Equivalence of the Apollonian and Its Inner Metric

Figure 2

The geodesic path 𝛾_𝑥𝑦 (w.r.t. the Apollonian inner metric 𝛼_𝐷) connecting 𝑥 and 𝑦 intersects 𝐵, and its modification 𝛾^′ from 𝑥₁ to 𝑥₂ along the circular part.