Mathematics Problem Archive
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_{501}
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_{502}
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_{503}
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_{701}
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_{702}
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_{703}
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_{802}
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_{803}
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 notesLet $\gamma$ be a smooth closed strictly convex plane curve, parametrized by arc length $s$, and let $d>0$. Form a sphere-like surface by gluing the t...
Open Problems on Billiards and Geometric Optics
v1.3 research notesLet $\gamma$ be a smooth closed strictly convex plane curve. The outer billiard map $T$ sends a point $A$ near $\gamma$ to the point $T(A)$ for which ...
Open Problems on Billiards and Geometric Optics
v1.3 research notesLet $\gamma$ be a smooth closed strictly convex curve and $\delta\in(0,\pi/2)$. Say that $\gamma$ has the $\delta$-Gutkin property if the curve of inc...