Wave Tracing with Elliptical Cones

• Wave tracing with elliptical cones is a geometric framework that models wave propagation using ellipsoidal cross-sections to capture anisotropy and finite beam extent.
• It unifies variational principles like Fermat’s principle with a generalized Snell’s law, extending classical ray dynamics to handle complex refractive interfaces.
• The approach supports both discrete interface-driven algorithms and continuous ODE integration, enabling efficient simulation for rendering and electromagnetic applications.

Searching arXiv for papers on wave tracing with elliptical cones and related cone-structure optics. Wave tracing with elliptical cones denotes a family of geometrical and variational methods in which wave propagation is represented not by infinitesimal rays alone, but by cone-shaped propagation domains whose cross-sections are ellipses or whose local speed sets are ellipsoids. In recent work, the term spans at least three closely related settings: weakly-local wave-optical transport using region-to-region elliptical cone envelopes in rendering and electromagnetic simulation (Steinberg et al., 24 Aug 2025); variational tracing of wave trajectories across interfaces in media whose local propagation cones are Lorentz–Finsler and, in the quadratic case, ellipsoidal (Javaloyes et al., 30 Sep 2025); and ray dynamics in bounded cavities with elliptical-cone geometry, where propagation and reflection organize inertial-wave attractors (Favier et al., 2024). Across these settings, the common structure is that anisotropy, finite beam extent, or both are encoded by ellipsoidal data, yielding explicit transport, refraction, and reflection laws that generalize classical ray tracing and Snell’s law.

1. Conceptual scope and geometric setting

In the Lorentz–Finsler formulation, a cone structure $C$ on an $(n+1)$-dimensional manifold $Q$ assigns to each point $p\in Q$ a smooth, salient, strongly convex hypersurface $C_p\subset T_pQ\setminus\{0\}$ separating causal from non-causal directions (Javaloyes et al., 30 Sep 2025). When $Q=\mathbb{R}\times N$ is split into time and space and $\ker dt$ carries a time-dependent Finsler metric $F^t(x,\cdot)$, the Lorentz–Finsler function

$L(v) = (v^0)^2 - (F^t(x, v^i\partial_{x^i}))^2$

vanishes exactly on the cone $C=\{L=0, dt>0\}$, so the cone encodes the admissible instantaneous propagation velocities (Javaloyes et al., 30 Sep 2025). In the quadratic specialization relevant to elliptical cones, the spatial indicatrix at $(n+1)$0 is an ellipsoid

$(n+1)$1

with $(n+1)$2 positive-definite and associated Finsler norm $(n+1)$3 (Javaloyes et al., 30 Sep 2025).

A closely related spacetime construction appears in the anisotropic rheonomic Huygens framework, where the instantaneous speed vectors satisfy

$(n+1)$4

and the corresponding lightlike condition is

$(n+1)$5

This defines a smooth embedded cone hypersurface in $(n+1)$6 and provides a Lorentz–Finsler metric $(n+1)$7 and a quadratic Lagrangian

$(n+1)$8

whose null set is the cone (Javaloyes et al., 2020).

In wave-optical rendering, elliptical cones appear in a different but structurally analogous role. There, transport is lifted from point-to-point paths to region-to-region steps $(n+1)$9, and each propagating Gaussian-beam primitive occupies a bounded support that evolves into an elliptical cone (Steinberg et al., 24 Aug 2025). The beam is parameterized by a mean origin $Q$0, mean direction $Q$1, wavefront semi-axes $Q$2, and half-opening angles $Q$3. The frustum at distance $Q$4 is

$Q$5

with $Q$6, $Q$7, and $Q$8 (Steinberg et al., 24 Aug 2025). Here the cone is not the light cone of a Lorentz–Finsler structure; it is an envelope of significant field amplitude for weakly-local transport. This suggests that “wave tracing with elliptical cones” is best understood as a unifying geometric idiom rather than a single formalism.

2. Variational foundations: Fermat, Huygens, and null pregeodesics

The generalized Fermat principle for cone structures states that a physically realized trajectory is a critical point of the arrival-time functional when source, interface, and receiver are modeled as submanifolds or curves of arbitrary causal character (Javaloyes et al., 30 Sep 2025). In the stationary Finsler picture, a candidate path $Q$9 has travel time

$p\in Q$0

Equivalently, in spacetime one considers lightlike curves $p\in Q$1 satisfying $p\in Q$2, and the arrival time is the $p\in Q$3-coordinate of the endpoint (Javaloyes et al., 30 Sep 2025).

The variational derivation uses admissible variations $p\in Q$4 that preserve endpoint constraints and the single interface crossing. For each smooth leg $p\in Q$5, the first variation satisfies

$p\in Q$6

where $p\in Q$7 is the fundamental tensor of $p\in Q$8 and $p\in Q$9 is the Chern covariant derivative (Javaloyes et al., 30 Sep 2025). Integration by parts yields the pregeodesic equations on each smooth segment,

$C_p\subset T_pQ\setminus\{0\}$0

so, up to reparametrization, each leg is a null pregeodesic of the relevant cone structure (Javaloyes et al., 30 Sep 2025).

The anisotropic rheonomic Huygens framework places the same geometry in PDE–ODE correspondence. The eikonal equation is expressed as

$C_p\subset T_pQ\setminus\{0\}$1

or equivalently $C_p\subset T_pQ\setminus\{0\}$2, and its characteristic curves satisfy the Hamilton system

$C_p\subset T_pQ\setminus\{0\}$3

(Javaloyes et al., 2020). The same propagation law can be written as null-geodesic equations of the Lorentzian metric

$C_p\subset T_pQ\setminus\{0\}$4

with the same null cones (Javaloyes et al., 2020). Thus the PDE for front evolution is reduced to ODE tracing of cone geodesics.

A common misconception is that elliptical-cone methods are purely heuristic beam constructions. The cone-structure literature shows that, at least for anisotropic and discontinuous media, the tracing law can be obtained from a full arrival-time variational principle rather than from ad hoc local matching alone (Javaloyes et al., 30 Sep 2025). Conversely, the rendering-oriented elliptical-cone method is not formulated as a Fermat principle; it is a weakly-local path-integral construction designed to preserve non-negativity of sampled contributions (Steinberg et al., 24 Aug 2025).

3. Generalized Snell’s law for ellipsoidal propagation

At an interface $C_p\subset T_pQ\setminus\{0\}$5, the generalized Snell law is formulated through tangent hyperplanes to the cone structures at the incident and transmitted velocities. If $C_p\subset T_pQ\setminus\{0\}$6 is the crossing point, $C_p\subset T_pQ\setminus\{0\}$7 is incident in $C_p\subset T_pQ\setminus\{0\}$8, and $C_p\subset T_pQ\setminus\{0\}$9 is refracted in $Q=\mathbb{R}\times N$0, then with $Q=\mathbb{R}\times N$1 one has the matching condition

$Q=\mathbb{R}\times N$2

In Lorentz–Finsler language this is equivalent to

$Q=\mathbb{R}\times N$3

(Javaloyes et al., 30 Sep 2025). If the trajectory remains in a single medium, the corresponding law of reflection is

$Q=\mathbb{R}\times N$4

equivalently

$Q=\mathbb{R}\times N$5

(Javaloyes et al., 30 Sep 2025).

For elliptical cones, these conditions acquire an explicit quadratic form. Let $Q=\mathbb{R}\times N$6 be the unit normal to the interface hyperplane $Q=\mathbb{R}\times N$7 in some background inner product. If $Q=\mathbb{R}\times N$8 is the incident direction on $Q=\mathbb{R}\times N$9, then tangential slowness matching gives

$\ker dt$0

for some scalar $\ker dt$1, and the refracted direction must satisfy $\ker dt$2 (Javaloyes et al., 30 Sep 2025). Writing $\ker dt$3 and $\ker dt$4, one obtains

$\ker dt$5

where $\ker dt$6 solves

$\ker dt$7

This is a scalar quadratic with discriminant

$\ker dt$8

The physically admissible root is the one placing $\ker dt$9 in the correct half-space (Javaloyes et al., 30 Sep 2025).

In the isotropic limit $F^t(x,\cdot)$0, the quadratic law reduces to

$F^t(x,\cdot)$1

and the condition $F^t(x,\cdot)$2 reproduces the usual total-reflection criterion $F^t(x,\cdot)$3 (Javaloyes et al., 30 Sep 2025). This reduction clarifies that the ellipsoidal formula is not an alternative to classical Snell refraction but its anisotropic quadratic generalization.

The existence and multiplicity structure is governed by convexity. Three cases are distinguished according to the causal character of $F^t(x,\cdot)$4: if $F^t(x,\cdot)$5 is timelike for $F^t(x,\cdot)$6, there is at most one proper refraction; if lightlike, at most one proper refraction plus possible exceptional tangential behavior; if spacelike, exactly two refractions, i.e. “double refraction” (Javaloyes et al., 30 Sep 2025). The locally time-minimizing branch is selected by an orientation test on the orthogonal half-hyperplanes; reflected rays are minimizing only in the trivial unbroken case, and for double refraction exactly one branch is minimizing (Javaloyes et al., 30 Sep 2025).

4. Algorithmic wave tracing in piecewise-quadratic anisotropic media

For piecewise-constant ellipsoidal data, the generalized Snell law leads directly to a tracing algorithm (Javaloyes et al., 30 Sep 2025). At each cell center one precomputes the ellipsoid matrix $F^t(x,\cdot)$7 and its inverse $F^t(x,\cdot)$8. Given a starting point $F^t(x,\cdot)$9 and direction $L(v) = (v^0)^2 - (F^t(x, v^i\partial_{x^i}))^2$0 satisfying $L(v) = (v^0)^2 - (F^t(x, v^i\partial_{x^i}))^2$1, one integrates the constant-$L(v) = (v^0)^2 - (F^t(x, v^i\partial_{x^i}))^2$2 geodesic

$L(v) = (v^0)^2 - (F^t(x, v^i\partial_{x^i}))^2$3

until it intersects a cell-wall hyperplane $L(v) = (v^0)^2 - (F^t(x, v^i\partial_{x^i}))^2$4 (Javaloyes et al., 30 Sep 2025). At the interface point $L(v) = (v^0)^2 - (F^t(x, v^i\partial_{x^i}))^2$5, the local normal $L(v) = (v^0)^2 - (F^t(x, v^i\partial_{x^i}))^2$6 is computed, the quadratic for $L(v) = (v^0)^2 - (F^t(x, v^i\partial_{x^i}))^2$7 is solved, non-real roots and directions pointing back into the old cell are discarded, and the new direction is set to

$L(v) = (v^0)^2 - (F^t(x, v^i\partial_{x^i}))^2$8

If necessary, one renormalizes by

$L(v) = (v^0)^2 - (F^t(x, v^i\partial_{x^i}))^2$9

before continuing into the next cell (Javaloyes et al., 30 Sep 2025).

The computational notes in the same formulation identify three implementation-level facts. Each interface crossing requires solving one quadratic in $C=\{L=0, dt>0\}$0. The orientation test for local minimization simplifies to checking the sign of $C=\{L=0, dt>0\}$1. Double roots, $C=\{L=0, dt>0\}$2, correspond to grazing rays, and slight grid perturbations may be used to avoid degenerate lightlike interface cases (Javaloyes et al., 30 Sep 2025).

A separate numerical recipe applies when the matrix field $C=\{L=0, dt>0\}$3 varies smoothly in time and space. There one discretizes the initial front by marker points, computes unit normal covectors $C=\{L=0, dt>0\}$4 by solving $C=\{L=0, dt>0\}$5, and integrates the Hamiltonian system

$C=\{L=0, dt>0\}$6

for each marker, for example by fourth-order Runge–Kutta (Javaloyes et al., 2020). At target times the set of evolved marker positions samples the propagated front, and the phase field may be reconstructed either as the lower envelope of tangent half-spaces or by an upwind Hamilton–Jacobi finite-difference solve (Javaloyes et al., 2020).

A plausible implication is that “wave tracing with elliptical cones” naturally bifurcates into two numerical regimes: interface-driven tracing in discretized media with piecewise-quadratic cone data (Javaloyes et al., 30 Sep 2025), and continuous characteristic integration in smoothly varying ellipsoidal media (Javaloyes et al., 2020). Both use the same underlying cone geometry but differ in where the primary computational effort lies: quadratic root selection at boundaries versus ODE integration in the bulk.

5. Weakly-local wave tracing and region-to-region elliptical cones

In wave-optical transport, the principal obstacle to a direct coherent path integral is the loss of non-negativity. The classical path tracing integral

$C=\{L=0, dt>0\}$7

has $C=\{L=0, dt>0\}$8 and corresponds to geometric optics, whereas the bilinear coherent integral

$C=\{L=0, dt>0\}$9

allows interference between pairs of paths but becomes difficult to sample efficiently because arbitrarily chosen paths can destructively cancel (Steinberg et al., 24 Aug 2025).

The elliptical-cone solution is to replace point-to-point paths by region-to-region steps

$(n+1)$00

and define a non-negative region-path contribution $(n+1)$01, so that

$(n+1)$02

(Steinberg et al., 24 Aug 2025). In operator notation,

$(n+1)$03

where $(n+1)$04 creates an emission distribution in $(n+1)$05, each $(n+1)$06 propagates the beam distribution from one region to the next, and the final inner product with $(n+1)$07 models sensor measurement (Steinberg et al., 24 Aug 2025). Because each operator is linear and each final measurement inner product is non-negative, the sampled quantity remains non-negative.

Elliptical cones enter because under free-space propagation each Gaussian-beam primitive evolves into a bounded spatial support with elliptical-cone envelope (Steinberg et al., 24 Aug 2025). Sampling then proceeds region by region. At each region $(n+1)$08, one chooses a new central axis $(n+1)$09 and envelope parameters $(n+1)$10 such that $(n+1)$11 and the beam solid angle is preserved. The geometric constraint is

$(n+1)$12

and with fixed ellipticity ratio $(n+1)$13, only the new axis $(n+1)$14 needs to be sampled (Steinberg et al., 24 Aug 2025). With proposal density $(n+1)$15, the Monte Carlo weight is

$(n+1)$16

where $(n+1)$17 (Steinberg et al., 24 Aug 2025).

The implementation described for this framework stores the scene in a BVH over triangles and edges, represents each beam envelope by $(n+1)$18, intersects the cone frustum with the BVH to gather all geometry in the current interaction region, and evaluates scattered power into candidate beams by combining single-primitive and pairwise terms (Steinberg et al., 24 Aug 2025). A ballistic segment optimization first shoots a pure ray for up to $(n+1)$19 wavelengths, reverting to cone tracing only if no nearby intersection occurs (Steinberg et al., 24 Aug 2025). The method also includes diffraction handling through a Fraunhofer-edge method and a UTD bending strategy (Steinberg et al., 24 Aug 2025).

A common misconception is that this renderer-like “wave tracing” is simply ray tracing with wider rays. The formal distinction is that the method is designed to capture within-region wave interference exactly while preserving a non-negative sampled integrand $(n+1)$20, something not achieved by classical shooting-bouncing rays alone (Steinberg et al., 24 Aug 2025).

6. Reflections, cavity dynamics, and related uses of elliptical-cone geometry

Elliptical-cone tracing also appears in bounded-domain wave dynamics. For inertial waves in a rotating fluid confined to a truncated elliptical cone, local wave packets propagate along straight ray segments at fixed angle $(n+1)$21 to the horizontal, with

$(n+1)$22

between reflections (Favier et al., 2024). Reflection on the sloping elliptic wall changes the horizontal propagation angle $(n+1)$23 according to

$(n+1)$24

where $(n+1)$25 is the azimuthal angle of the wall normal projected into the horizontal plane and $(n+1)$26 (Favier et al., 2024). This is a reflection law for characteristics in an elliptical-cone cavity, not a Snell law across media, but it again shows how cone geometry yields explicit local update formulas.

For $(n+1)$27, linearization around the axisymmetric $(n+1)$28D attractor in the plane $(n+1)$29 produces a $(n+1)$30 map

$(n+1)$31

with eigenvalues

$(n+1)$32

(Favier et al., 2024). Convergence requires $(n+1)$33, and in that case all rays contract toward the plane $(n+1)$34 and onto the same quadrilateral cycle, producing a one-dimensional closed-curve “super-attractor” localized in the bulk of the fluid (Favier et al., 2024). This use of ray tracing in an elliptical cone is dynamically rather than variationally motivated, but it illustrates the broader relevance of cone-based wave tracing.

A different, microlocal use of elliptical-cone terminology occurs in the study of manifolds with conic singularities. The diffractive wave trace on such manifolds has singularities at the lengths of closed diffractive geodesics, and for a strictly diffractive closed geodesic striking a cone point once, the trace near $(n+1)$35 is represented as

$(n+1)$36

with $(n+1)$37 a classical symbol of order $(n+1)$38 (Ford et al., 2014). In the special case of an elliptical cone in $(n+1)$39, the diffraction coefficient and Jacobian factors can be written in terms of the curvature $(n+1)$40 of the elliptical link, and the principal diffractive amplitude depends explicitly on that curvature (Ford et al., 2014). This is not a tracing algorithm, but it demonstrates that elliptical-cone geometry also enters spectral and diffractive analysis.

7. Relations, limitations, and interpretive boundaries

Three distinct but mathematically adjacent meanings of “wave tracing with elliptical cones” can therefore be separated.

Setting	Elliptical-cone role	Primary law
Piecewise anisotropic media	Local speed indicatrix or propagation cone	Generalized Snell/reflection law (Javaloyes et al., 30 Sep 2025)
Weakly-local wave optics	Beam envelope for region-to-region transport	Non-negative region-path integral (Steinberg et al., 24 Aug 2025)
Rotating-fluid cavity dynamics	Physical domain boundary and characteristic geometry	3D inertial-wave reflection map (Favier et al., 2024)

The first setting is rooted in generalized Fermat’s principle and null pregeodesics of Lorentz–Finsler cone structures (Javaloyes et al., 30 Sep 2025). The second is rooted in a path-integral reformulation in which elliptical cones define bounded regions over which wave interference is aggregated locally, restoring Monte Carlo tractability (Steinberg et al., 24 Aug 2025). The third concerns ray dynamics in a specific non-axisymmetric cavity geometry and yields a global attractor structure under repeated reflections (Favier et al., 2024).

Several interpretive cautions follow. First, “elliptical cone” may refer either to a cone of admissible velocities, an envelope of significant beam support, or a geometric cavity. These should not be conflated. Second, generalized Snell laws for cone structures govern transmission and reflection at interfaces in anisotropic media; they do not, by themselves, provide a full wave-optical interference model (Javaloyes et al., 30 Sep 2025). Third, region-to-region wave tracing with elliptical cones is not a direct consequence of Lorentz–Finsler variational optics, even though both frameworks rely on ellipsoidal geometry (Steinberg et al., 24 Aug 2025).

At the same time, the formal parallels are substantial. In each case, anisotropy or finite wave extent is encoded by quadratic or near-quadratic structure. In each case, transport proceeds by updating geometric objects richer than a single Euclidean ray. And in each case, the elliptical cone functions as a compact way to carry directional, spatial, and interface information through propagation, whether for wavefront evolution (Javaloyes et al., 2020), refraction in discontinuous anisotropic media (Javaloyes et al., 30 Sep 2025), weakly-local wave-optical rendering (Steinberg et al., 24 Aug 2025), or cavity-wave asymptotics (Favier et al., 2024).

This suggests a broader interpretation: wave tracing with elliptical cones is an emergent cross-disciplinary methodology in which ellipsoidal local models provide explicit propagation and scattering laws while remaining compatible with variational optics, characteristic methods, and practical simulation.