Create a triangle with specific angles in Hyperbolic space, eg., Upper Halfspace Plane

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I'm trying to create a triangle in hyperbolic space, specifically in the Upper Halfspace Plane, using SageMath. I would like to specify the angles of the triangle and generate its corresponding geometric representation.

Could anyone provide guidance on how to achieve this using SageMath? Any example code or resources would be greatly appreciated. Thank you!

Examlple

rotations = [pi / 3, pi / 3, pi / 4]
base_point = CC(0.5 + 0.5 * I)
pts = [base_point]
_z = base_point
for _phi in rotations:
    _z = e ^ (I * _phi) * _z
    pts.append(_z)

P = Graphics()
g = hyperbolic_polygon(pts=pts, model="PD", fill=True)
P += g.plot()
P.show(axes=True)