Source-linked AI summary
Delayed-Light Rendering for Superluminal Objects
David Bizzozero
TL;DR
Finite-speed imaging in a Newtonian setting can show multiple simultaneous images of bodies that outrun the signal, but finding those images requires solving for past emission times. The paper enumerates emission-condition roots over recorded history and reports interactive-rate measurements alongside independent implementation checks.
Problem
Finite-speed imaging of moving bodies can produce multiple simultaneous images, but standard fixed-point iteration diverges for many superluminal bodies.
Source: §1 · p. 1, §6.1 · p. 4Method
The renderer enumerates roots of the emission condition over recorded state history, using piecewise-linear segments and a quadratic solve.
Source: §1 · p. 1Results
8.4 ms per frame walked and 2.8 ms under tick-aligned cadence for 120 bodies and 20 anchors; an independent port reproduced 150 images with 0 missed and 0 spurious.
Source: §11.4 · pp. 13–14Takeaways & Limitations
The delay landscape accepts arbitrary observation events, and the method was deployed in a released real-time strategy game.
Source: §1 · p. 2The imaging model omits occlusion, and visibility questions for a 3D renderer remain out of scope.
Source: §12 · p. 15
Abstract
from arXiv · showhide
We present a real-time rendering method for scenes perceived through signals of finite speed $c$ in a non-relativistic setting: $c$ is a property of the imaging signal, not the causal speed limit, so bodies may move faster than $c$. A superluminal body presents several simultaneous images; pairs are created and annihilated in caustic flashes, and some branches play backward in time. Rather than approximating these effects, we enumerate them as roots of an emission condition posed against recorded state history, exact at every point solved, up to one stated approximation for moving observers. Because a fixed-step simulation records piecewise-linear history, the emission condition restricted to one history segment is a quadratic whose discriminant detects image-pair creation and whose slope classifies each image's playback direction, rate, and brightness. Perception is not tied to a single reference point: a global perceived state is built for an arbitrary finite set of observation events, and the delay landscape is the upper envelope of their backward light cones. A monotonic emission clamp guarantees the picture never regresses to images older than those already shown. Per-vertex solves shear bodies straddling delay gradients, and pre-generated frame-sequence assets are adapted by treating them as $(x,y,t)$ volumes sliced by the solved emission surface, so playback rate, reversal, and intra-body de-phasing arise with no animation-specific code. The exposition and implementations are planar: the emission condition and its per-segment solve are norm conditions, independent of dimension, but the visibility and occlusion questions a 3D renderer must answer are out of scope. The method is deployed in a released real-time strategy game; we describe the optimizations that make it run at interactive rates, and measure the algorithms through independent reference implementations.
1 Introduction
The paper renders finite-speed perception in a Newtonian setting where bodies can outrun the imaging signal, producing multiple simultaneous images. It models perception across arbitrary observation events and presents an interactive root-enumeration method for these effects.
1 Introduction
- The renderer treats finite-speed perception as root finding over recorded state history, with arbitrary observation events defining a shared delay landscape.The landscape is the minimum delay over the observation events’ backward light cones; the contributions also include per-vertex delayed-time shear and frame-sequence asset adaptation.Source: §1 · p. 1
2 Light speed versus causality speed
- The imaging signal is not the causal speed limit, so Newtonian bodies can outrun it and produce fully visible delay distortions and multiple images.The paper distinguishes this regime from relativity, where kinematic effects partially cancel light-travel-time distortion.Source: §2 · p. 2, Table 1 · p. 3
3 Related work
- Unlike relativistic visualization and transient rendering, this method enumerates images of moving superluminal bodies against history, extending established phenomena with an interactive rendering algorithm.Prior relativistic systems exclude superluminal motion, while transient rendering resolves travel time through essentially static scenes.Source: §3 · p. 2
4 Architecture: two clocks and a state-history ring
A renderer reads tick-indexed state history independently of the simulation clock and assigns each scene point the freshest emission time available from any observation event. A monotonic clamp prevents coverage loss from making the displayed picture regress.
4 Architecture: two clocks and a state-history ring
- The simulation advances deterministically at fixed ticks while the renderer runs independently and reads snapshots from a state-history ring buffer.The ring retains enough history for the longest possible perception delay, sized against the slowest permitted signal speed.Source: §4 · p. 3
5 The perceived delay model
- Each observation event contributes a backward light cone, and the perceived delay is the minimum total of light-travel time and event age.The winning event need not be the nearest anchor; arbitrary event placement, including routed relay networks, is an input to the model.Source: §5 · pp. 3–4
5.1 The monotonic emission clamp
- A running-maximum emission clamp prevents relay loss from rewinding displayed images, holding them until fresher direct-light information arrives.The clamp is applied per solved point and, for superluminal bodies, per image branch; no held image outlasts D/c.Source: §5.1 · p. 4
6 The delayed-time solve
The delayed-time solve finds past emission events consistent with signal travel, including multiple roots for superluminal motion. It replaces divergent fixed-point iteration with a segment-wise quadratic walk and adds safeguards for moving relay observers.
6 The delayed-time solve
- An emission root is the past time when the vertex’s recorded position emitted the signal arriving at the present, and that root may be non-unique.The delayed-time condition evaluates the point at its historical location, rather than simply applying a delay at its current position.Source: §6 · p. 4
6.1 Why fixed-point iteration fails
- Superluminal motion can make fixed-point iteration diverge and produce multiple simultaneous image roots, so the solve must enumerate roots rather than assume convergence.The emission condition remains bracketed when at least one anchor reports the point over a finite route; if all routes are lost, the monotonic clamp preserves the last displayed emission time.Source: §6.1 · p. 4
6.2 The segment-exact quadratic walk
- Piecewise-linear history reduces the emission condition on each segment to a quadratic per anchor, whose roots are validated and searched from freshest to oldest.A zero discriminant marks image-pair tangency; the walk also handles coverage-boundary kinks, and resolves fully degenerate intervals to their freshest end.Source: §6.2 · pp. 4–5
6.3 The nested relay refinement and the safety net
- The quadratic walk is exact for fixed-position anchors, while mobile relay anchors use a nested fixed-point condition and secant validation.Validation accepts roots with |F| ≤ 3 ms; a one-history-segment safety net checks the nested condition, and the freshest direct-light root acts as a floor against stale relay artifacts.Source: §6.3 · pp. 5–6
7 Bifurcation: multiple images, playback rate, brightness
The per-segment solve enumerates simultaneous images as roots: a caustic creates or annihilates a pair, and root slope determines playback direction and rate. Brightness follows a capped, stylized response to that rate.
7 Bifurcation: multiple images, playback rate, brightness
- For a zigzagging source moving at v ≫ c, the retained window can contain on the order of v/c simultaneous images.The enumeration mode emits every validated root in the retained window rather than stopping at the freshest root.Source: §7 · p. 6
- The apparent playback rate is 1/|F′(t′)|, and F′(t′) > 0 marks a backward-playing branch whose recorded motion is replayed in reverse.For that branch, emission time decreases as render time advances; F′(t′) < 0 marks forward playback.Source: §7 · p. 6
- Opacity is proportional to playback rate under a capped, clamped brightness response that reproduces qualitative focusing but is not radiometric.It includes no geometric-spreading term and conserves no energy; root pairs within approximately one tick of a caustic coalesce into a saturated flash.Source: §7 · pp. 6–7
- At the caustic, a tangent root creates an image pair; afterward, two roots represent forward- and backward-playing images.The figure relates the tangency to the discriminant of the per-segment quadratic, so the solver cannot step over pair creation.Source: Figure 3 · p. 7, §7 · p. 8
- The monotonic clamp applies per branch only to delay-field regression: backward-playing branches can move to older emission times, while caustic annihilation renders as merge-and-vanish.Only the freshest surviving branch receives the shear grid and drives overlays and hit-testing; secondary images render rigidly under pooled caps with hysteresis.Source: §7 · p. 8
8 Per-vertex shear
Per-vertex emission-time solves deform bodies across delay gradients, with grid interpolation approximating the delay field. The interpolation bound depends on field curvature and does not apply across a boundary where the winning anchor changes.
8 Per-vertex shear
- Bodies imaged at a distance use an N×M grid of solves, from 2×2 to 4×4, with vertex emission times bilinearly interpolated from grid cells.Within a cell, the delay-field interpolation error has a curvature bound proportional to L^2; for a single cone its transverse curvature is 1/(cd), and dividing by |F′| converts the bound to emission-time error.Source: §8 · p. 8
- At a coverage boundary the winning anchor changes, so the delay field is continuous but nondifferentiable and the quadratic interpolation bound no longer applies.This is the configuration that shears a body most visibly; vertices at different past times stretch and shear it, while their historical facings can produce a winding orientation skew.Source: §8 · p. 8
- Animated sub-parts use each vertex’s own emission time, so spinning parts shear across steep gradients and periodic lights de-phase with distance.One emission time determines one pose.Source: §8 · p. 8
9 Frame-sequence assets: slicing the (𝑥,𝑦,𝑡) volume
Stored frame sequences become time-stacked image volumes sliced at solved emission times, so each sample reads the frame associated with its own past. This reuses the shear lattice and requires no animation-specific asset code.
9 Frame-sequence assets: slicing the (𝑥,𝑦,𝑡) volume
- A looping or transient frame deck is sliced along the solved emission-time surface, with samples reading stored frames at their own emission times.The technique reuses the shear lattice’s interpolated emission times and adds a two-fetch mix per sample.Source: §9 · p. 8, Figure 5 · p. 10
- Animation phase depends only on emission time, so playback slows on receding images and runs backward on backward-playing branches without animation-specific code.The phase velocity is −1/(𝑃𝐹′), and the animation phase is a pure function of solved emission time.Source: §9 · pp. 8–9
- When bulk motion creates an image pair, both branches begin from the same emission-time frame and then de-phase as their roots separate.Frames are indexed by emission time, not render time.Source: §9 · p. 9
- A steep delay gradient can make different parts of one body display different frames at once; across a coverage boundary, frame indexing stays seamless while playback rate changes.The solved emission time is continuous at the boundary, but its gradient jumps.Source: §9 · p. 9
- The frame deck is projected imagery, so motion depicted inside its frames has no worldline for the solver and cannot itself bifurcate, flash, or multiply.Those effects come only from bulk-object motion; extracting per-texel trajectories is not attempted.Source: §9 · p. 9
- Linear frame mixing can ghost when inter-frame displacement exceeds roughly one pixel, while motion above half the feature spacing is unrecoverable from the frames alone.The latter sampling limit is independent of the interpolator.Source: §9 · p. 9
- A supplemental browser implementation applies the frame-deck method to user-supplied looping animations under an adjustable worldline.It includes superluminal branches.Source: §9 · p. 10
10 Optimizations
The optimizations reduce full emission-condition evaluations through certified root-free skips, cached-root advancement, and staggered walks. Hard cases return to the full walk, preserving correctness while allowing some images to be discovered late.
10 Optimizations
- Certified strides let the walk skip root-free stretches, making far objects cost a handful of evaluations without bridging a skipped stretch with a false bracket.For fixed anchors, the stride follows the |𝐹|/(1 + 𝑣/𝑐) bound; mobile-relay strides account for the flat-versus-nested discrepancy.Source: §10 · p. 10
- Within a simulation tick, cached roots shift exactly under the present-pinned-apex convention; across ticks, transported roots are only warm starts corrected and accepted against the current full 𝐹.For fixed-spacetime anchors, 𝐹 is instead independent of render time and roots do not move between ticks.Source: §10 · pp. 10–11
- The rotating walk schedule amortizes full walks across display frames, while a per-frame budget supports low frame rates and scenes with hundreds of bodies.Full-𝐹 evaluations dominate cost, and each includes a scan over mobile relay anchors.Source: §10 · pp. 10–11
- Flashes, streak images, floor hits, non-convergence, and roots nearing coalescence return to the full walk; missed discoveries may be delayed a few ticks, but false images are never shown.The advance changes how a root is found, not the full walk’s correctness test.Source: §10 · p. 11
- Near image-pair creation, quadratic solves cost the most; per anchor and history segment, setting up and solving a quadratic takes roughly 30 floating-point operations.Far, slowly moving bodies typically settle to a handful of evaluations per frame.Source: §10 · p. 11
11 Validation and performance
The section validates root enumeration against dense sweeps and closed-form oracles, then measures interpolation and cadence costs; a browser port demonstrates interactive solver timings.
11.1 Enumeration against an exhaustive sweep, and a code-free oracle
- 150 images were enumerated across three scenes, with 0 missed, 0 spurious, and a maximum accepted residual of 0.071 ms.The maximum residual is an order of magnitude below the 3 ms tolerance; the static-anchor scenes agree exactly with the sweep.Source: §11.1 · p. 11, Table 2 · p. 12
- The exhaustive sweep cannot certify image pairs narrower than its sampling and tangency window, which is also the walk’s flash-coalescence width.The reported reference timings are interpreted, single-threaded measurements, not frame-rate claims for a shipped product.Source: §11 · p. 11, §11.1 · p. 12, §11.3 · p. 13
- The affine flyby oracle matched 84 roots within 1.3 × 10^-12 s, while the nested relay oracle matched 140 roots with 0 unmatched and roots within 1.7 ms.The flyby oracle is evaluated from scene constants rather than solver code; the nested oracle factors the relay and body solves into chained quadratics.Source: §11.1 · p. 12
- As relay speed rose from 0.1c to 0.9c, enumeration remained complete, while the corrected slope stayed within 1.6 × 10^-3 of the nested oracle.The segment-only slope error grew from 0.17 to 1.00 across that sweep.Source: §11.1 · p. 12
11.2 Interpolation error against the shear bound
- Halving lattice cell span reduced emission-time error by approximately fourfold, and measured error was 1.13–1.44 times the bound in equation (11).The measurement covered four lattice sizes and three anchor distances; the error also fell with anchor distance as predicted.Source: §11.2 · p. 13
11.4 Rendered results and a real-time measurement
- The JavaScript port reproduced the enumeration experiment’s 150 images with no missed or spurious images and a worst residual of 0.071 ms.The reference implementation is independent of the production system, but shared generative-AI assistance means agreement alone cannot rule out a shared misreading.Source: §11 · p. 11, §11.4 · p. 13
11.3 Cost, and what the cadence carries
- The tick-aligned cadence advanced 68.75% of solves and reduced cost per frame by 2.88× at the largest configuration without accepting advances outside walk tolerance.Of six advances handed back to a full walk, five exhausted iterations above tolerance and one left the retained window; none hit the near-caustic guard.Source: §11.3 · p. 13, Table 3 · p. 14
11.4 Rendered results and a real-time measurement
- 8.4 ms per frame when walked and 2.8 ms under cadence: the 120-body, 20-anchor scene ran at 119 and 351 solver frames per second, respectively.These single-threaded Chromium solver timings exclude drawing and are reported as method bounds, not product timings.Source: §11.4 · pp. 14–15
12 Limitations and scope
The exposition and implementation are planar, and the account identifies visibility and occlusion as substantive limits on extending the renderer to 3D. Exactness is also qualified by interpolation, recorded-history resolution, moving-relay approximation, and cadence latency.
12 Limitations and scope
- Light-leg occlusion is not implemented; removing cones at shadow boundaries can make delay discontinuous, leaving the monotonic clamp to carry the transition.The model leaves sweeping shadow edges and coverage changes as open phenomena.Source: §12 · p. 15
- The exposition and implementation are planar; although norm-based delay and emission solves extend unchanged to 3D, visibility makes 3D a genuine extension.A volumetric shear lattice increases solved points with depth resolution, and frame-deck memory grows by the depth axis; moving occluders require visibility queries against recorded history.Source: §12 · p. 15
- The solve is exact only at solved points and against recorded piecewise-linear history, while between-point interpolation and finite history resolution remain bounded errors.Mobile relays use their present position as an approximation, then are polished to a stated tolerance with a strided safety net.Source: §12 · p. 15
- Tick-aligned cadence trades computation for discovery latency: image pairs created between full walks may be reported a few simulation ticks late.The stated qualifications bound rather than eliminate interpolation error, history resolution, relay polish, and discovery delay.Source: §12 · p. 16
- Motion depicted within a frame-sequence asset cannot independently bifurcate, flash, or multiply; only bulk-body motion can produce those effects.Promoting depicted motion into the solver is left as a further costed extension.Source: §12 · p. 16
13 Summary
The renderer computes perceived images by solving an emission condition against recorded state history over an envelope of backward light cones. Its segment-wise quadratics enumerate image branches and support interactive rendering while clamps prevent regression to staler imagery.
13 Summary
- The renderer solves an emission condition against recorded state history at each vertex, using an upper envelope of backward light cones from arbitrary observation events.The implementation can use observers joined by a faster reporting channel.Source: §13 · p. 16
- Piecewise-linear history reduces root-finding to per-segment quadratics, whose discriminants detect image-pair creation; certified skip bounds and tick-aligned Newton cadence reduce redundant work.A monotonic emission clamp, direct-light floor, and temporal grid clamp prevent any branch from regressing to staler imagery than already shown.Source: §13 · p. 16
- Solved roots yield held ghosts, sheared bodies, and reversed or multiplied images of superluminal bodies, including caustic flashes, rather than effects added afterward.The non-relativistic setting leaves these distortions without kinematic cancellation.Source: §13 · p. 16
- Backward-playing images replay recorded history in reverse while render time and every system clock continue forward.Source: §13 · p. 16
Supplemental material
The accompanying materials provide implementations, scripts, browser demonstrations, exported records, and a supplemental video to reproduce the paper's non-closed-form results and visual examples.
Supplemental material
- The reference implementation and experiment driver reproduce the delay field, solver variants, shear lattice, experiments, and reported tables and figures.These include correctness checks, closed-form oracles, relay-speed and hold-and-resume experiments, branch events, shear error, and cost.Source: Supplemental material · p. 16
- Figure-generation scripts recreate Figures 1–5 from textual closed forms, while a separate script composes Figure 7 from demonstration frame grabs.Two self-contained browser demonstrations provide live solver validation and real-time measurement, and frame-deck rendering for looping animations.Source: Supplemental material · p. 16
- A supplemental video presents demonstration scenes including pair creation, simultaneous images, relay behavior, animated-body slicing, and a 40-body crowd.A script deterministically re-renders the video.Source: Supplemental material · p. 16
- All five accompanying items run without a graphics engine; the first two require Python with NumPy and Matplotlib, while the demonstrations require only a browser.Source: Supplemental material · p. 16
A Derivations
The appendix derives the emission-time slope along a mobile-relay branch and a certified stride for safely skipping root-free intervals despite the difference between flat and nested relay conditions.
A.1 The mobile-relay slope
- A.1 The mobile-relay slope gives the reception-time change per emission-time change from the body and relay velocities projected along the relay-to-emitter direction.Setting relay velocity to zero recovers the fixed-anchor slope.Source: §A.1 · p. 17
A.2 The flat-versus-nested discrepancy, and the certified stride
- A.2 The flat-versus-nested discrepancy is bounded using relay displacement, the largest recorded relay speed, and the flat-condition slope bound.The routed-distance bound is preserved by taking the minimum over anchors.Source: §A.2 · p. 17
- A stride satisfying the strict positivity condition certifies that the skipped interval contains no root of the nested condition, not merely the flat one.With no mobile relays, the stride reduces to the flat certificate.Source: §A.2 · p. 17
A.3 The nested oracle for affine worldlines
For affine body and relay worldlines, the nested emission condition has closed-form roots because it passes through a single intermediate reception event. The resulting candidates can be checked against the delay envelope without solver code.
A.3 The nested oracle for affine worldlines
- Affine body and relay worldlines yield closed-form roots because both nested conditions have the segment-condition form and meet at one relay reception event.The reference observer receives the relay's report at that event.Source: §A.3 · p. 17
- At render time, solve relay reception first, then the body's emission time against that fixed event, retaining only roots with t_r ≤ t and t' ≤ t_r.The inner condition treats the body's position against the relay reception event as a fixed anchor.Source: §A.3 · pp. 17–18
- A candidate is valid for the enveloped condition only when its branch attains the minimum arrival time; closed forms also supply the competing branch for this test.Slopes are evaluated on the same closed geometry.Source: §A.3 · p. 18
- The closed-form oracle serves as a code-free reference for the branch approximated by the walk and is re-derived across relay speeds.Source: §A.3 · p. 18
B Generative AI disclosure
- Claude assisted with drafting, editing, and test-code generation under author specifications and review; authors checked derivations and validated against closed-form oracles that used no generated code.Source: B Generative AI disclosure · p. 18