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Matthias Koeppe
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src/doc/en/thematic_tutorials/geometry/polyhedra_tutorial.rst: Add # optional - sage.rings.number_field, sage.symbolic
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‎src/doc/en/thematic_tutorials/geometry/polyhedra_tutorial.rst

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@@ -162,17 +162,22 @@ The following example demonstrates the limitations of :code:`RDF`.
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::
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sage: D = polytopes.dodecahedron()
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sage: D
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A 3-dimensional polyhedron in (Number Field in sqrt5 with defining polynomial x^2 - 5 with sqrt5 = 2.236067977499790?)^3 defined as the convex hull of 20 vertices
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sage: D_RDF = Polyhedron(vertices = [n(v.vector(),digits=6) for v in D.vertices()], base_ring=RDF)
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sage: D = polytopes.dodecahedron() # optional - sage.rings.number_field
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sage: D # optional - sage.rings.number_field
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A 3-dimensional polyhedron
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in (Number Field in sqrt5 with defining polynomial x^2 - 5
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with sqrt5 = 2.236067977499790?)^3
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defined as the convex hull of 20 vertices
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sage: vertices_RDF = [n(v.vector(),digits=6) for v in D.vertices()] # optional - sage.rings.number_field
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sage: D_RDF = Polyhedron(vertices=vertices_RDF, base_ring=RDF) # optional - sage.rings.number_field
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doctest:warning
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...
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UserWarning: This polyhedron data is numerically complicated; cdd
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could not convert between the inexact V and H representation
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without loss of data. The resulting object might show
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inconsistencies.
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sage: D_RDF = Polyhedron(vertices = sorted([n(v.vector(),digits=6) for v in D.vertices()]), base_ring=RDF)
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sage: D_RDF = Polyhedron(vertices=sorted(vertices_RDF), base_ring=RDF) # optional - sage.rings.number_field
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Traceback (most recent call last):
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...
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ValueError: *Error: Numerical inconsistency is found. Use the GMP exact arithmetic.
@@ -189,15 +194,15 @@ automatically converts the data to :code:`RDF`:
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.. end of output
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It is also possible to define polyhedron over algebraic numbers.
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It is also possible to define a polyhedron over algebraic numbers.
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::
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sage: sqrt_2 = AA(2)^(1/2)
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sage: cbrt_2 = AA(2)^(1/3)
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sage: timeit('Polyhedron(vertices = [[sqrt_2, 0], [0, cbrt_2]])') # random
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sage: sqrt_2 = AA(2)^(1/2) # optional - sage.rings.number_field
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sage: cbrt_2 = AA(2)^(1/3) # optional - sage.rings.number_field
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sage: timeit('Polyhedron(vertices = [[sqrt_2, 0], [0, cbrt_2]])') # optional - sage.rings.number_field # random
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5 loops, best of 3: 43.2 ms per loop
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sage: P4 = Polyhedron(vertices = [[sqrt_2, 0], [0, cbrt_2]]); P4
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sage: P4 = Polyhedron(vertices = [[sqrt_2, 0], [0, cbrt_2]]); P4 # optional - sage.rings.number_field
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A 1-dimensional polyhedron in AA^2 defined as the convex hull of 2 vertices
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.. end of output
@@ -206,11 +211,11 @@ There is another way to create a polyhedron over algebraic numbers:
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::
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sage: K.<a> = NumberField(x^2 - 2, embedding=AA(2)**(1/2))
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sage: L.<b> = NumberField(x^3 - 2, embedding=AA(2)**(1/3))
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sage: timeit('Polyhedron(vertices = [[a, 0], [0, b]])') # random
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sage: K.<a> = NumberField(x^2 - 2, embedding=AA(2)**(1/2)) # optional - sage.rings.number_field
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sage: L.<b> = NumberField(x^3 - 2, embedding=AA(2)**(1/3)) # optional - sage.rings.number_field
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sage: timeit('Polyhedron(vertices = [[a, 0], [0, b]])') # optional - sage.rings.number_field # random
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5 loops, best of 3: 39.9 ms per loop
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sage: P5 = Polyhedron(vertices = [[a, 0], [0, b]]); P5
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sage: P5 = Polyhedron(vertices = [[a, 0], [0, b]]); P5 # optional - sage.rings.number_field
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A 1-dimensional polyhedron in AA^2 defined as the convex hull of 2 vertices
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.. end of output
@@ -219,11 +224,15 @@ If the base ring is known it may be a good option to use the proper :meth:`sage.
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::
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sage: J = K.composite_fields(L)[0]
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sage: timeit('Polyhedron(vertices = [[J(a), 0], [0, J(b)]])') # random
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sage: J = K.composite_fields(L)[0] # optional - sage.rings.number_field
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sage: timeit('Polyhedron(vertices = [[J(a), 0], [0, J(b)]])') # optional - sage.rings.number_field # random
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25 loops, best of 3: 9.8 ms per loop
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sage: P5_comp = Polyhedron(vertices = [[J(a), 0], [0, J(b)]]); P5_comp
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A 1-dimensional polyhedron in (Number Field in ab with defining polynomial x^6 - 6*x^4 - 4*x^3 + 12*x^2 - 24*x - 4 with ab = -0.1542925124782219?)^2 defined as the convex hull of 2 vertices
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sage: P5_comp = Polyhedron(vertices = [[J(a), 0], [0, J(b)]]); P5_comp # optional - sage.rings.number_field
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A 1-dimensional polyhedron
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in (Number Field in ab with defining polynomial
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x^6 - 6*x^4 - 4*x^3 + 12*x^2 - 24*x - 4
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with ab = -0.1542925124782219?)^2
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defined as the convex hull of 2 vertices
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.. end of output
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@@ -232,9 +241,9 @@ It is not possible to define a polyhedron over it:
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::
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sage: sqrt_2s = sqrt(2)
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sage: cbrt_2s = 2^(1/3)
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sage: Polyhedron(vertices = [[sqrt_2s, 0], [0, cbrt_2s]])
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sage: sqrt_2s = sqrt(2) # optional - sage.symbolic
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sage: cbrt_2s = 2^(1/3) # optional - sage.symbolic
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sage: Polyhedron(vertices = [[sqrt_2s, 0], [0, cbrt_2s]]) # optional - sage.symbolic
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Traceback (most recent call last):
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...
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ValueError: no default backend for computations with Symbolic Ring
@@ -567,12 +576,12 @@ but not algebraic or symbolic values:
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::
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sage: P4_cdd = Polyhedron(vertices = [[sqrt_2, 0], [0, cbrt_2]], backend='cdd')
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sage: P4_cdd = Polyhedron(vertices = [[sqrt_2, 0], [0, cbrt_2]], backend='cdd') # optional - sage.rings.number_field
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Traceback (most recent call last):
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ValueError: No such backend (=cdd) implemented for given basering (=Algebraic Real Field).
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sage: P5_cdd = Polyhedron(vertices = [[sqrt_2s, 0], [0, cbrt_2s]], backend='cdd')
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sage: P5_cdd = Polyhedron(vertices = [[sqrt_2s, 0], [0, cbrt_2s]], backend='cdd') # optional - sage.symbolic
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Traceback (most recent call last):
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...
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ValueError: No such backend (=cdd) implemented for given basering (=Symbolic Ring).
@@ -646,9 +655,11 @@ An example with quadratic field:
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::
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sage: V = polytopes.dodecahedron().vertices_list()
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sage: Polyhedron(vertices=V, backend='polymake') # optional - polymake
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A 3-dimensional polyhedron in (Number Field in sqrt5 with defining polynomial x^2 - 5)^3 defined as the convex hull of 20 vertices
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sage: V = polytopes.dodecahedron().vertices_list() # optional - sage.rings.number_field
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sage: Polyhedron(vertices=V, backend='polymake') # optional - polymake # optional - sage.rings.number_field
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A 3-dimensional polyhedron
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in (Number Field in sqrt5 with defining polynomial x^2 - 5)^3
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defined as the convex hull of 20 vertices
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.. end of output
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@@ -775,8 +786,11 @@ polytope is already defined!
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::
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sage: A = polytopes.buckyball(); A # can take long
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A 3-dimensional polyhedron in (Number Field in sqrt5 with defining polynomial x^2 - 5 with sqrt5 = 2.236067977499790?)^3 defined as the convex hull of 60 vertices
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sage: A = polytopes.buckyball(); A # can take long # optional - sage.rings.number_field
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A 3-dimensional polyhedron
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in (Number Field in sqrt5 with defining polynomial x^2 - 5
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with sqrt5 = 2.236067977499790?)^3
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defined as the convex hull of 60 vertices
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sage: B = polytopes.cross_polytope(4); B
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A 4-dimensional polyhedron in ZZ^4 defined as the convex hull of 8 vertices
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sage: C = polytopes.cyclic_polytope(3,10); C

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