# Comparison triangle > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Comparison_triangle > Markdown URL: https://mediated.wiki/source/Comparison_triangle.md > Source: https://en.wikipedia.org/wiki/Comparison_triangle > Source revision: 1332571703 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) In [metric geometry](/source/Metric_geometry), comparison triangles are constructions used to define [higher bounds on curvature](/source/CAT(k)_space) in the framework of [locally geodesic metric spaces](/source/Geodesic_metric_space), thereby playing a similar role to that of higher bounds on [sectional curvature](/source/Sectional_curvature) in [Riemannian geometry](/source/Riemannian_geometry). ## Definitions ### Comparison triangles Let M_{0}^{2} = \mathbb{E}^2 be the [euclidean plane](/source/Euclidean_plane), M_{1}^{2} = \mathbb{S}^2 be the [unit 2-sphere](/source/Unit_sphere), and M_{-1}^{2} = \mathbb{H}^2 be the [hyperbolic plane](/source/Hyperbolic_geometry). For k > 0, let M_{k}^{2} and M_{-k}^{2} denote the spaces obtained, respectively, from M_{1}^{2} and M_{-1}^{2} by multiplying the distance by \frac{1}{\sqrt{|k|}}. For any k\in \R, M_{k}^{2} is the unique [complete](/source/Complete_manifold), [simply-connected](/source/Simply_connected_space), 2-dimensional [Riemannian manifold](/source/Riemannian_manifold) of constant sectional curvature k. Let X be a [metric space](/source/Metric_space). Let T be a geodesic triangle in X, i.e. three points p, q and r and three geodesic segments [p, q], [q, r] and [r, p]. A **comparison triangle** T* in M_{k}^{2} for T is a [geodesic triangle](/source/Geodesic_triangle) in M_{k}^{2} with vertices p', q' and r' such that d(p,q) = d(p',q'), d(p,r) = d(p',r') and d(r,q) = d(r',q'). Such a triangle, when it exists, is unique up to [isometry](/source/Isometry). The existence is always true for k\le 0. For k > 0, it can be ensured by the additional condition d(p, q) + d(q, r) + d(r, p) \le \frac{2\pi}{\sqrt{k}} (i.e. the length of the triangle does not exceed that of a [great circle](/source/Great_circle) of the sphere M_{k}^{2}). #### Comparison angles The [interior angle](/source/Interior_angle) of T* at p' is called the **comparison angle** between q and r at p. This is well-defined provided q and r are both distinct from p, and only depends on the lengths d(p, q), d(q, r), d(p, r). Let it be denoted by \overline{\angle}_{p, q, r}^{(k)}. Using inverse trigonometry, one has the formulas: \cos(\overline{\angle}_{p, q, r}^{(0)}) = \frac{d(q, r)^2 - d(p, q)^2 - d(p, r)^2}{2d(p, q)d(p, r)}, \cos(\overline{\angle}_{p, q, r}^{(k)}) = \frac{\cos(\sqrt{k}d(q, r)) - \cos(\sqrt{k}d(p, q))\cos(\sqrt{k}d(p, r))}{\sin(\sqrt{k}d(p, q))\sin(\sqrt{k}d(p, r))} ~~ \text{for} ~~ k > 0, \cos(\overline{\angle}_{p, q, r}^{(k)}) = \frac{\cosh(\sqrt{-k}d(p, q))\cosh(\sqrt{-k}d(p, r)) - \cosh(\sqrt{-k}d(q, r))}{\sinh(\sqrt{-k}d(p, q))\sinh(\sqrt{-k}d(p, r))} ~~ \text{for} ~~ {k < 0}. #### Alexandrov angles Comparison angles provide notions of angles between geodesics that make sense in arbitrary metric spaces. The **Alexandrov angle**, or **outer angle**, between two nontrivial geodesics c, c' with c(0) = c'(0) is defined as \angle_{c, c'} = \limsup_{t, t' \rightarrow 0} \overline{\angle}_{c(0), c(t), c'(t')}. ### Comparison tripods See also: [Hyperbolic metric space](/source/Hyperbolic_metric_space) The following similar construction, which appears in certain possible definitions of Gromov-hyperbolicity, may be regarded as a limit case when k\rightarrow -\infty. For three points x, y, z in a metric space X, the [Gromov product](/source/Gromov_product) of x and y at z is half of the [triangle inequality](/source/Triangle_inequality) defect: (x, y)_z = \frac{1}{2}(d(x, z) + d(y, z) - d(x, y)) Given a geodesic triangle \Delta in X with vertices (p, q, r), the **comparison tripod** T_\Delta for \Delta is the metric graph obtained by gluing three segments [p', c_p], [q', c_q], [r', c_r] of respective lengths (q, r)_p, (r, p)_q, (p, q)_r along a vertex c, setting c_p = c_q = c_r = c. One has d(p', q') = d(p, q),~~d(q', r') = d(q, r),~~d(r', p') = d(r, p), and T_\Delta is the union of the three unique geodesic segments [p', q'], [q', r'], [r', p']. Furthermore, there is a well-defined comparison map f_\Delta: \Delta \longrightarrow T_\Delta with f_\Delta(p) = p', f_\Delta(q) = q', f_\Delta(r) = r', such that f_\Delta is [isometric](/source/Isometry) on each side of \Delta. The vertex c is called the **center** of T_\Delta, and its preimage under f_\Delta is called the **center** of \Delta, its points the **internal points** of \Delta, and its [diameter](/source/Metric_space#Diameter_of_a_metric_space) the **insize** of \Delta. One way to formulate Gromov-hyperbolicity is to require f_\Delta not to change the distances by more than a constant \delta \ge 0. Another way is to require the insizes of triangles \Delta to be bounded above by a uniform constant \delta' \ge 0. Equivalently, a tripod is a comparison triangle in a universal [real tree](/source/Real_tree) of valence \ge 3. Such trees appear as [ultralimits](/source/Ultralimit#Ultralimit_of_metric_spaces_with_specified_base-points) of the M_{k}^{2} as k\rightarrow -\infty.[1] ## The CAT(k) condition Main article: [CAT(k) space](/source/CAT(k)_space) ## The Alexandrov lemma In various situations, the **Alexandrov lemma** (also called the **triangle gluing lemma**) allows one to decompose a geodesic triangle into smaller triangles for which proving the CAT(k) condition is easier, and then deduce the CAT(k) condition for the bigger triangle. This is done by gluing together comparison triangles for the smaller triangles and then "unfolding" the figure into a comparison triangle for the bigger triangle. ## References 1. DruĊ£u, Cornelia & Kapovich, Michael (2018-03-28). ["Geometric Group Theory"](http://www.ams.org/books/coll/063/). *American Mathematical Society*. Retrieved 2024-12-10. - M Bridson & [A Haefliger](/source/A_Haefliger) - *Metric Spaces Of Non-Positive [Curvature](/source/Curvature)*, ISBN 3-540-64324-9 --- Adapted from the Wikipedia article [Comparison triangle](https://en.wikipedia.org/wiki/Comparison_triangle) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Comparison_triangle?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.