21 Matching across cuffs: positions and directions

The pants viewpoint suggests that orthogeodesic pieces should glue when their boundary data match. There are two levels of matching.

First consider a positive-length cuff $a>0$, with $m\geq1$ distinct endpoint positions $x_1,\ldots,x_m$ and $y_1,\ldots,y_m$ on the two sides. Choose compatible coordinates in $\mathbb R/a\mathbb Z$, reversing one induced boundary orientation as needed. For a prescribed cyclic matching $\sigma$, a twist $\tau$ must satisfy $$y_{\sigma(j)}-x_j\equiv\tau\pmod a \qquad(1\le j\le m).$$(21.1) One endpoint determines $\tau$ modulo $a$; the remaining $m-1$ equations require the cyclic spacings to agree. Integer winding data must also be retained if the curve is marked, rather than only considered modulo a twist. These congruences are necessary and sufficient for the prescribed position matching. They do not supply the direction data of the pieces.

Second, orient a chain by its direction of travel and measure all tangential components in the common cuff coordinate just chosen. A transverse unit-speed geodesic segment has a signed tangential velocity component $p\in(-1,1)$ at an endpoint. If $S$ is the sum of segment lengths, the first variation at a glued endpoint has the form $$\frac{\partial S}{\partial x}=p_{\mathrm{in}}-p_{\mathrm{out}}.$$(21.2) Here $p_{\mathrm{in}}$ is the terminal tangential component of the incoming segment and $p_{\mathrm{out}}$ the initial component of the outgoing segment. The pieces must approach and leave on the opposite sides of the cut. Equality of their tangential components then also matches the normal components, whose magnitudes are $\sqrt{1-p^2}$ and whose signs are fixed by this crossing condition. Stationarity therefore imposes full tangent matching as well as position matching. For orthogonal segments the tangential components vanish, so smooth gluing is automatic once the endpoints match. A generic closed geodesic does not satisfy this orthogonality condition along an arbitrarily chosen pants decomposition.

Equivalently, boundary data live in a phase space $(s,p)$: endpoint position and tangential component. The length of an arc is a generating function, with $$\mathop{}\!\mathrm dL=-p_s\,\mathop{}\!\mathrm ds+p_t\,\mathop{}\!\mathrm dt, \qquad\text{and cuff gluing acts by }(s,p)\longmapsto(s+\tau,p).$$ A nonperipheral periodic chain with these position and direction matches is smooth and locally geodesic, hence is its closed geodesic representative. Conversely, cutting a transverse closed geodesic gives such a chain. Tangent segments lying on a cuff require separate treatment and are not included in the transverse phase space. The orthospectrum records the special $p=0$ representatives. It is excellent input for gap identities and local geometry, but endpoint positions and directions are needed to reconstruct arbitrary glued curves.