Hyperplanes#
Note
If you want to learn about Sage’s hyperplane arrangements then you
should start with
sage.geometry.hyperplane_arrangement.arrangement
. This
module is used to represent the individual hyperplanes, but you
should never construct the classes from this module directly (but
only via the
HyperplaneArrangements
.
A linear expression, for example, \(3x+3y-5z-7\) stands for the
hyperplane with the equation \(x+3y-5z=7\). To create it in Sage, you
first have to create a
HyperplaneArrangements
object to define the variables \(x\), \(y\), \(z\):
sage: H.<x,y,z> = HyperplaneArrangements(QQ)
sage: h = 3*x + 2*y - 5*z - 7; h
Hyperplane 3*x + 2*y - 5*z - 7
sage: h.coefficients()
[-7, 3, 2, -5]
sage: h.normal()
(3, 2, -5)
sage: h.constant_term()
-7
sage: h.change_ring(GF(3))
Hyperplane 0*x + 2*y + z + 2
sage: h.point()
(21/38, 7/19, -35/38)
sage: h.linear_part()
Vector space of degree 3 and dimension 2 over Rational Field
Basis matrix:
[ 1 0 3/5]
[ 0 1 2/5]
Another syntax to create hyperplanes is to specify coefficients and a constant term:
sage: V = H.ambient_space(); V
3-dimensional linear space over Rational Field with coordinates x, y, z
sage: h in V
True
sage: V([3, 2, -5], -7)
Hyperplane 3*x + 2*y - 5*z - 7
Or constant term and coefficients together in one list/tuple/iterable:
sage: V([-7, 3, 2, -5])
Hyperplane 3*x + 2*y - 5*z - 7
sage: v = vector([-7, 3, 2, -5]); v
(-7, 3, 2, -5)
sage: V(v)
Hyperplane 3*x + 2*y - 5*z - 7
Note that the constant term comes first, which matches the notation
for Sage’s Polyhedron()
sage: Polyhedron(ieqs=[(4,1,2,3)]).Hrepresentation()
(An inequality (1, 2, 3) x + 4 >= 0,)
The difference between hyperplanes as implemented in this module and hyperplane arrangements is that:
hyperplane arrangements contain multiple hyperplanes (of course),
linear expressions are a module over the base ring, and these module structure is inherited by the hyperplanes.
The latter means that you can add and multiply by a scalar:
sage: h = 3*x + 2*y - 5*z - 7; h
Hyperplane 3*x + 2*y - 5*z - 7
sage: -h
Hyperplane -3*x - 2*y + 5*z + 7
sage: h + x
Hyperplane 4*x + 2*y - 5*z - 7
sage: h + 7
Hyperplane 3*x + 2*y - 5*z + 0
sage: 3*h
Hyperplane 9*x + 6*y - 15*z - 21
sage: h * RDF(3)
Hyperplane 9.0*x + 6.0*y - 15.0*z - 21.0
Which you can’t do with hyperplane arrangements:
sage: arrangement = H(h, x, y, x+y-1); arrangement
Arrangement <y | x | x + y - 1 | 3*x + 2*y - 5*z - 7>
sage: arrangement + x
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for +:
'Hyperplane arrangements in 3-dimensional linear space
over Rational Field with coordinates x, y, z' and
'Hyperplane arrangements in 3-dimensional linear space
over Rational Field with coordinates x, y, z'
- class sage.geometry.hyperplane_arrangement.hyperplane.AmbientVectorSpace(base_ring, names=())#
Bases:
sage.geometry.linear_expression.LinearExpressionModule
The ambient space for hyperplanes.
This class is the parent for the
Hyperplane
instances.- Element#
alias of
Hyperplane
- change_ring(base_ring)#
Return a ambient vector space with a changed base ring.
INPUT:
base_ring
– a ring; the new base ring
OUTPUT:
A new
AmbientVectorSpace
.EXAMPLES:
sage: M.<y> = HyperplaneArrangements(QQ) sage: V = M.ambient_space() sage: V.change_ring(RR) 1-dimensional linear space over Real Field with 53 bits of precision with coordinate y
- dimension()#
Return the ambient space dimension.
OUTPUT:
An integer.
EXAMPLES:
sage: M.<x,y> = HyperplaneArrangements(QQ) sage: x.parent().dimension() 2 sage: x.parent() is M.ambient_space() True sage: x.dimension() 1
- symmetric_space()#
Construct the symmetric space of
self
.Consider a hyperplane arrangement \(A\) in the vector space \(V = k^n\), for some field \(k\). The symmetric space is the symmetric algebra \(S(V^*)\) as the polynomial ring \(k[x_1, x_2, \ldots, x_n]\) where \((x_1, x_2, \ldots, x_n)\) is a basis for \(V\).
EXAMPLES:
sage: H.<x,y,z> = HyperplaneArrangements(QQ) sage: A = H.ambient_space() sage: A.symmetric_space() Multivariate Polynomial Ring in x, y, z over Rational Field
- class sage.geometry.hyperplane_arrangement.hyperplane.Hyperplane(parent, coefficients, constant)#
Bases:
sage.geometry.linear_expression.LinearExpression
A hyperplane.
You should always use
AmbientVectorSpace
to construct instances of this class.INPUT:
parent
– the parentAmbientVectorSpace
coefficients
– a vector of coefficients of the linear variablesconstant
– the constant term for the linear expression
EXAMPLES:
sage: H.<x,y> = HyperplaneArrangements(QQ) sage: x+y-1 Hyperplane x + y - 1 sage: ambient = H.ambient_space() sage: ambient._element_constructor_(x+y-1) Hyperplane x + y - 1
For technical reasons, we must allow the degenerate cases of an empty space and of a full space:
sage: 0*x Hyperplane 0*x + 0*y + 0 sage: 0*x + 1 Hyperplane 0*x + 0*y + 1 sage: x + 0 == x + ambient(0) # because coercion requires them True
- dimension()#
The dimension of the hyperplane.
OUTPUT:
An integer.
EXAMPLES:
sage: H.<x,y,z> = HyperplaneArrangements(QQ) sage: h = x + y + z - 1 sage: h.dimension() 2
- intersection(other)#
The intersection of
self
withother
.INPUT:
other
– a hyperplane, a polyhedron, or something that defines a polyhedron
OUTPUT:
A polyhedron.
EXAMPLES:
sage: H.<x,y,z> = HyperplaneArrangements(QQ) sage: h = x + y + z - 1 sage: h.intersection(x - y) A 1-dimensional polyhedron in QQ^3 defined as the convex hull of 1 vertex and 1 line sage: h.intersection(polytopes.cube()) A 2-dimensional polyhedron in QQ^3 defined as the convex hull of 3 vertices
- linear_part()#
The linear part of the affine space.
OUTPUT:
Vector subspace of the ambient vector space, parallel to the hyperplane.
EXAMPLES:
sage: H.<x,y,z> = HyperplaneArrangements(QQ) sage: h = x + 2*y + 3*z - 1 sage: h.linear_part() Vector space of degree 3 and dimension 2 over Rational Field Basis matrix: [ 1 0 -1/3] [ 0 1 -2/3]
- linear_part_projection(point)#
Orthogonal projection onto the linear part.
INPUT:
point
– vector of the ambient space, or anything that can be converted into one; not necessarily on the hyperplane
OUTPUT:
Coordinate vector of the projection of
point
with respect to the basis oflinear_part()
. In particular, the length of this vector is one less than the ambient space dimension.EXAMPLES:
sage: H.<x,y,z> = HyperplaneArrangements(QQ) sage: h = x + 2*y + 3*z - 4 sage: h.linear_part() Vector space of degree 3 and dimension 2 over Rational Field Basis matrix: [ 1 0 -1/3] [ 0 1 -2/3] sage: p1 = h.linear_part_projection(0); p1 (0, 0) sage: p2 = h.linear_part_projection([3,4,5]); p2 (8/7, 2/7) sage: h.linear_part().basis() [ (1, 0, -1/3), (0, 1, -2/3) ] sage: p3 = h.linear_part_projection([1,1,1]); p3 (4/7, 1/7)
- normal()#
Return the normal vector.
OUTPUT:
A vector over the base ring.
EXAMPLES:
sage: H.<x, y, z> = HyperplaneArrangements(QQ) sage: x.normal() (1, 0, 0) sage: x.A(), x.b() ((1, 0, 0), 0) sage: (x + 2*y + 3*z + 4).normal() (1, 2, 3)
- orthogonal_projection(point)#
Return the orthogonal projection of a point.
INPUT:
point
– vector of the ambient space, or anything that can be converted into one; not necessarily on the hyperplane
OUTPUT:
A vector in the ambient vector space that lies on the hyperplane.
In finite characteristic, a
ValueError
is raised if the the norm of the hyperplane normal is zero.EXAMPLES:
sage: H.<x,y,z> = HyperplaneArrangements(QQ) sage: h = x + 2*y + 3*z - 4 sage: p1 = h.orthogonal_projection(0); p1 (2/7, 4/7, 6/7) sage: p1 in h True sage: p2 = h.orthogonal_projection([3,4,5]); p2 (10/7, 6/7, 2/7) sage: p1 in h True sage: p3 = h.orthogonal_projection([1,1,1]); p3 (6/7, 5/7, 4/7) sage: p3 in h True
- plot(**kwds)#
Plot the hyperplane.
OUTPUT:
A graphics object.
EXAMPLES:
sage: L.<x, y> = HyperplaneArrangements(QQ) sage: (x+y-2).plot() # optional - sage.plot Graphics object consisting of 2 graphics primitives
- point()#
Return the point closest to the origin.
OUTPUT:
A vector of the ambient vector space. The closest point to the origin in the \(L^2\)-norm.
In finite characteristic a random point will be returned if the norm of the hyperplane normal vector is zero.
EXAMPLES:
sage: H.<x,y,z> = HyperplaneArrangements(QQ) sage: h = x + 2*y + 3*z - 4 sage: h.point() (2/7, 4/7, 6/7) sage: h.point() in h True sage: H.<x,y,z> = HyperplaneArrangements(GF(3)) sage: h = 2*x + y + z + 1 sage: h.point() (1, 0, 0) sage: h.point().base_ring() Finite Field of size 3 sage: H.<x,y,z> = HyperplaneArrangements(GF(3)) sage: h = x + y + z + 1 sage: h.point() (2, 0, 0)
- polyhedron()#
Return the hyperplane as a polyhedron.
OUTPUT:
A
Polyhedron()
instance.EXAMPLES:
sage: H.<x,y,z> = HyperplaneArrangements(QQ) sage: h = x + 2*y + 3*z - 4 sage: P = h.polyhedron(); P A 2-dimensional polyhedron in QQ^3 defined as the convex hull of 1 vertex and 2 lines sage: P.Hrepresentation() (An equation (1, 2, 3) x - 4 == 0,) sage: P.Vrepresentation() (A line in the direction (0, 3, -2), A line in the direction (3, 0, -1), A vertex at (0, 0, 4/3))
- primitive(signed=True)#
Return hyperplane defined by primitive equation.
INPUT:
signed
– boolean (optional, default:True
); whether to preserve the overall sign
OUTPUT:
Hyperplane whose linear expression has common factors and denominators cleared. That is, the same hyperplane (with the same sign) but defined by a rescaled equation. Note that different linear expressions must define different hyperplanes as comparison is used in caching.
If
signed
, the overall rescaling is by a positive constant only.EXAMPLES:
sage: H.<x,y> = HyperplaneArrangements(QQ) sage: h = -1/3*x + 1/2*y - 1; h Hyperplane -1/3*x + 1/2*y - 1 sage: h.primitive() Hyperplane -2*x + 3*y - 6 sage: h == h.primitive() False sage: (4*x + 8).primitive() Hyperplane x + 0*y + 2 sage: (4*x - y - 8).primitive(signed=True) # default Hyperplane 4*x - y - 8 sage: (4*x - y - 8).primitive(signed=False) Hyperplane -4*x + y + 8
- to_symmetric_space()#
Return
self
considered as an element in the corresponding symmetric space.EXAMPLES:
sage: L.<x, y> = HyperplaneArrangements(QQ) sage: h = -1/3*x + 1/2*y sage: h.to_symmetric_space() -1/3*x + 1/2*y sage: hp = -1/3*x + 1/2*y - 1 sage: hp.to_symmetric_space() Traceback (most recent call last): ... ValueError: the hyperplane must pass through the origin