Polygons and triangles in hyperbolic geometry¶
AUTHORS:
Hartmut Monien (2011-08)
Vincent Delecroix (2014-11)
- class sage.plot.hyperbolic_polygon.HyperbolicPolygon(pts, model, options)[source]¶
Bases:
HyperbolicArcCore
Primitive class for hyperbolic polygon type.
See
hyperbolic_polygon?
for information about plotting a hyperbolic polygon in the complex plane.INPUT:
pts
– coordinates of the polygon (as complex numbers)options
– dictionary of valid plot options to pass to constructor
EXAMPLES:
Note that constructions should use
hyperbolic_polygon()
orhyperbolic_triangle()
:sage: from sage.plot.hyperbolic_polygon import HyperbolicPolygon sage: print(HyperbolicPolygon([0, 1/2, I], "UHP", {})) Hyperbolic polygon (0.000000000000000, 0.500000000000000, 1.00000000000000*I)
>>> from sage.all import * >>> from sage.plot.hyperbolic_polygon import HyperbolicPolygon >>> print(HyperbolicPolygon([Integer(0), Integer(1)/Integer(2), I], "UHP", {})) Hyperbolic polygon (0.000000000000000, 0.500000000000000, 1.00000000000000*I)
- sage.plot.hyperbolic_polygon.hyperbolic_polygon(pts, model='UHP', resolution=200, alpha=1, fill=False, thickness=1, rgbcolor='blue', zorder=2, linestyle='solid', **options)[source]¶
Return a hyperbolic polygon in the hyperbolic plane with vertices
pts
.Type
?hyperbolic_polygon
to see all options.INPUT:
pts
– list or tuple of complex numbers
OPTIONS:
model
– (default:UHP
) Model used for hyperbolic planealpha
– (default: 1)fill
– (default:False
)thickness
– (default: 1)rgbcolor
– (default:'blue'
)linestyle
– (default:'solid'
) the style of the line, which is one of'dashed'
,'dotted'
,'solid'
,'dashdot'
, or'--'
,':'
,'-'
,'-.'
, respectively
EXAMPLES:
Show a hyperbolic polygon with coordinates \(-1\), \(3i\), \(2+2i\), \(1+i\):
sage: hyperbolic_polygon([-1,3*I,2+2*I,1+I]) Graphics object consisting of 1 graphics primitive
>>> from sage.all import * >>> hyperbolic_polygon([-Integer(1),Integer(3)*I,Integer(2)+Integer(2)*I,Integer(1)+I]) Graphics object consisting of 1 graphics primitive
With more options:
sage: hyperbolic_polygon([-1,3*I,2+2*I,1+I], fill=True, color='red') Graphics object consisting of 1 graphics primitive
>>> from sage.all import * >>> hyperbolic_polygon([-Integer(1),Integer(3)*I,Integer(2)+Integer(2)*I,Integer(1)+I], fill=True, color='red') Graphics object consisting of 1 graphics primitive
With a vertex at \(\infty\):
sage: hyperbolic_polygon([-1,0,1,Infinity], color='green') Graphics object consisting of 1 graphics primitive
>>> from sage.all import * >>> hyperbolic_polygon([-Integer(1),Integer(0),Integer(1),Infinity], color='green') Graphics object consisting of 1 graphics primitive
Poincare disc model is supported via the parameter
model
. Show a hyperbolic polygon in the Poincare disc model with coordinates \(1\), \(i\), \(-1\), \(-i\):sage: hyperbolic_polygon([1,I,-1,-I], model='PD', color='green') Graphics object consisting of 2 graphics primitives
>>> from sage.all import * >>> hyperbolic_polygon([Integer(1),I,-Integer(1),-I], model='PD', color='green') Graphics object consisting of 2 graphics primitives
With more options:
sage: hyperbolic_polygon([1,I,-1,-I], model='PD', color='green', fill=True, linestyle='-') Graphics object consisting of 2 graphics primitives
>>> from sage.all import * >>> hyperbolic_polygon([Integer(1),I,-Integer(1),-I], model='PD', color='green', fill=True, linestyle='-') Graphics object consisting of 2 graphics primitives
Klein model is also supported via the parameter
model
. Show a hyperbolic polygon in the Klein model with coordinates \(1\), \(e^{i\pi/3}\), \(e^{i2\pi/3}\), \(-1\), \(e^{i4\pi/3}\), \(e^{i5\pi/3}\):sage: p1 = 1 sage: p2 = (cos(pi/3), sin(pi/3)) sage: p3 = (cos(2*pi/3), sin(2*pi/3)) sage: p4 = -1 sage: p5 = (cos(4*pi/3), sin(4*pi/3)) sage: p6 = (cos(5*pi/3), sin(5*pi/3)) sage: hyperbolic_polygon([p1,p2,p3,p4,p5,p6], model='KM', fill=True, color='purple') Graphics object consisting of 2 graphics primitives
>>> from sage.all import * >>> p1 = Integer(1) >>> p2 = (cos(pi/Integer(3)), sin(pi/Integer(3))) >>> p3 = (cos(Integer(2)*pi/Integer(3)), sin(Integer(2)*pi/Integer(3))) >>> p4 = -Integer(1) >>> p5 = (cos(Integer(4)*pi/Integer(3)), sin(Integer(4)*pi/Integer(3))) >>> p6 = (cos(Integer(5)*pi/Integer(3)), sin(Integer(5)*pi/Integer(3))) >>> hyperbolic_polygon([p1,p2,p3,p4,p5,p6], model='KM', fill=True, color='purple') Graphics object consisting of 2 graphics primitives
Hyperboloid model is supported partially, via the parameter
model
. Show a hyperbolic polygon in the hyperboloid model with coordinates \((3,3,\sqrt(19))\), \((3,-3,\sqrt(19))\), \((-3,-3,\sqrt(19))\), \((-3,3,\sqrt(19))\):sage: pts = [(3,3,sqrt(19)),(3,-3,sqrt(19)),(-3,-3,sqrt(19)),(-3,3,sqrt(19))] sage: hyperbolic_polygon(pts, model='HM') Graphics3d Object
>>> from sage.all import * >>> pts = [(Integer(3),Integer(3),sqrt(Integer(19))),(Integer(3),-Integer(3),sqrt(Integer(19))),(-Integer(3),-Integer(3),sqrt(Integer(19))),(-Integer(3),Integer(3),sqrt(Integer(19)))] >>> hyperbolic_polygon(pts, model='HM') Graphics3d Object
Filling a hyperbolic_polygon in hyperboloid model is possible although jaggy. We show a filled hyperbolic polygon in the hyperboloid model with coordinates \((1,1,\sqrt(3))\), \((0,2,\sqrt(5))\), \((2,0,\sqrt(5))\). (The doctest is done at lower resolution than the picture below to give a faster result.)
sage: pts = [(1,1,sqrt(3)), (0,2,sqrt(5)), (2,0,sqrt(5))] sage: hyperbolic_polygon(pts, model='HM', resolution=50, ....: color='yellow', fill=True) Graphics3d Object
>>> from sage.all import * >>> pts = [(Integer(1),Integer(1),sqrt(Integer(3))), (Integer(0),Integer(2),sqrt(Integer(5))), (Integer(2),Integer(0),sqrt(Integer(5)))] >>> hyperbolic_polygon(pts, model='HM', resolution=Integer(50), ... color='yellow', fill=True) Graphics3d Object
- sage.plot.hyperbolic_polygon.hyperbolic_triangle(a, b, c, model='UHP', **options)[source]¶
Return a hyperbolic triangle in the hyperbolic plane with vertices
(a,b,c)
.Type
?hyperbolic_polygon
to see all options.INPUT:
a
,b
,c
– complex numbers in the upper half complex plane
OPTIONS:
alpha
– (default: 1)fill
– (default:False
)thickness
– (default: 1)rgbcolor
– (default:'blue'
)linestyle
– (default:'solid'
) the style of the line, which is one of'dashed'
,'dotted'
,'solid'
,'dashdot'
, or'--'
,':'
,'-'
,'-.'
, respectively
EXAMPLES:
Show a hyperbolic triangle with coordinates \(0\), \(1/2 + i\sqrt{3}/2\) and \(-1/2 + i\sqrt{3}/2\):
sage: hyperbolic_triangle(0, -1/2+I*sqrt(3)/2, 1/2+I*sqrt(3)/2) Graphics object consisting of 1 graphics primitive
>>> from sage.all import * >>> hyperbolic_triangle(Integer(0), -Integer(1)/Integer(2)+I*sqrt(Integer(3))/Integer(2), Integer(1)/Integer(2)+I*sqrt(Integer(3))/Integer(2)) Graphics object consisting of 1 graphics primitive
A hyperbolic triangle with coordinates \(0\), \(1\) and \(2+i\) and a dashed line:
sage: hyperbolic_triangle(0, 1, 2+i, fill=true, rgbcolor='red', linestyle='--') Graphics object consisting of 1 graphics primitive
>>> from sage.all import * >>> hyperbolic_triangle(Integer(0), Integer(1), Integer(2)+i, fill=true, rgbcolor='red', linestyle='--') Graphics object consisting of 1 graphics primitive
A hyperbolic triangle with a vertex at \(\infty\):
sage: hyperbolic_triangle(-5,Infinity,5) Graphics object consisting of 1 graphics primitive
>>> from sage.all import * >>> hyperbolic_triangle(-Integer(5),Infinity,Integer(5)) Graphics object consisting of 1 graphics primitive
It can also plot a hyperbolic triangle in the Poincaré disk model:
sage: z1 = CC((cos(pi/3),sin(pi/3))) sage: z2 = CC((0.6*cos(3*pi/4),0.6*sin(3*pi/4))) sage: z3 = 1 sage: hyperbolic_triangle(z1, z2, z3, model='PD', color='red') Graphics object consisting of 2 graphics primitives
>>> from sage.all import * >>> z1 = CC((cos(pi/Integer(3)),sin(pi/Integer(3)))) >>> z2 = CC((RealNumber('0.6')*cos(Integer(3)*pi/Integer(4)),RealNumber('0.6')*sin(Integer(3)*pi/Integer(4)))) >>> z3 = Integer(1) >>> hyperbolic_triangle(z1, z2, z3, model='PD', color='red') Graphics object consisting of 2 graphics primitives
sage: hyperbolic_triangle(0.3+0.3*I, 0.8*I, -0.5-0.5*I, model='PD', color='magenta') Graphics object consisting of 2 graphics primitives
>>> from sage.all import * >>> hyperbolic_triangle(RealNumber('0.3')+RealNumber('0.3')*I, RealNumber('0.8')*I, -RealNumber('0.5')-RealNumber('0.5')*I, model='PD', color='magenta') Graphics object consisting of 2 graphics primitives