Comparison angles of hinges, model triangle angles, and the Alexandrov upper angle

(1) Comparison angle of a length triple. For real lengths $b,c>0$ and $a\ge 0$ satisfying $\mathrm{\mid }b-c\mathrm{\mid }\le a\le b+c$, define ${\theta }_E(a;b,c):=\arccos \text{\hspace{0.17em}⁣}\left(\frac{b^2+c^2-a^2}{2bc}\right)\mathrm{.}$ This is a number in $[0,\pi ]$ (Principal inverse sine and inverse cosine): the two assumed length inequalities give $(b-c{)}^2\le a^2\le (b+c{)}^2$, hence $-2bc\le b^2+c^2-a^2\le 2bc$. For spherical adjacent lengths $b,c\in (0,\pi )$ and an opposite length $a\ge 0$, put $z_S(a;b,c):=\frac{\cos a-\cos b\cos c}{\sin b\sin c}\mathrm{.}$ The denominator is positive (Pi is the first positive zero of sine). Whenever $z_S(a;b,c)\in [-1,1]$, define the spherical comparison angle by ${\theta }_S(a;b,c):=\arccos z_S(a;b,c)\in [0,\pi ]$. No spherical angle is assigned by this definition when that validity condition fails. In either model the notation chooses the unique angle in $[0,\pi ]$ having the indicated cosine.

(2) Comparison angle of a metric hinge. In a metric space $(X,d)$ (Metric space: $d(x,y)=0$ iff $x=y$, symmetry, and the triangle inequality; pseudometric and ultrametric), let $p,x,y\in X$ with $x\mathrm{\ne }p$ and $y\mathrm{\ne }p$; $x=y$ is allowed. The Euclidean comparison angle at $p$ is ${\tilde{\mathrm{\angle }}}_p(x,y):={\theta }_E(d(x,y);d(p,x),d(p,y))\mathrm{.}$ Its length triple meets (1): the metric triangle inequality gives $d(x,y)\le d(p,x)+d(p,y)$ and, applied with each of $x,y$ as the middle point, $\mathrm{\mid }d(p,x)-d(p,y)\mathrm{\mid }\le d(x,y)$.

(3) Alexandrov upper angle. Let $c:[0,L]\to X$ and $c^{\mathrm{\prime }}:[0,L^{\mathrm{\prime }}]\to X$ be unit-speed geodesic segments with $L,L^{\mathrm{\prime }}>0$ and $c(0)=c^{\mathrm{\prime }}(0)$ (Geodesics and geodesic metric spaces). For $\epsilon >0$ set $H(\epsilon ):=\sup \{{\tilde{\mathrm{\angle }}}_{c(0)}(c(s),c^{\mathrm{\prime }}(t)):0<s\le \min (\epsilon ,L),0<t\le \min (\epsilon ,L^{\mathrm{\prime }})\},$ and define their Alexandrov upper angle by $\mathrm{\angle }(c,c^{\mathrm{\prime }}):={\inf }_{\epsilon >0}H(\epsilon )\mathrm{.}$ Each set in the supremum is nonempty and contained in $[0,\pi ]$, so its supremum exists and also lies in $[0,\pi ]$ (Upper bound, least upper bound, and strict upper bound, The Cauchy-sequence reals have the least-upper-bound property). The set of these suprema is likewise nonempty and bounded; its infimum exists by applying the same least-upper-bound property to its negatives. Thus $\mathrm{\angle }(c,c^{\mathrm{\prime }})\in [0,\pi ]$ is well defined. This is the two-variable upper-limit convention, without asserting existence of an ordinary limit, monotonicity of the comparison angles in $s,t$, invariance under reparametrization, or a CAT inequality.

(4) Triangle-angle conventions. In a geodesic triangle with distinct vertices and chosen sides (Comparison triangles, the CAT(0) and CAT(1) inequalities, local CAT, local geodesics and round circles), the angle at a vertex means (3) for the unit-speed parametrizations of the two sides starting there. For a model triangle in ${\mathbb{M}}_{\kappa }^2$, $\kappa \in \{0,1\}$, with distinct vertices, the model angle at a vertex is defined by (1) from its two adjacent side lengths and opposite side length. In the spherical case the side lengths must be in $(0,\pi )$ and the validity condition in (1) must hold. Model angles are therefore side-length comparison angles; the identification with geometric angles, and any further relations among upper angles, require their own arguments.

A comparison angle in (2) is a quantity determined by the three metric distances, while (3) uses arbitrarily short initial portions of the two geodesics. The definition supplies these quantities and their domains; it does not by itself prove the upper-angle triangle inequality or the angle comparison consequences of curvature bounds.