# Polygons and triangles in hyperbolic geometry¶

AUTHORS:

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

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 – 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


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

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


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