Delayed-Light Rendering for Superluminal Objects

Abstract

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.