Gajmer Ray-Casting Algorithm (GRCA)

• GRCA is an emitter-centric, triangle-driven algorithm designed to optimize real-time LiDAR simulation in highly dynamic scenes.
• It inverts the conventional ray-centric query by determining, for each triangle, the potential intersecting rays using geometric approximations like GACP.
• The method achieves significant speedups over BVH-based approaches, with performance boosts up to 14.55× on CPU and 7.97× on GPU in dynamic scenarios.

Searching arXiv for the specified paper to ground the article. The Gajmer Ray-Casting Algorithm (GRCA) is a compute-based, vendor-agnostic ray-casting algorithm introduced for real-time LiDAR simulation in arbitrarily dynamic scenes, particularly where large numbers of moving and deformable objects make per-frame acceleration-structure rebuilds costly (Gajmer et al., 11 May 2026). Its defining move is an emitter-centric inversion: rather than asking, for each emitted ray, which triangle it hits first, GRCA asks, for each triangle, which rays could possibly intersect it. In the formulation reported for spinning LiDAR, this inversion is coupled to geometric models of the emitter as rotation-traced cones or planes and to emitter-centric apparent-area culling, enabling per-triangle filtering of channels and rays without any acceleration structure. Although developed for LiDAR, the method is presented as a general-purpose ray-casting algorithm for settings in which rays originate from a known position.

1. Problem formulation and the limits of ray-centric acceleration structures

The target problem is the per-frame identification of the closest intersecting triangle for every emitted ray from a spinning LiDAR operating in a dynamic scene. A spinning LiDAR casts up to $\Omega \cdot \gamma \cdot \chi$ rays into a scene of $\tau$ triangles each frame. A brute-force loop therefore requires $\Omega \cdot \gamma \cdot \chi \cdot \tau$ ray–triangle intersection checks (RTIC); the reported example is

$$1 \times 128 \times 4096 \times 10^7 \approx 5.2 \times 10^{12}$$

tests per frame (Gajmer et al., 11 May 2026).

The paper positions BVH-based methods such as OptiX and Embree as the dominant acceleration-structure approaches and identifies three structural shortcomings in real-time, highly dynamic scenes. First, in dynamic scenes, full per-frame BVH rebuilds are $O(\tau)$ regardless of how much geometry moved. Second, for $360^\circ$ LiDARs, rays cover every direction, eliminating spatial coherence and preventing amortization of node tests. Third, BVH traversal performs culling only at ray time; out-of-range or back-facing triangles still incur per-ray tests.

These points define the specific niche of GRCA. The algorithm is not framed merely as an implementation optimization over existing BVH traversal, but as a reorganization of work whose scaling behavior is intended to remain favorable when scene dynamics are pervasive, sensor resolution is high, and multiple sensors operate simultaneously. A plausible implication is that the method is especially aligned with workloads where acceleration-structure maintenance becomes the dominant cost rather than the intersection kernel itself.

2. Emitter-centric inversion

GRCA replaces the conventional ray-centric query with a triangle-centric query. Traditional ray casting asks, for each ray, “which triangle does it hit first?” GRCA instead asks, for each triangle $T$, “which rays could possibly intersect it?” (Gajmer et al., 11 May 2026). The paper states that, by answering per triangle rather than per ray, work scales with scene geometry, $O(\tau)$, and with per-triangle contributions, not with $\Omega \cdot \gamma \cdot \chi$.

This inversion is central to the method’s identity. In a ray-centric pipeline, every ray triggers traversal and candidate testing. In GRCA, a triangle is first filtered against the set of channels and rays that can geometrically reach it, and only those candidates are processed further. The result is a workflow in which the emitter’s structure is exploited before individual ray tests are executed.

A common misconception is to treat GRCA as eliminating ray–triangle tests altogether. The paper does not make that claim. Individual rays remain half-lines, and the pipeline still performs RTIC through update_hit(T,o_n,C,R) using Möller–Trumbore; what changes is the candidate-generation mechanism. Another common misconception is to regard the method as intrinsically LiDAR-specific. The paper explicitly describes the emitter-centric inversion as applying to any setting where rays originate from a known position, not only LiDAR.

3. Geometric modeling of spinning emitters

For a sensor at origin $o$ with basis vectors $\tau$0, $\tau$1, and $\tau$2, the paper indexes vertical channels by $\tau$3 and horizontal rays by $\tau$4. Elevation and azimuth are defined as

$\tau$5

with $\tau$6 and elevation spanning the specified channel range. The ray direction is

$\tau$7

The geometric surrogate used by GRCA is the GACP: Geometrically Approximated Cone or Plane (Gajmer et al., 11 May 2026). Because all rays in a channel share elevation $\tau$8, they satisfy

$\tau$9

As azimuth varies over $\Omega \cdot \gamma \cdot \chi \cdot \tau$0, the union of half-lines

$\Omega \cdot \gamma \cdot \chi \cdot \tau$1

is the right circular cone around axis $\Omega \cdot \gamma \cdot \chi \cdot \tau$2 with half-angle $\Omega \cdot \gamma \cdot \chi \cdot \tau$3, or degenerates to the horizontal plane when $\Omega \cdot \gamma \cdot \chi \cdot \tau$4. For any world point $\Omega \cdot \gamma \cdot \chi \cdot \tau$5, the cone equation is

$\Omega \cdot \gamma \cdot \chi \cdot \tau$6

The paper states that this single surface exactly approximates all rays in a channel, while individual rays remain half-lines. In operational terms, the GACP provides a channel-level geometric test that precedes ray-level work. This is the mathematical basis for GRCA’s binary search over channel spans and for its later precise edge-by-edge refinement on deferred cases.

4. Apparent-area culling, SAT/BAT classification, and the two-pass pipeline

GRCA combines emitter-centric inversion with apparent-area reasoning. For a triangle $\Omega \cdot \gamma \cdot \chi \cdot \tau$7 with area $\Omega \cdot \gamma \cdot \chi \cdot \tau$8, centroid $\Omega \cdot \gamma \cdot \chi \cdot \tau$9, normal $$1 \times 128 \times 4096 \times 10^7 \approx 5.2 \times 10^{12}$$0, and sensor origin $$1 \times 128 \times 4096 \times 10^7 \approx 5.2 \times 10^{12}$$1, the classical origin-centric apparent area is

$$1 \times 128 \times 4096 \times 10^7 \approx 5.2 \times 10^{12}$$2

where $$1 \times 128 \times 4096 \times 10^7 \approx 5.2 \times 10^{12}$$3 (Gajmer et al., 11 May 2026). This quantity supports early filtering together with back-face and range tests.

GRCA then predicts an integer channel span $$1 \times 128 \times 4096 \times 10^7 \approx 5.2 \times 10^{12}$$4 and ray span $$1 \times 128 \times 4096 \times 10^7 \approx 5.2 \times 10^{12}$$5. Defining

$$1 \times 128 \times 4096 \times 10^7 \approx 5.2 \times 10^{12}$$6

and letting $$1 \times 128 \times 4096 \times 10^7 \approx 5.2 \times 10^{12}$$7 be the maximum $$1 \times 128 \times 4096 \times 10^7 \approx 5.2 \times 10^{12}$$8 over channels in $$1 \times 128 \times 4096 \times 10^7 \approx 5.2 \times 10^{12}$$9, the emitter-centric apparent area is approximated by

$O(\tau)$0

The triangle is classified as SAT (smaller-appearing triangle) if $O(\tau)$1 and $O(\tau)$2 and the triangle does not straddle the scan seam; otherwise it is BAT. The paper reports that empirically $O(\tau)$3 balances early versus late work. SAT triangles are culled conservatively and processed inline, whereas BAT triangles are deferred for precise spans.

The overall pipeline has two passes. In the early pass, performed per triangle and per sensor origin, GRCA executes cheap origin-centric filters, computes $O(\tau)$4, skips triangles outside $O(\tau)$5, finds the channel span by binary search, derives a fast conservative ray span from three vertex projections, and then either processes SAT triangles inline or records BAT triangles for later handling. In the late pass, for each deferred entry, the method performs precise GACP–triangle edge-by-edge intersection, solving for edge parameters $O(\tau)$6 satisfying

$O(\tau)$7

collects up to two intersection points per edge, derives exact per-channel azimuth spans, disambiguates seam arcs if needed, and then runs RTIC over the precise spans.

The complexity reported for this organization is $O(\tau)$8 in the early pass plus $O(\tau)$9 in the late pass; more explicitly, the late pass is

$360^\circ$0

quadratic solves plus Möller–Trumbore over the resulting hit spans (Gajmer et al., 11 May 2026). This partition makes the SAT/BAT boundary a key workload-shaping mechanism: most triangles are intended to be resolved with conservative, inexpensive spans, while only a small deferred subset pays for exact per-channel reconstruction.

5. Extensions and quantitative performance

Two extensions are described as independent. The first is range culling, which allows realistic near- and far-range limits $360^\circ$1 and performs an early exit for triangles whose $360^\circ$2 falls outside that window. The second is a hybrid static/dynamic pipeline, in which a static BVH is built once for purely static geometry, GRCA is applied only to the per-frame dynamic subset, and per-ray hits are merged by taking the minimum distance between static BVH queries and GRCA results (Gajmer et al., 11 May 2026).

The benchmark regime evaluates 2–8 simultaneous 128×4096-ray LiDARs with 360°/180° coverage over complex dynamic scenes, including the case of two sensors casting approximately one million rays per frame. All tests are reported at 100 Hz simulation, using a full-sphere 128×4096 LiDAR in dynamic scenes. With range culling inactive, GRCA reaches up to 7.97× over hardware-accelerated OptiX on GPU and 14.55× over Embree on CPU. With range culling at realistic deployment ranges of 10–100 m, the reported boosts are up to 7.02× on GPU and 9.33× on CPU. In the hybrid static/dynamic configuration, the reported speedups reach up to 10.5× on GPU and 19.2× on CPU (Gajmer et al., 11 May 2026).

The largest highlighted hybrid demonstration is Power Plant + 30 dyn (22 M tris, 8 LiDARs), with ~22M triangles, ~9M of which are dynamic, where the hybrid pipeline attains 10.5× GPU and 19.2× CPU speedup versus standalone rebuild-only OptiX/Embree under scene-wide deformation. Robustness studies further report that, in static-only scenes with no rebuild, GRCA is still faster in 4 of 6 test meshes, at approximately 0.17×–0.65× the BVH trace cost. Under mesh subdivision up to 64× (19 M triangles), GRCA GPU cost grows only +20% as SAT triangles vanish, whereas BVH rebuilds scale super-linearly. Across 216 configurations on five hardware platforms, mean frame-time variance remains within ±20%, and more than 55% of frames finish faster than the mean.

These results are reported specifically against OptiX on RTX RT cores and against single-threaded Embree with AVX2/SSE4.1. The significance claimed by the paper is not merely raw throughput but the avoidance of per-frame rebuild overhead in highly dynamic regimes, where conventional acceleration structures pay costs that cannot be amortized by limited motion.

6. Scope, generality, and interpretive boundaries

The paper presents GRCA as broader than a LiDAR-specialized simulator kernel (Gajmer et al., 11 May 2026). Any bounded origin of rays, including point lights, spot lights, area lights, depth cameras, ultrasound sensors, or radar sensors, is stated to be modelable by an emitter-centric GACP or plane. For shadow rays to $360^\circ$3 lights, the paper proposes applying GRCA per light or cluster so that, per triangle, one predicts which lights can cast shadows on it without per-frame BVH updates. For solid-state LiDARs with arbitrary patterns, the suggested approach is to mask the $360^\circ$4 grid or use multiple co-located origins with varied orientations to approximate non-monotonic patterns.

The generality claim has clear limits. The reported geometric construction depends on rays originating from a known position, and the efficiency mechanism depends on predicting per-triangle channel and ray spans from that emitter-centric geometry. This suggests that the algorithm is most natural in acquisition or rendering settings with structured, origin-bounded ray families rather than fully unconstrained ray fields.

The paper’s summary states that GRCA replaces per-ray BVH traversal with per-triangle emitter-centric filtering, requires no per-frame acceleration-structure rebuild, bounds work by apparent area, and scales $360^\circ$5 in dynamic scenes. Within those boundaries, GRCA can be understood as a reformulation of ray casting in which the emitter is not merely the source of rays but the organizing geometry of the entire filtering process.