Projective Billiards: Geometry and Integrability

• Projective billiards are dynamical systems defined by reflection laws based on projective geometry, using harmonic conditions and involutive mappings.
• They generalize classical, spherical, and pseudo-Euclidean billiards by encoding reflections via transverse line fields and demonstrating rigidity through conic and quadric boundaries.
• Integrability in these systems emerges from rational invariants and duality principles, linking projective dynamics with Hamiltonian, mechanical, and Finsler billiard models.

Searching arXiv for recent and foundational papers on projective billiards and related integrable/projective-mechanical billiard systems. arXiv search query: "projective billiards" Projective billiards are billiard systems whose reflection law is formulated in projective-geometric rather than purely metric terms. In the most classical formulation, due to Tabachnikov, a projective billiard is a domain with piecewise smooth boundary equipped with a field of transverse lines, and the reflection at a boundary point is defined by a projective involution, equivalently by a harmonicity condition on four lines through the impact point (Fierobe, 2020). This framework contains ordinary Euclidean billiards as a special case, because taking the transverse line field to be the normal line recovers specular reflection (Fierobe, 2020, Fierobe, 2023). It also encompasses billiards induced from space forms and metrics projectively equivalent to Euclidean geometry, and it interacts with several adjacent theories: caustics in higher-dimensional projective and pseudo-Euclidean billiards (Fierobe, 2023), rationally integrable dual and projective billiards (Glutsyuk, 2021, Glutsyuk, 2023), Hamiltonian and Minkowski-projective rigidity phenomena (Glutsyuk, 2024, Glutsyuk et al., 2024), and a distinct but related projective-dynamical approach to mechanical billiards such as Kepler, two-center, and Lagrange billiards (Zhao, 2020, Takeuchi et al., 2022, Pinzari et al., 24 Apr 2025).

1. Definitions and reflection laws

In Tabachnikov’s formulation, a projective billiard in $\mathbb{R}^n$ is a domain $\Omega\subset\mathbb{R}^n$ with piecewise smooth boundary, equipped with a field of transverse lines $L$ along $\partial\Omega$ (Fierobe, 2023). At a smooth point $p\in\partial\Omega$, the line $L_p$ passes through $p$ and is transverse to $T_p\partial\Omega$ (Fierobe, 2023). The dynamics acts on oriented lines inside $\Omega$: the ray travels as a straight line until it hits $\partial\Omega$, then is reflected according to a projective law determined by $\Omega\subset\mathbb{R}^n$0 (Fierobe, 2023).

The reflection law admits two equivalent descriptions. In linear terms, the reflected velocity $\Omega\subset\mathbb{R}^n$1 is obtained from the incoming velocity $\Omega\subset\mathbb{R}^n$2 by the unique nontrivial linear involution

$\Omega\subset\mathbb{R}^n$3

that preserves the tangent hyperplane $\Omega\subset\mathbb{R}^n$4 and the line $\Omega\subset\mathbb{R}^n$5 directed by $\Omega\subset\mathbb{R}^n$6 (Fierobe, 2023). In projective terms, if $\Omega\subset\mathbb{R}^n$7 is the 2-plane spanned by the incoming line and $\Omega\subset\mathbb{R}^n$8, and $\Omega\subset\mathbb{R}^n$9, then the quadruple $L$0 of lines through $L$1 is harmonic (Fierobe, 2023). In dimension two this becomes the basic planar definition: two lines through $L$2 are symmetric if the quadruple $L$3 is harmonic (Fierobe, 2020).

In coordinates on the pencil of lines through a point, the projective symmetry is represented by a Möbius involution. If $L$4 are two distinct lines through $L$5, with azimuths $L$6, then the symmetry around $L$7 is

$L$8

Two lines are symmetric with respect to $L$9 precisely when their azimuths are related by this involution (Fierobe, 2020). This formula is central in local analytic treatments of projective billiards, because it converts harmonicity into an explicit rational transformation of line directions.

Ordinary Euclidean billiards are recovered by taking $\partial\Omega$0 to be the normal line at $\partial\Omega$1; then the projective harmonic reflection coincides with specular reflection (Fierobe, 2020). More generally, projective billiards unify Euclidean, pseudo-Euclidean, and Riemannian billiards whose geodesics are straight lines, because in each case the reflection law is encoded by a distinguished normal line field (Fierobe, 2023).

2. Geometric models and projective origins

Projective billiards arise naturally from projective models of space forms. Space forms such as the sphere and the hyperbolic plane project to projective billiards in $\partial\Omega$2: geodesics become straight lines and normals become projective line fields (Fierobe, 2020). This produces projective images of spherical and hyperbolic billiards and explains why projective billiards are regarded as a common generalization of billiards on surfaces of constant curvature (Glutsyuk, 2021, Glutsyuk, 14 Sep 2025).

A fundamental planar example is the right-spherical billiard. Let $\partial\Omega$3 be three non-collinear points. On side $\partial\Omega$4, attach the line $\partial\Omega$5 at each point $\partial\Omega$6; on $\partial\Omega$7, attach $\partial\Omega$8; on $\partial\Omega$9, attach $p\in\partial\Omega$0 (Fierobe, 2020). This defines a projective billiard structure on the triangle $p\in\partial\Omega$1, called the right-spherical billiard (Fierobe, 2020). Geometrically, it is the projective image of a spherical billiard on a right triangle in $p\in\partial\Omega$2 bounded by three orthogonal great circles (Fierobe, 2020). It is 3-reflective: there exists an open 2-dimensional family of triangular orbits (Fierobe, 2020).

A different projective origin appears in projective dynamics of mechanical systems. Central projection between affine planes and spheres can send unparametrized trajectories of one natural mechanical system to those of another, after time reparametrization (Zhao, 2020). In the Kepler and two-center settings, this produces billiard correspondences between planar, spherical, and hyperbolic mechanical billiards, and the corresponding energies become extra first integrals (Zhao, 2020, Takeuchi et al., 2022, Pinzari et al., 24 Apr 2025). This suggests a broader usage of the term “projective billiards,” namely billiard systems whose integrability is rooted in projective correspondences between different ambient geometries and mechanical systems (Pinzari et al., 24 Apr 2025).

3. Reflective configurations and the projective Ivrii problem

In Euclidean billiards, Ivrii’s conjecture predicts that the set of periodic trajectories has measure zero, hence no billiard should be $p\in\partial\Omega$3-reflective, meaning that the set of $p\in\partial\Omega$4-periodic orbits has non-empty interior in phase space (Fierobe, 2020). Projective billiards admit an analogous notion: a projective billiard is $p\in\partial\Omega$5-reflective if the set of its $p\in\partial\Omega$6-periodic orbits has non-empty interior in the appropriate phase space (Fierobe, 2020). The projective analogue of Ivrii’s conjecture asks for a classification, or nonexistence, of such $p\in\partial\Omega$7-reflective projective billiards (Fierobe, 2020).

For triangular orbits, the planar classification is complete. The only 3-reflective planar projective billiard with piecewise smooth boundary is the right-spherical billiard (Fierobe, 2020). In analytic terms, the local 3-reflective real or complex analytic planar projective billiards are exactly the real or complex right-spherical billiards (Fierobe, 2020). In higher dimensions there are no 3-reflective local analytic real or complex projective billiards, and in the smooth real case there are no 3-pseudo-reflective local $p\in\partial\Omega$8-smooth projective billiards in dimension $p\in\partial\Omega$9 (Fierobe, 2020). Thus the only open family of triangular trajectories in the projective setting comes from the spherical model projected to the plane (Fierobe, 2020).

The proof uses a local phase-space formalism in the bundle $L_p$0, line-framed curves and hypersurfaces, and a singular analytic distribution called Birkhoff’s distribution (Fierobe, 2020). Integral 2-surfaces of this distribution correspond to 2-parameter families of triangular orbits (Fierobe, 2020). In the planar analytic case, one first shows that if the classical boundaries are lines, 3-reflectivity forces the right-spherical structure; then one excludes non-linear boundaries by an integrability argument for the distribution and a degeneration analysis near tangencies (Fierobe, 2020).

A complementary construction shows that projective billiards can violate Ivrii-type expectations very strongly. Explicit polygonal examples exhibit 2-parameter families of $L_p$1-periodic orbits, with $L_p$2 equal to $L_p$3 or any even number greater than $L_p$4 (Fierobe, 2020). These examples include the right-spherical triangle and centrally-projective polygons endowed with line fields defined by a fixed center $L_p$5 (Fierobe, 2020). In this polygonal class, the set of periodic trajectories has positive measure in phase space, showing that Ivrii’s measure-zero phenomenon is not stable under passage from Euclidean to projective reflection laws (Fierobe, 2020).

4. Integrability, duality, and conic geometry

A major branch of the theory concerns rationally integrable projective billiards. A planar projective billiard is rationally $L_p$6-homogeneously integrable if its flow admits a non-constant first integral that is a rational function of the velocity, homogeneous of degree $L_p$7, with coefficients depending on the position (Glutsyuk, 2021). Via orthogonal polarity, projective billiards on a curve $L_p$8 correspond to dual billiards on the dual curve $L_p$9, and rational $p$0-homogeneous integrability of the projective billiard is equivalent to rational integrability of the dual billiard (Glutsyuk, 2021).

In the $p$1-smooth connected nonlinear planar case, rational $p$2-homogeneous integrability is rigid: the boundary curve is a conic (Glutsyuk, 2021, Glutsyuk, 2023). The full classification on a conic contains two types. The first type is induced by a pencil of conics; in this case the integral is quadratic (Glutsyuk, 2021). The second type consists of exotic rationally integrable dual and projective billiards on a conic, including two infinite series and several isolated examples, with arbitrarily high even minimal degree (Glutsyuk, 2021). This establishes a projective analogue of the Birkhoff conjecture under the strong assumption of rational integrability: in the closed strictly convex case, the only rationally $p$3-homogeneously integrable projective billiards are space form billiards in projective disguise, with boundaries and invariant curves coming from a pencil of conics (Glutsyuk, 2021).

The piecewise smooth classification is subtler. For piecewise $p$4-smooth rationally $p$5-homogeneously integrable projective billiards whose boundary contains a nonlinear arc, the nonlinear pieces are constrained by the same conic rigidity, but line segments can also occur (Glutsyuk, 2023). Unexpectedly, a projective billiard associated to a dual pencil of conics may have minimal integral degree $p$6, $p$7, or $p$8 (Glutsyuk, 2023). The proof proceeds through dual multibilliards, where piecewise smooth projective billiards become piecewise smooth dual systems with several local involutions acting on tangent lines (Glutsyuk, 2023).

Complex algebraic caustics provide another route to rigidity. For ordinary billiards on a real planar curve $p$9, a complex algebraic curve $T_p\partial\Omega$0 is a complex caustic if every complex line through a boundary point $T_p\partial\Omega$1 tangent to $T_p\partial\Omega$2 is mapped by the complexified reflection at $T_p\partial\Omega$3 to another tangent line to $T_p\partial\Omega$4 (Glutsyuk, 14 Sep 2025). If a nonlinear $T_p\partial\Omega$5-smooth connected embedded planar curve has a complex caustic, then the boundary is a conic and the caustic is either a single confocal conic or a finite union of such confocal conics (Glutsyuk, 14 Sep 2025). The same holds on the sphere and in the hyperbolic plane, with the appropriate notion of confocality, except for the absolute in the spherical and hyperbolic cases (Glutsyuk, 14 Sep 2025). For projective billiards, the existence of at least two different complex caustics forces the boundary to be a conic and the projective billiard to be rationally $T_p\partial\Omega$6-homogeneously integrable (Glutsyuk, 14 Sep 2025). This suggests that complex-algebraic caustics strongly constrain projective billiards toward the conic and dual-pencil paradigm.

5. Caustics and higher-dimensional rigidity

In dimensions $T_p\partial\Omega$7, the theory of caustics becomes especially rigid. A caustic is a smooth hypersurface $T_p\partial\Omega$8 such that if a billiard trajectory is tangent to $T_p\partial\Omega$9 once, then it remains tangent to $\Omega$0 after all subsequent reflections (Fierobe, 2023). For projective billiards, one works locally with a line-framed hypersurface $\Omega$1 and a pair of local caustic pieces $\Omega$2: oriented lines tangent to $\Omega$3, reflected by $\Omega$4, remain tangent to $\Omega$5 (Fierobe, 2023).

The main higher-dimensional result is Theorem 17 of Fierobe’s paper. Let $\Omega$6, let $\Omega$7 be a $\Omega$8-smooth embedded hypersurface with non-degenerate second fundamental form, and let $\Omega$9 be a smooth field of transverse lines along $\partial\Omega$0. If $\partial\Omega$1 is a piece of caustic for the projective billiard $\partial\Omega$2, then the following are equivalent: $\partial\Omega$3 and $\partial\Omega$4 are open subsets of quadrics, and $\partial\Omega$5 is $\partial\Omega$6-symmetric (Fierobe, 2023). Moreover, if either condition holds, then $\partial\Omega$7 and $\partial\Omega$8 lie in the same quadric (Fierobe, 2023). Thus under the natural symmetry condition, only quadrics can appear as pseudo-caustics in projective billiards (Fierobe, 2023).

This theorem specializes to several metric settings. For Riemannian billiards in metrics projectively equivalent to the Euclidean metric, a quadratic caustic exists if and only if the domain is $\partial\Omega$9-symmetric (Fierobe, 2023). For pseudo-Euclidean billiards, every space-time hypersurface is automatically $\Omega\subset\mathbb{R}^n$00-symmetric, so if a pseudo-Euclidean billiard has a caustic, then both the boundary and the caustic are quadrics belonging to the same pseudo-confocal family (Fierobe, 2023). The paper leaves open whether projective billiards not satisfying $\Omega\subset\mathbb{R}^n$01-symmetry can have non-quadratic caustics in higher dimensions (Fierobe, 2023).

A plausible implication is that higher-dimensional projective billiards behave analogously to Euclidean higher-dimensional billiards: quadrics and their confocal or pseudo-confocal families occupy a distinguished and possibly exhaustive place among billiard tables with robust caustic structures.

6. Projective dynamics and mechanical billiards

A separate but increasingly important usage of “projective billiards” comes from projective dynamics of mechanical systems. In this setting the free motion is not along straight lines but along trajectories of a natural Hamiltonian system, and projective maps relate one mechanical system to another while preserving unparametrized orbits (Zhao, 2020).

A prominent example is the projective explanation of integrable Kepler and Boltzmann billiards. Central projection between a plane and a hemisphere maps the planar Kepler–Coulomb problem to the spherical Kepler–Coulomb problem, after time reparametrization (Zhao, 2020). For a planar Kepler billiard with a line wall, the extra first integral arises from the spherical energy of the corresponding system; in normalized coordinates it takes the form

$\Omega\subset\mathbb{R}^n$02

where $\Omega\subset\mathbb{R}^n$03 is the angular momentum, $\Omega\subset\mathbb{R}^n$04 is a component of the Laplace–Runge–Lenz vector, and $\Omega\subset\mathbb{R}^n$05 is the signed distance from the Kepler center to the wall (Zhao, 2020). This yields a projective-dynamical explanation of the integrability of the Boltzmann billiard model (Zhao, 2020).

The same approach extends to “Projective Integrable Mechanical Billiards,” where the underlying mechanical system is the Lagrange problem: the superposition of two Kepler problems and a Hooke problem, with the Hooke center at the midpoint of the Kepler centers (Takeuchi et al., 2022). Takeuchi and Zhao show that in the plane, on the sphere, and in the hyperbolic plane, mechanical billiards with the Lagrange problem as free dynamics and with any combinations of confocal conic sections with foci at the Kepler centers as reflection wall are integrable (Takeuchi et al., 2022). This covers free, Hooke, Kepler, two-center, and Hooke–Kepler billiards as subcases (Takeuchi et al., 2022). The projective method here is not Tabachnikov’s harmonic reflection law on straight-line billiards, but a projective correspondence between mechanical systems transporting trajectories, reflection laws, and first integrals (Takeuchi et al., 2022).

This line of work culminates, in the provided corpus, in the 2025 paper connecting three-body secular theory and Kepler billiards. There a particular first integral

$\Omega\subset\mathbb{R}^n$06

appears both in partially averaged secular three-body dynamics and in integrable Kepler billiards (Pinzari et al., 24 Apr 2025). The common origin is the projective dynamics of the two-center problem: central projection links Euclidean and spherical two-center or Lagrange systems, and the spherical energy becomes a nontrivial first integral of the Euclidean system (Pinzari et al., 24 Apr 2025). The same projective invariant $\Omega\subset\mathbb{R}^n$07 is preserved by reflections at confocal conic walls, so it governs both secular three-body motion and Kepler billiards (Pinzari et al., 24 Apr 2025). The paper then uses this projective picture to define integrable billiard systems on surfaces of constant curvature, including secular billiards based on partially averaged Hamiltonians (Pinzari et al., 24 Apr 2025). In this broader sense, projective billiards are systems whose integrability is rooted in projective geometry, central projection, and projective invariants rather than solely in local reflection symmetry (Pinzari et al., 24 Apr 2025).

7. Hamiltonian, Minkowski, and Finsler projective billiards

Projective reflection laws also appear in symplectic and Finsler generalizations. For bounded strictly convex bodies $\Omega\subset\mathbb{R}^n$08 and $\Omega\subset\mathbb{R}^n$09, the boundary $\Omega\subset\mathbb{R}^n$10 carries a characteristic line field defined by the standard symplectic form (Glutsyuk, 2024). Its projections to $\Omega\subset\mathbb{R}^n$11 give the $\Omega\subset\mathbb{R}^n$12-billiard in $\Omega\subset\mathbb{R}^n$13, which is a Minkowski/Finsler billiard when $\Omega\subset\mathbb{R}^n$14 is centrally symmetric (Glutsyuk, 2024). The natural question is when the corresponding reflection law is projective in Tabachnikov’s sense.

The answer is rigid. If the $\Omega\subset\mathbb{R}^n$15-billiard reflection law is projective, then $\Omega\subset\mathbb{R}^n$16 is an ellipsoid (Glutsyuk, 2024). Equivalently, all the $\Omega\subset\mathbb{R}^n$17-billiards are simultaneously affine equivalent to Euclidean billiards (Glutsyuk, 2024). This leads to corollaries for Finsler billiards: if a Minkowski Finsler billiard has projective reflection law, then its unit sphere is an ellipsoid, so the metric is Euclidean up to affine transformation (Glutsyuk, 2024). If a simply connected projectively flat Finsler structure has projective reflection law, then, up to scaling, it is isometric to a domain in Euclidean space, the unit sphere, or hyperbolic space (Glutsyuk, 2024).

A later note simplifies and strengthens this rigidity. If a billiard in a convex domain is simultaneously Minkowski and projective, then it is the standard Euclidean billiard in an appropriate Euclidean structure (Glutsyuk et al., 2024). The proof works in $\Omega\subset\mathbb{R}^n$18-smoothness and also yields semi-local and local versions: in the local setting, the hypersurfaces involved must be quadrics (Glutsyuk et al., 2024). This suggests that projective billiards, when compatible with Minkowski reflection laws, are rigidly forced back to the Euclidean case.

Another projective-Finsler direction comes from the Funk metric. The projective co-nomadic Finsler structure attached to a convex body is represented by the reverse Funk metric, and its symmetrization is the Hilbert metric (Faifman, 2020). The associated Funk billiards are projectively invariant: both the reflection law and the lengths of periodic orbits are independent of the choice of hyperplane at infinity (Faifman, 2020). In the ellipsoidal case, Funk billiards recover hyperbolic billiards in the Beltrami–Klein model (Faifman, 2020). Moreover, dual periodic orbits in $\Omega\subset\mathbb{R}^n$19 and $\Omega\subset\mathbb{R}^n$20 have equal Funk length (Faifman, 2020), extending Gutkin–Tabachnikov duality for Minkowski billiards. This constitutes yet another notion of projective billiards: billiard dynamics arising from projectively natural non-reversible Finsler metrics on convex domains (Faifman, 2020).

8. Relations to conics, quadrics, and open problems

Across the diverse meanings of projective billiards, a common geometric theme is the privileged role of conics and quadrics. In the planar smooth rationally integrable setting, the boundary must be a conic (Glutsyuk, 2021). In the piecewise smooth rationally integrable setting, nonlinear pieces are still conic arcs (Glutsyuk, 2023). In 4-reflective complex analytic billiards, the only nonlinear mirrors are confocal conics, up to the classified line configurations (Glutsyuk, 2014). In higher dimensions, under $\Omega\subset\mathbb{R}^n$21-symmetry, the only pseudo-caustics are quadrics (Fierobe, 2023). In Hamiltonian and Minkowski-projective settings, projective reflection laws force ellipsoids or quadrics (Glutsyuk, 2024, Glutsyuk et al., 2024). This recurring rigidity suggests that conics and quadrics are not merely examples but structural fixed points of projective billiard geometry.

Several open questions remain. In the projective Ivrii problem, the classification of $\Omega\subset\mathbb{R}^n$22-reflective projective billiards is known for $\Omega\subset\mathbb{R}^n$23 in the planar case, but higher periods remain largely open (Fierobe, 2020). Polygonal reflective examples exist for $\Omega\subset\mathbb{R}^n$24 and even $\Omega\subset\mathbb{R}^n$25, but the classification of smooth or analytic $\Omega\subset\mathbb{R}^n$26-reflective projective billiards beyond triangles is unresolved (Fierobe, 2020). In the higher-dimensional caustic problem, it is open whether projective billiards lacking $\Omega\subset\mathbb{R}^n$27-symmetry can admit non-quadratic caustics (Fierobe, 2023). In the Birkhoff-type rigidity direction, the removal of the rationality assumption from dual/projective billiard integrability remains a major challenge (Glutsyuk, 2021). The 2025 work on complex algebraic caustics raises a concrete problem: whether an algebraically integrable real billiard must possess a rational caustic whose complexification is a complex caustic (Glutsyuk, 14 Sep 2025). An affirmative answer would yield strong new results toward the Birkhoff conjecture in the algebraic category (Glutsyuk, 14 Sep 2025).

A plausible synthesis is that “projective billiards” is best regarded not as a single theory but as a family of closely related theories centered on projective invariance of reflection, orbit geometry, or integrability. In one branch, the reflection law itself is projective, defined by harmonic pencils and line fields (Fierobe, 2020, Fierobe, 2023). In another, projective duality and rational integrability organize the classification of caustics and conic boundaries (Glutsyuk, 2021, Glutsyuk, 2023, Glutsyuk, 14 Sep 2025). In a third, projective dynamics relates mechanical billiards across different geometries and explains common first integrals (Zhao, 2020, Takeuchi et al., 2022, Pinzari et al., 24 Apr 2025). In a fourth, projective Finsler and symplectic models expose rigidity of reflection laws and duality phenomena (Faifman, 2020, Glutsyuk, 2024, Glutsyuk et al., 2024). Taken together, these developments place projective billiards at a crossroads of dynamical systems, projective differential geometry, convexity, and integrable Hamiltonian mechanics.