Particle-family geometry

The displayed graph also explains the absence of an additional branch event. For $s\le 0$, $h_{\mathrm{lin}}\ge s+k_{\mu }$. For $s>0$, the new triangles are bounded by the vertices $B_i,C_i$; lem:bounds separates them from the cap.

For reference, the induced areas of compact faces are

\operatorname{Area}(V_i)&=
 \begin{cases}
 S_\alpha r^2/6,&s\le0,\\
 \dfrac{S_\alpha}{3}(r^2/2-BCs^2),&s>0,
 \end{cases}\nonumber\\
 \operatorname{Area}(U_i^\pm)&=\frac25S_\alpha BCs^2\quad(s>0),&
 \operatorname{Area}(Q_i)&=\frac12S_\alpha(16-r^2).
 \label{eq:areas}

These follow by multiplying projected areas by the constant spacelike area factors $1/3$, $4/5$, and $1$, respectively. The outgoing star lies strictly inside the incoming core triangle, so the displayed active areas are positive.