Fig. 20 | Discrete & Computational Geometry

Fig. 20

From: Nets of Lines with the Combinatorics of the Square Grid and with Touching Inscribed Conics

Fig. 20

The two polygonal chains \(p_{i-1}, p_i,p_{i+1}\) and \(q_{j-1}, q_j, q_{j+1}\) are billiards that are inscribed in a conic \(\mathscr {D}\). Suppose that the two billiards have the same confocal caustic. Then, by the Graves–Chasles theorem, there exists a circle that is tangent to the four dotted lines. The centre of the circle is the intersection point of the tangent lines of \(\mathscr {D}\) at \(p_i\) and \(q_j\). These tangent lines generate a line grid with touching inscribed conics

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