center, radius, absolute)[source]

Create a Cayley-Klein sphere from center, radius and absolute.

Let b be the inner product induced by the absolute. A Cayley-Klein sphere is defined as the solution set of:

b(center, x) ** 2 - radius * b(center, center) * b(x, x) = 0

Note that if center lies on the absolute, you will just get the polar hyperplane of center.

Parameters:

centerPoint
radiusfloat
absoluteQuadric

Returns:

Quadric

Raises:

ValueError: If center lies on absolute.

ddg.geometry.quadrics.cayley_klein_horosphere(center, radius, absolute)[source]

Create Cayley-Klein Horosphere from center, “radius” and absolute.

Let b be the inner product induced by the absolute. A Cayley-Klein sphere is defined as the solution set of:

b(center, x) ** 2 - radius * b(x, x) = 0

This allows centers to lie on the absolute quadric, at the expense of depending on the representative vector of center. The equation of a regular Cayley-Klein sphere can still be obtained by using the new radius radius * b(center, center).

Parameters:

centerPoint
radiusfloat
absoluteQuadric

Returns:

Quadric

ddg.geometry.quadrics.axis(cone, affine_component=-1)[source]

Find the axis of a non-parabolic cone.

More precisely: The axis of a non-parabolic quadric with signature (n-1, 1, 1), where n is the dimension of the containing subspace.

Parameters:

coneQuadric
affine_componentint (default=-1)

Returns:

Subspace

Raises:

ValueError: If Quadric does not have the signature mentioned above.

ddg.geometry.quadrics.normalization(quadric, affine=False, affine_component=-1)[source]

Signature and normalizing transformation.

Approximately satisfies:

F.T @ quadric.matrix @ F == sgn.matrix

See ddg.math.symmetric_matrices.Signature and ddg.math.symmetric_matrices.AffineSignature for more information.

Note that this is a transformation within the subspace as opposed to a transformation of the ambient space and thus cannot be used with quadric.transform.

Parameters:

quadricQuadric
affinebool (default=False): Whether to normalize to projective or affine normal form.
affine_componentint (default=-1)

Returns:

sgnSignature or AffineSignature
Fnumpy.ndarray of shape (ambient_dimension+1, dimension+2): Transformation that normalizes the quadric

ddg.geometry.quadrics.intersect_quadric_subspace(quadric, subspace)[source]

Intersect quadric with subspace

Parameters:

quadricQuadric
subspaceSubspace

Returns:

Quadric

ddg.geometry.quadrics.join_quadric_subspace(quadric, subspace)[source]

Join a quadric and a subspace.

The result will be the projective cone with top S and basis Q.

Parameters:

quadricQuadric
subspaceSubspace

Returns:

Quadric or Subspace

Raises:

ValueError

If quadric.subspace and subspace are not disjoint.

NotImplementedError

If quadric.subspace and subspace are not disjoint and subspace is not a point
If the join is an object like a “solid cone”, which we can’t represent.

ddg.geometry.quadrics.intersect_quadrics(quadric1, quadric2, affine=False, affine_component=-1, atol=1e-09, rtol=0.0)[source]

Intersection curve of certain quadrics.

Currently only works for two quadrics that are both contained in a pencil

u1 * diag([1, 1, 0, -1]) + u2 * diag([0, k, 1, -1])
     \_______  ________/        \_______  ________/
             \/                         \/
             D1                         D2

for some k in [0, 1]. The quadrics can be contained in any 3D subspace as long as they are given w.r.t. the same basis. The function checks if both quadrics are contained in a pencil like this.

If you know your two quadrics with matrices Q1, Q2 are contained in a transformed version of such a pencil, i.e.

u1 * F.T @ D1 @ F + u2 * F.T @ D2 @ F

for some known invertible F, you can make it work as well: Let B be quadric1.subspace.matrix. Then the quadrics

Quadric(inv(F.T) @ Q1 @ inv(F), subspace=(B @ inv(F)).T)
Quadric(inv(F.T) @ Q1 @ inv(F), subspace=(B @ inv(F)).T)

can be intersected with this function. Automatic detection of this is not yet implemented.

Parameters:

quadric1, quadric2Quadric
affinebool (default=False): Whether the resulting curve should output affine or homogeneous coordinates.
affine_componentint (default=-1): Used to dehomogenize the output curve.
atolfloat (default=1e-9)
rtolfloat(default=0.0)

Returns:

NetCollection

Raises:

ValueError

If the two quadrics are given in different coordinate systems.
If quadric1 and quadric2 do not span the above-mentioned pencil.

Notes

For visualization purposes, not all subspace bases work well, because the curve might pass through infinity.

This function uses the global tolerance defaults if atol or rtol are set to None. See ddg.abc.NonExact for details.

ddg.geometry.quadrics.polarize(obj, quadric)[source]

Polarize a geometric object with respect to a quadric.

Parameters:

objGeometric object: Currently supported: Subspace, Quadric.
quadricQuadric

Returns:

type(obj)

Raises:

TypeError: If the type of object is not supported.

ddg.geometry.quadrics.signature(quadric, subspace=None, affine=False, affine_component=-1)[source]

Signature of a quadric, optionally restricted to a subspace.

Parameters:

subspaceSubspace or None (default=None): Subspace contained in self.subspace
affinebool (default=False): Whether to return projective or affine signature
affine_componentint (default=-1)

Returns:

Signature or AffineSignature

Raises:

ValueError: If subspace is not contained in self.subspace.

See also

ddg.math.symmetric_matrices.Signature
ddg.math.symmetric_matrices.AffineSignature