Mathematics Problem Archive
Elliptic-billiard invariant k_{304}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{307}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{401}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{402}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{403,a}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{403,b}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{404}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{405}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{406,a}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{406,b}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{407}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{601}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{602}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{603}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{605,a}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{606}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{607}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{608}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{609}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{610}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{804,a}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{804,b}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{805}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{806,a}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{806,b}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{807}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{808}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{809}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{810}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{811}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{812,a}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{812,b}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{813}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{814}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{815}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{816}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{817}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{818}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{903,a}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{904,a}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{905}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{906}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{907,a}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{907,b}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{908,a}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Elliptic-billiard invariant k_{908,b}
v1.3 research notesFix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...
Open Problems on Billiards and Geometric Optics
v1.3 research notesFor a Birkhoff billiard inside a closed smooth strictly convex hypersurface $S\subset\mathbb{R}^d$, let $T$ act on the space $\mathbb{A}$ of oriented ...
Open Problems on Billiards and Geometric Optics
v1.3 research notesLet $\gamma$ be a smooth convex plane billiard table symmetric about an axis $l$, and let $C$ be a convex caustic. Must $C$ be symmetric about $l$? Pr...
Open Problems on Billiards and Geometric Optics
v1.3 research notesSuppose a bounded strictly convex planar billiard has two nested closed caustics such that the smaller caustic is itself a caustic for the billiard in...
Open Problems on Billiards and Geometric Optics
v1.3 research notesLet $C\subset\mathbb{R}^2$ be a curve and translate it through an $\varepsilon$-square lattice, recording a click whenever it meets a lattice point. C...