The angular path metric, the Euclidean cone and spherical joins

(1) Angular path metric. Let $L$ be a set with an extended metric $d_{\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h}}:L\times L\to [0,\mathrm{\infty }]$, symmetric, vanishing exactly on the diagonal and satisfying the triangle inequality in the extended reals. For the angular link ${\mathrm{Lk}}_X(F)$ of a face of a finite spherical complex (Spherical Gram simplices and angular links of Euclidean faces, Gram realisations, radial normalisation, finite spherical complexes and link Gram formulas) the metric $d_{\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h}}$ is the componentwise intrinsic path distance: the infimum of lengths of finite chains of directions inside a common cell, with $d_{\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h}}(x,y)=+\mathrm{\infty }$ for $x,y$ in different components. The value $+\mathrm{\infty }$ is auxiliary notation only and is never passed to the published definition of a metric space (Metric space: $d(x,y)=0$ iff $x=y$, symmetry, and the triangle inequality; pseudometric and ultrametric).

(2) Truncated angular metric. With the convention $\min \{\pi ,\mathrm{\infty }\}:=\pi$, put $d_{\pi }(x,y):=\min \{\pi ,d_{\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h}}(x,y)\}$. This finite-valued function on $L\times L$ is the angular metric; it has values in $[0,\pi ]$ and its metric axioms are proved in The cone and join metrics and the local product chart of a polyhedral gluing ↗ (Sine and cosine defined by their real power series, Pi is the first positive zero of sine, Principal inverse sine and inverse cosine).

(3) The Euclidean cone. The Euclidean cone on the angular link $L$ is the set $C(L):=\{o\}\bigsqcup ((0,\mathrm{\infty })\times L),$ where $o$ is the apex, with $d_C(o,o):=0,d_C(o,(r,x)):=r,d_C((r,x),(s,y){)}^2:=r^2+s^2-2rs\cos d_{\pi }(x,y)\mathrm{.}$ In particular $C(\mathrm{\varnothing })=\{o\}$ is a one-point space, not the empty space; and if $d_{\pi }(x,y)=\pi$ — which happens in particular when $x,y$ lie in different components of $L$ — then $d_C((r,x),(s,y))=r+s$, the length of the path through the apex. The infinite value of $d_{\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h}}$ is never an ordinary metric value, and the cone receives only $d_{\pi }$.

(4) Angular CAT(1) convention. A link $L$ is $D_{\pi }$-geodesic if every pair of points at distance $<\pi$ is joined by a minimizing segment. Angular CAT(1) statements about a link concern only triangles of perimeter $<2\pi$ and their comparison in the unit sphere $S^2$ (Euclidean spheres and closed balls as subspaces of ${\mathbb{R}}^n$); every such test lies in one intrinsic component and agrees with the componentwise intrinsic test of $d_{\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h}}$ whenever no side equals $\pi$, while a side of length $\pi$ is realised in the model sphere by an antipodal pair; this is proved in The cone and join metrics and the local product chart of a polyhedral gluing ↗.

(5) Spherical join. Let $L_1,L_2$ be angular links with metrics $d_{\pi }^1,d_{\pi }^2$. The spherical join $L_1\ast L_2$ is the quotient of $L_1\times L_2\times [0,\pi \mathrm{/}2]$ by the identifications $(x,y,0)\sim (x,y^{\mathrm{\prime }},0)$ and $(x,y,\pi \mathrm{/}2)\sim (x^{\mathrm{\prime }},y,\pi \mathrm{/}2)$; write $(\cos \theta )x+(\sin \theta )y$ for the class of $(x,y,\theta )$. The distance of $x=(\cos \theta )x_1+(\sin \theta )x_2$ and $x^{\mathrm{\prime }}=(\cos {\theta }^{\mathrm{\prime }})x_1^{\mathrm{\prime }}+(\sin {\theta }^{\mathrm{\prime }})x_2^{\mathrm{\prime }}$ is the unique number in $[0,\pi ]$ with $\cos d(x,x^{\mathrm{\prime }})=\cos \theta \cos {\theta }^{\mathrm{\prime }}\cos d_{\pi }^1(x_1,x_1^{\mathrm{\prime }})+\sin \theta \sin {\theta }^{\mathrm{\prime }}\cos d_{\pi }^2(x_2,x_2^{\mathrm{\prime }})\mathrm{.}$ Conventions: $L\ast \mathrm{\varnothing }:=L$, $\mathrm{\varnothing }\ast L:=L$ and $\mathrm{\varnothing }\ast \mathrm{\varnothing }:=\mathrm{\varnothing }$; these are consistent with the product-cone isometry $C(L_1)\times C(L_2)\cong C(L_1\ast L_2)$, under which $L_1\ast L_2$ is the unit link of the product cone. Quotient descent to the identified endpoints, the triangle inequality, associativity, the face metrics and the isometry with the unit link are conclusions of The cone and join metrics and the local product chart of a polyhedral gluing ↗; this item asserts only the construction, the formula and the conventions.

• What is construction and what is theorem. Clauses (1)–(5) fix notation and conventions only: the truncated metric, the cone with its apex and the join with its empty conventions are defined here, while the metric axioms of $d_{\pi }$, the cone metric and its geodesics, the quotient descent and triangle inequality of the join, associativity and the product-cone isometry are all proved in The cone and join metrics and the local product chart of a polyhedral gluing ↗. This item is the justification target of that theorem and asserts none of its conclusions.
• Why the truncation. The Euclidean cone formula requires a finite angular distance bounded by $\pi$: the value $+\mathrm{\infty }$ of $d_{\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h}}$ across distinct components of a link is replaced by $\pi$, and the geodesics between the corresponding rays then pass through the apex. The link of a finite spherical complex is the case in which $d_{\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h}}$ is the componentwise intrinsic path distance of Gram realisations, radial normalisation, finite spherical complexes and link Gram formulas(vi).
• The square-sum convention. By the product-cone isometry, $C(L_1)\times C(L_2)$ carries the square-sum product metric $d^2=d_1^2+d_2^2$ and $L_1\ast L_2$ is its unit link; the displayed cosine formula is the law of cosines of that cone, not an independent claim of the definition.