Ptolemy diagrams in a finite polygon $P$, see [@HJR:torsionpairsA thm. A]. Again there is an analogue for $\cC( \cZ )$, and again, convergence plays a crucial role. Hence the following definition.

\[Conditions PC1 and PC2\] \[def:PC\] We can impose the following conditions on a set ${\mathscr{X}}$ of diagonals of ${\mathscr{Z}}$, see Figure \[fig:PC\].

(0,0) circle(0.5cm);

at (5:0.60)[$x_0^i$]{}; at (60:0.60)[$p$]{}; at (115:0.60)[$x'^i_0$]{};

at (155:0.60)[$x_1^i$]{}; at (210:0.60)[$q$]{}; at (265:0.60)[$x'^i_1$]{};

(10:0.60) arc (10:55:0.60); (65:0.60) arc (65:110:0.60);

(160:0.60) arc (160:205:0.60); (215:0.60) arc (215:260:0.60);

(210:0.5) node\[fill=white,circle,inner sep=0.065cm\] circle (0.02cm); (60:0.5) node\[fill=white,circle,inner sep=0.065cm\] circle (0.02cm);

(5:0.48) – (5:0.52); (155:0.48) – (155:0.52); (5:0.5) .. controls (5:0.3) and (155:0.3) .. (155:0.5);

(25:0.48) – (25:0.52); (175:0.48) – (175:0.52); (25:0.5) .. controls (25:0.3) and (175:0.3) .. (175:0.5);

(40:0.48) – (40:0.52); (190:0.48) – (190:0.52); (40:0.5) .. controls (40:0.3) and (190:0.3) .. (190:0.5);

(50:0.48) – (50:0.52); (200:0.48) – (200:0.52); (50:0.5) .. controls (50:0.3) and (200:0.3) .. (200:0.5);

(115:0.48) – (115:0.52); (265:0.48) – (265:0.52); (115:0.5) .. controls (115:0.3) and (265:0.3) .. (265:0.5);

(95:0.48) – (95:0.52); (245:0.48) – (245:0.52); (95:0.5) .. controls (95:0.3) and (245:0.3) .. (245:0.5);

(80:0.48) – (80:0.52); (230:0.48) – (230:0.52); (80:0.5) .. controls (80:0.3) and (230:0.3) .. (230:0.5);

(70:0.48) – (70:0.52); (220:0.48) – (220:0.52); (70:0.5) .. controls (70:0.3) and (220:0.3) .. 