# Comparison triangle

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Define as the 2-dimensional metric space of constant curvature . So, for example, is the Euclidean plane, is the surface of the unit sphere, and is the hyperbolic plane.

Let be a metric space. Let be a triangle in , with vertices , and . A **comparison triangle** in for is a triangle in with vertices , and such that , and .

Such a triangle is unique up to isometry.

The interior angle of at is called the **comparison angle** between and at . This is well-defined provided and are both distinct from .

## References

- M Bridson & A Haefliger -
*Metric Spaces Of Non-Positive Curvature*, ISBN 3-540-64324-9