Polygons and triangles in hyperbolic geometry

AUTHORS:

  • Hartmut Monien (2011-08)
  • Vincent Delecroix (2014-11)
class sage.plot.hyperbolic_polygon.HyperbolicPolygon(pts, options)

Bases: sage.plot.bezier_path.BezierPath

Primitive class for hyberbolic 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 – dict of valid plot options to pass to constructor

EXAMPLES:

Note that constructions should use hyperbolic_polygon() or hyperbolic_triangle():

sage: from sage.plot.hyperbolic_polygon import HyperbolicPolygon
sage: print(HyperbolicPolygon([0, 1/2, I], {}))
Hyperbolic polygon (0.000000000000000, 0.500000000000000, 1.00000000000000*I)
sage.plot.hyperbolic_polygon.hyperbolic_polygon(pts, rgbcolor='blue', thickness=1, zorder=2, alpha=1, linestyle='solid', fill=False, **options)

Return a hyperbolic polygon in the hyperbolic plane with vertices pts.

Type ?hyperbolic_polygon to see all options.

INPUT:

  • pts – a list or tuple of complex numbers

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 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
../../_images/hyperbolic_polygon-1.svg

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
../../_images/hyperbolic_polygon-2.svg
sage.plot.hyperbolic_polygon.hyperbolic_triangle(a, b, c, **options)

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
../../_images/hyperbolic_polygon-3.svg

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
../../_images/hyperbolic_polygon-4.svg