Shadow Accrual Maps in Urban and Gravitational Contexts

• Shadow Accrual Maps are dual-purpose frameworks that aggregate discrete shadow contributions to enable efficient city-scale shadow analysis and quantify black-hole shadow responses.
• In urban planning, they leverage solar motion coherence to reduce computational cost while accurately estimating gross and continuous shadow durations across complex cityscapes.
• In gravitational theory, they employ linear kernel transforms to assess the sensitivity of black-hole shadows to matter perturbations, offering insights into critical scaling and optical observables.

Shadow Accrual Maps designate two distinct constructions in recent literature. In computer graphics and urban analysis, they provide an efficient way to accumulate, analyze, and visualize city-scale shadows over user-defined periods without brute-force sampling at dense time steps. In gravitational theory, the same term denotes a linear kernel transform that accrues compact, conserved thermodynamic or effective-medium perturbations of $\Delta T^{\mu}{}_{\nu}$ into shifts of black-hole shadow observables such as the shadow radius and photon-sphere frequency. The shared expression therefore refers not to a single methodology, but to two mathematically structured accrual frameworks operating in different domains: temporal accumulation of urban direct-sun shadow occupancy, and linear-response propagation from matter susceptibilities to geometric optics observables (Miranda et al., 2019, Wu et al., 24 Jan 2026).

1. Terminological scope and conceptual commonality

The city-scale formulation was introduced in "Shadow Accrual Maps: Efficient Accumulation of City-Scale Shadows Over Time" and published in 2019. It addresses the problem of determining the effects of building shadows across different periods in a year, with applications including thermal comfort, vegetation growth, planning and zoning, right-to-light, and solar access. Its core idea is to exploit short-interval coherence in solar motion so that shadow movement can be tracked between interval endpoints rather than sampled minute by minute (Miranda et al., 2019).

The black-hole formulation appeared in 2026 in "From Thermodynamic Criticality to Geometric Criticality: A Linear Kernel Map from Matter Susceptibilities to Black-Hole Shadows." There, a Shadow Accrual Map is an explicit linear map from compact, conserved thermodynamic or effective-medium perturbations of the stress-energy tensor to the metric response, and from there to the shadow radius and photon-sphere frequency. The response is written through $L^{1}$-bounded kernels in a piecewise "local + tail" form (Wu et al., 24 Jan 2026).

The common conceptual element is accrual. In the urban setting, accrual refers to integrating or summing direct-shadow occupancy over time on receivers such as ground planes, parks, and façades. In the gravitational setting, accrual refers to integrating radial source perturbations against kernels that encode sensitivity of the shadow observable. This suggests that the phrase names a structural pattern—aggregation by a map from elementary shadow-generating contributions to a cumulative observable—rather than a domain-specific algorithm.

2. Urban Shadow Accrual Maps: mathematical formulation and solar-geometry model

In the urban literature, the central quantity is the direct-shadow indicator $I(x,t)\in\{0,1\}$ at location $x$ and time $t$. Accrual over a time set $T$ with temporal weights $w(t)$ is defined by

$S(x)=\int_{t\in T} w(t)\,I(x,t)\,dt,$

with discrete approximation

$$S(x)=\sum_{k=1}^{K} w_k\,I(x,t_k), \qquad \widehat{S}(x)=\frac{S(x)}{\sum_{k=1}^{K} w_k}.$$

The paper distinguishes gross shadow, meaning total time or fraction in shadow over $T$, from continuous shadow, meaning the longest contiguous duration in shadow over $L^{1}$0 (Miranda et al., 2019).

The computational difficulty arises because a naïve baseline computes shadows for many time steps. The paper gives the example that 3 hours per day over a week at 1-minute resolution implies 1260 sun directions. At city scale, with thousands of buildings and millions of triangles, such dense temporal sampling is prohibitively expensive and impedes interactivity. The key observation is that over short intervals of approximately 1 hour, the sun’s direction changes nearly linearly, so shadow occupancy can be obtained by tracking boundary motion and effectively “sweeping” shadows across the receiver.

The solar model is stated in conventional astronomical terms. For day-of-year $L^{1}$1, approximate declination is

$L^{1}$2

and with local solar time $L^{1}$3, the hour angle is

$L^{1}$4

Given latitude $L^{1}$5, solar zenith is

$L^{1}$6

elevation is $L^{1}$7, and the sun direction vector is

$L^{1}$8

A notable empirical result is that, in New York City, the cosine similarity between actual and interpolated sun directions is approximately $L^{1}$9 for intervals up to 60 minutes, with mean approximately $I(x,t)\in\{0,1\}$0 and $I(x,t)\in\{0,1\}$1 at 60 minutes; the error grows beyond an hour. The method therefore segments analyses into short intervals and uses linear interpolation within each one.

Geometrically, hard shadows are cast by directional light onto a receiver that may be a ground plane $I(x,t)\in\{0,1\}$2 or an arbitrary plane. For a scene point $I(x,t)\in\{0,1\}$3 on an occluder, its shadow projection on the receiver along $I(x,t)\in\{0,1\}$4 is obtained from the parametric line

$I(x,t)\in\{0,1\}$5

The swept region $I(x,t)\in\{0,1\}$6 traces a curve that is approximately straight under linear sun motion. Rasterization of this sweep yields occupancy across the interval, and occlusions are resolved by depth tests that keep the farthest depth per texel per slice.

3. Extensions to shadow maps and ray tracing

The 2019 paper develops two extensions: a shadow-map-based SAM and a ray-tracing-based Inverse Accrual Maps (IAM) method. In the shadow-map formulation, conventional shadow mapping would compute a separate shadow map per time step, whereas SAM computes a single 3D texture whose slices correspond to all steps in a short interval in one pass. For each scene point, the method projects to $I(x,t)\in\{0,1\}$7 and $I(x,t)\in\{0,1\}$8 on a shadow plane orthogonal to the start direction $I(x,t)\in\{0,1\}$9, and for intermediate slice $x$0 uses

$x$1

where $x$2 and $x$3. The paper states a theorem that, under linear sun movement, the 3D accrual texture is equivalent to computing $x$4 shadow maps at each time step; SAM consolidates $x$5 passes into one pass with no quality loss (Miranda et al., 2019).

The shadow-map implementation uses image atomics in OpenGL 4.3 to store the maximum depth per texel per slice in one rendering pass. Trapezoidal shadow map warping improves texel density near the view frustum, and a depth bias mitigates shadow acne. Gross and continuous shadow are then computed from per-pixel bitvectors over slices.

IAM addresses flat receivers such as parks and sidewalks. Its motivation is that ray tracing yields higher-quality results and avoids shadow map aliasing. The method traces, for each receiver pixel at $x$6, rays along $x$7 to record up to $x$8 intersection points, each treated as a potential shadow source. The mapping from $x$9 to $t$0 is stored in a 3D texture whose slice $t$1 corresponds to the $t$2-th closest source. To capture stretching and contracting shadow behavior, the method computes IAM both forward $t$3 and reverse $t$4. Accrual then proceeds by line rasterization: if a subsegment corresponding to time step $t$5 crosses a pixel, the bit $t$6 is set to 1.

The source-level tradeoff is explicit. Larger $t$7 increases accuracy at dawn and dusk, when many occluders may matter, but increases cost. The paper reports that $t$8 suffices with mean area error less than $t$9 and maximum error approximately $T$0, with the worst cases at dawn and dusk. This places IAM as a higher-fidelity method for flat receivers, while SAM remains the interactive method for broader urban exploration.

4. Implementation, validation, and urban-planning use

The methods were implemented in the interactive visual analysis system Shadow Profiler, targeted at city planners and architects. The reported architecture combines C++, OpenGL 4.3 for GPU shadow accrual maps and image atomics, OpenCL 1.2 for bit operations and progressive accumulation, GLSL shaders for IAM ray traversal, and a 3D grid acceleration structure for ray tracing. The interface includes a 3D map widget, date and time selection, gross or continuous accumulation type, averaging over days, progressive computation and visualization, polygonal ROI selection, and a divergent colormap for positive or negative differences in minutes. Two modes are defined: exploration, based on SAM, and analysis, based on IAM (Miranda et al., 2019).

The Manhattan case study uses OpenStreetMap geometry extruded to heights, with approximately 1.5 million triangles across 43k buildings. Experiments were conducted at resolutions from $T$1 to $T$2, with shadow-map resolutions up to $T$3, on a workstation with Intel Xeon E5-2620, 128 GB RAM, and an NVIDIA GTX 1080 8 GB. At $T$4, SAM uses approximately 240 MB and IAM approximately 56 MB; memory scales proportionally with resolution, reaching 960 MB for SAM and 224 MB for IAM at $T$5.

The reported validation covers linearity of sun motion, clustering of directions, source truncation in IAM, and performance. With a direction graph using bins bounded by $T$6, mean direction similarity is $T$7 with $T$8. Shadow area computed with cluster representatives has mean absolute error approximately $T$9 of total area, median approximately $w(t)$0, and approximately $w(t)$1 of pixels misclassified as shadow or non-shadow on average. For performance over 6 hours per day, 10 random positions, and 20 random days, SAM is approximately $w(t)$2 faster than minute-sampled shadow maps, and IAM is approximately $w(t)$3 faster than minute-sampled ray tracing. For annual accumulation over 6 hours per day, the direction graph reduces hourly-interval computations from 2190 to 299 edges, more than $w(t)$4 fewer, and end-to-end speedups over $w(t)$5 are observed with increasing time periods.

The planning-oriented case studies center on Manhattan. For Central Park skyscraper scenarios, the baseline impact of the Time Warner Center is described as localized, while new supertall towers cast widespread long shadows but keep the net increase above 30 minutes concentrated. An alternate scenario with shorter, wider towers of equal total area yields stronger near-base impact, especially under Boston-style thresholds of at least 60 minutes. The paper also reports that most Manhattan neighborhoods average more than 4 hours per day in shadow due to density, while wide streets, plazas, and parks fare better. Weighted “shadow score” maps are mostly negative overall in dense areas, with pockets of positive scores in lower-density regions. This suggests that the method supports comparisons between summer shade benefits and winter shadow penalties in planning practice.

5. Assumptions, limitations, and relation to adjacent urban methods

The urban formulation makes several assumptions explicit. It models hard shadows only; penumbrae and soft-shadow integration are not included. It models direct sunlight only; diffuse skylight, atmospheric scattering, and refraction are not considered. IAM assumes flat receivers, so arbitrary topography and vertical receivers require extensions, including hybrid approaches or Monte Carlo. Accuracy declines at dawn and dusk because of rapid shadow motion and the presence of multiple occluders when $w(t)$6 is small. The paper also notes sensitivity to user-defined temporal weights $w(t)$7, so planning conclusions must be contextualized by the weighting scheme (Miranda et al., 2019).

The method is also conditioned by data quality and implementation choices. Building geometry errors, missing heights, and overhangs affect accuracy. Shadow-map artifacts require warping and bias tuning. The empirical validation of 60-minute segmentation is specific to New York City, and other latitudes or climates may require parameter tuning. Practical guidance in the paper therefore includes hourly intervals per short segment, direction clustering with at most $w(t)$8 bins for multi-day or seasonal analyses, and spatial resolutions around $w(t)$9 to $S(x)=\int_{t\in T} w(t)\,I(x,t)\,dt,$0 for city-scale exploration. For parks and streets, IAM at $S(x)=\int_{t\in T} w(t)\,I(x,t)\,dt,$1 with $S(x)=\int_{t\in T} w(t)\,I(x,t)\,dt,$2 is recommended in the reported experiments.

Relative to earlier approaches, the contrast is methodological. Traditional shadow mapping requires separate shadow maps per time step, while SAM uses a single pass to populate all slices by exploiting sun-motion coherence. Minute-by-minute ray tracing is expensive, while IAM maps shadow-source events and rasterizes lines instead of sampling every minute. Voxel occupancy and precomputed occlusion intervals involve heavier precomputation and are less flexible under scene changes. Fernando’s occlusion interval maps are identified as prior art, but SAM and IAM are event-based and reuse direction bins across days. Deep shadow maps, adaptive or warped shadow maps, and shadow volumes remain per-light computations; the paper locates SAM’s novelty in leveraging time-coherent sun movement to accelerate temporal accumulation.

A common misconception is that SAM is a general daylighting model. The cited formulation is narrower: it is a direct-sun, hard-shadow accumulation framework. Another misconception is that IAM is a universal replacement for shadow maps. In the paper’s own system design, SAM is the interactive approach for broad regions, whereas IAM is the higher-fidelity analysis mode for flat receivers.

6. Shadow Accrual Map in black-hole optics and critical phenomena

In the 2026 gravitational literature, the term denotes a linear kernel functional rather than a rendering algorithm. The background is a static, spherically symmetric spacetime in areal-radius gauge with

$S(x)=\int_{t\in T} w(t)\,I(x,t)\,dt,$3

where

$S(x)=\int_{t\in T} w(t)\,I(x,t)\,dt,$4

Linearization is performed around a vacuum background with $S(x)=\int_{t\in T} w(t)\,I(x,t)\,dt,$5, $S(x)=\int_{t\in T} w(t)\,I(x,t)\,dt,$6, and $S(x)=\int_{t\in T} w(t)\,I(x,t)\,dt,$7. Matter perturbations are diagonal,

$S(x)=\int_{t\in T} w(t)\,I(x,t)\,dt,$8

with compact support in a shell $S(x)=\int_{t\in T} w(t)\,I(x,t)\,dt,$9, smooth profiles, and the conservation constraint $$S(x)=\sum_{k=1}^{K} w_k\,I(x,t_k), \qquad \widehat{S}(x)=\frac{S(x)}{\sum_{k=1}^{K} w_k}.$$0 (Wu et al., 24 Jan 2026).

The optical observables are defined through the photon sphere. An unstable circular photon orbit satisfies

$$S(x)=\sum_{k=1}^{K} w_k\,I(x,t_k), \qquad \widehat{S}(x)=\frac{S(x)}{\sum_{k=1}^{K} w_k}.$$1

and the corresponding shadow radius for a static observer at infinity is

$$S(x)=\sum_{k=1}^{K} w_k\,I(x,t_k), \qquad \widehat{S}(x)=\frac{S(x)}{\sum_{k=1}^{K} w_k}.$$2

The photon-sphere frequency is

$$S(x)=\sum_{k=1}^{K} w_k\,I(x,t_k), \qquad \widehat{S}(x)=\frac{S(x)}{\sum_{k=1}^{K} w_k}.$$3

Linearizing Einstein’s equations yields

$$S(x)=\sum_{k=1}^{K} w_k\,I(x,t_k), \qquad \widehat{S}(x)=\frac{S(x)}{\sum_{k=1}^{K} w_k}.$$4

After integration by parts, the metric response takes the integral-kernel form

$$S(x)=\sum_{k=1}^{K} w_k\,I(x,t_k), \qquad \widehat{S}(x)=\frac{S(x)}{\sum_{k=1}^{K} w_k}.$$5

Propagation to the shadow shift then gives the Shadow Accrual Map

$$S(x)=\sum_{k=1}^{K} w_k\,I(x,t_k), \qquad \widehat{S}(x)=\frac{S(x)}{\sum_{k=1}^{K} w_k}.$$6

The kernels are written in a "local + tail" decomposition. Local contributions arise from mass loading and from the pressure term entering through $$S(x)=\sum_{k=1}^{K} w_k\,I(x,t_k), \qquad \widehat{S}(x)=\frac{S(x)}{\sum_{k=1}^{K} w_k}.$$7 for $$S(x)=\sum_{k=1}^{K} w_k\,I(x,t_k), \qquad \widehat{S}(x)=\frac{S(x)}{\sum_{k=1}^{K} w_k}.$$8, while tail contributions contain $$S(x)=\sum_{k=1}^{K} w_k\,I(x,t_k), \qquad \widehat{S}(x)=\frac{S(x)}{\sum_{k=1}^{K} w_k}.$$9 or $T$0 and encode sensitivity to the photon-sphere vicinity and to far-zone shells, including AdS tails. For compact sources, the kernels are absolutely integrable, and the paper states the $T$1-bound

$T$2

The principal theoretical result concerns critical scaling. If the thermodynamic or effective-medium susceptibilities satisfy power-law divergence with analytic corrections on fixed compact support, then dominated convergence transfers the thermodynamic exponent to the geometric susceptibility: $T$3 Because $T$4, the frequency channel obeys the universal amplitude ratio

$T$5

This is not a visual shadow map in the graphics sense. It is a sensitivity map across radius, assigning each shell a weight in the linear response of the black-hole shadow observable.

7. Numerical pipeline, asymptotic control, and scope of the black-hole construction

The 2026 paper presents a reproducible numerical pipeline on a finite radial domain $T$6, with fixed compact support $T$7 across $T$8. A smooth bump is chosen for $T$9, and an anisotropy closure may be imposed through

$L^{1}$00

Conservation is enforced by solving the two-point boundary value problem

$L^{1}$01

using shooting from $L^{1}$02 to both edges with bisection or secant updates, attaining relative $L^{1}$03 residuals smaller than $L^{1}$04. The far-zone diagnostics check $L^{1}$05 over the outer 20% of the domain and monitor

$L^{1}$06

for slopes $L^{1}$07 in the asymptotically flat case and $L^{1}$08 in AdS on log–log plots (Wu et al., 24 Jan 2026).

The photon sphere is extracted by fitting a smoothing spline to $L^{1}$09, solving $L^{1}$10, and evaluating $L^{1}$11 with $L^{1}$12 interpolation. Susceptibility is estimated via spline differentiation of $L^{1}$13, and scaling fits use OLS or Theil–Sen, with grid-refinement convergence $L^{1}$14 for $L^{1}$15 and domain-truncation convergence $L^{1}$16 in the flat case or $L^{1}$17 in AdS.

The physical interpretation is radial sensitivity. Near the photon sphere, local terms reflect direct changes in the depth and slope of the optical potential at $L^{1}$18. In AdS, far-zone tails are suppressed as $L^{1}$19, so shells with $L^{1}$20 have strongly bounded impact on $L^{1}$21. The paper gives an explicit thin-shell example with $L^{1}$22 and $L^{1}$23, showing that $L^{1}$24 is approximately a weighted sum of $L^{1}$25 and $L^{1}$26. It also gives outside-support bounds in AdS with constants depending only on smooth background data at $L^{1}$27 and shell geometry.

The scope is sharply delimited. The regime of validity is static, spherically symmetric backgrounds, linear metric response, compact conserved sources with smooth profiles, and support $L^{1}$28 independent of the control parameter. The paper lists extensions to slowly rotating spacetimes, non-spherical perturbations, different boundary conditions, higher-derivative gravity, charged or regular black holes, and anisotropic effective media. It also notes caveats: strongly nonlinear regimes require higher-order terms, rapidly spinning backgrounds complicate treatment of photon rings and caustic structure, and fine-tuned cancellations in $L^{1}$29 can mask the leading susceptibility and elevate subleading corrections.

The two research programs therefore share a name but not an application domain. In one case, Shadow Accrual Maps reformulate temporal shadow accumulation in urban environments by exploiting motion coherence of the sun; in the other, they reformulate geometric shadow response as a linear kernel transform from matter perturbations. The commonality is formal accrual, while the concrete objects being accrued—time-dependent urban shadow occupancy versus radial source sensitivity of black-hole shadows—are fundamentally different.