Recalling Ivory's theorem

Abstract

Ivory's Theorem states that in each curvilinear quadrangle of a confocal net of conics the two diagonals have the same lengths. This theorem is valid not only in the Euclidean plane, but also in planar hyperbolic, spherical and pseudo-Euclidean (or Minkowski) geometry, and similar statements hold in all dimensions. Recent publications on this theorem and its generalizations on surfaces are the reason to focus again on this topic and to show a few algebraic consequences

The Geometry of Billiards in Ellipses and their Poncelet Grids

“…Remark 3.9. The Theorems 3.5 and 3.6 as well as the constancy of the length w according to Lemma 3.8 are also valid in spherical geometry (note [23]) and in hyperbolic geometry. On the sphere (see Figure 7), the caustic consists of a pair of opposite components, and for N -periodic billiards the confocal spherical ellipse e (j) coincides with e (N −2−j) w.r.t.…”

“…Then we obtain a curvilinear Ivory quadrangle P 1 T 1 T P 1 with diagonals of equal lengths. On the other hand, the lines [P 1 , T ] and [P 1 , T 1 ] must contact the same confocal conic (see, e.g., [7, p. 153] or [23,Lemma 1]). Hence, the billiard through the point P 1 ∈ e with caustic c contains one side on the line [P 1 , T ].…”

The geometry of billiards in ellipses and their poncelet grids

“…Remark 3.9. The Theorems 3.5 and 3.6 as well as the constancy of the length w according to Lemma 3.8 are also valid in spherical geometry (note [26]) and in hyperbolic geometry. On the sphere (see Fig.…”

“…. Note that there are j sides between those which intersect at S (j) i , and 'in the middle' of these j sides there is for 3 An extended version of this theorem in [26] addresses the symmetry between the ranges Rc and Rt. This generalizes the statement that in the case of confocal conics c and c 1 the quadrilateral A 1 A 2 B 1 B 2 has an incircle d (Figs.…”

“…Then we obtain a curvilinear Ivory quadrangle P 1 T 1 T P 1 with diagonals of equal lengths. On the other hand, the lines [P 1 , T ] and [P 1 , T 1 ] must contact the same confocal conic (see, e.g., [7, p. 153] or [26,Lemma 1]). Hence, the billiard through the point P 1 ∈ e with caustic c contains one side on the line [P 1 , T ].…”