Shadow-map bias is a physical length along the incident light ray, applied immediately before the depth comparison. Too short a length misclassifies an analytically lit receiver as shadowed—surface acne. Too long a length lifts the contact shadow of a thin occluder off its geometric foot—peter-panning. This note measures that trade-off on one map, one sun, and one constant-bias sweep.

Discrete depth encoding, winner-islands, and reverse-Z belong to Z-fighting is geometric compression plus quantization. That note defers the acne-versus-peter-pan distinction here and keeps fight-fraction metrics in its own scope. We fix a single shadow map and do not sweep a depth code. The shared geometric idea is that a depth failure can be stated as a length: there an isolatable interval in front of the camera; here a bias $b$ in metres along the sun vector.

Scene

The cover is a service-yard curb: asphalt apron, 160 mm concrete curb, poured wash ramp at $20^\circ$, and a 12 mm steel plate, in one $1280\times 720$ framebuffer. One directional sun at altitude $32^\circ$ feeds a single $512^2$ depth-compare map with an 8 m orthographic frustum. The shadow map is rendered once per frame. Across the ladder, bias is the only independent variable.

Beauty runs on Mesa llvmpipe in scene-referred linear Rec.709, then through the display path in Tone Mapping: Scene-Referred to Display-Referred: Khronos PBR Neutral with $e=1.00$, $F_{90}=0.04$, $K_s=0.76$, $K_d=0.15$, and an sRGB OETF on the CPU. Exposure is locked at $K=3.40$ for every bias sample. Neutral only assigns display codes; it does not enter the acne fraction or the contact gap.

At the operating point $b^\star=36\,\mathrm{mm}$ ($0.036000\,\mathrm{m}$), as $b$ runs from $0$ through $b^\star$ to $b_{\max}=120\,\mathrm{mm}$, ramp acne $A_{\mathrm{ramp}}$ falls from $0.509434$ through $0.018868$ to $0$ (lede abbreviations $0.509\to 0.019\to 0$), and apron acne $A_{\mathrm{apron}}$ falls from $0.482759$ to $0$ (lede $0.483\to 0$). The contact gap $G$ opens from $0$ through $15\,\mathrm{mm}$ to $91\,\mathrm{mm}$. Slope-scale at $k_s=1$, $b_c=0$ clears both acne fractions with $G=13\,\mathrm{mm}$. The smallest constant bias that also clears the ramp is $b_{\mathrm{match}}=40\,\mathrm{mm}$, where $G$ is already $20\,\mathrm{mm}$. The run prints 11 pass / 0 fail. Sweep rows, the full meter table, and gate notes live in the appendices.

Method: bias as a length on one map

A shadow map stores one depth per texel—the front-most surface from the light. The compare asks whether a shaded point lies farther from the light than that stored depth. On a surface tilted relative to the texel plane, continuous shaded points fall deeper than the winning fragment, so the compare fails. Acne is that mismatch: analytic visibility toward the sun is 1, yet the compare marks the sample shadowed.

Under this orthographic frustum, texel slope dominates the 24-bit depth step. Apron slope is $0.025005\,\mathrm{m}$ per shadow texel; the quantization step is $4.564028\times 10^{-7}\,\mathrm{m}$. The ratio is $54787.622775$ (lede $54788$), well above the gate floor of $50$. The acne in the cover sequence is therefore a texel-slope failure; winner-islands stay in the z-fighting note.

Bias $b$ subtracts a length along the ray toward the sun before the compare, sliding the shaded point toward the light. Large enough $b$ swallows the texel mismatch and acne falls. The same slide moves a thin occluder's shadow boundary away from its foot: the lit band between foot and shadow is the contact gap $G$, measured along the apron.

Two receivers keep the trade-off from collapsing to one slope. The wash ramp sees grazing incidence at $12^\circ$; the apron sees the sun altitude $32^\circ$. A constant bias large enough to kill ramp acne opens a large gap at the plate. Slope-scale applies a longer offset where the texel slope is steeper (ramp) and a shorter one on the apron, leaving the constant-bias curve.

The environment is right-handed, $Y$ up, in metres.

One light camera serves the whole ladder. Depth increases along $-s$. The ortho square has width $W_l=8\,\mathrm{m}$:

This run uses $n_l=-3.722583\,\mathrm{m}$ and $f_l=3.934586\,\mathrm{m}$, so $t=0.015625\,\mathrm{m}$ ($15.625\,\mathrm{mm}$). The 24-bit step is $4.564028\times 10^{-7}\,\mathrm{m}$. Texel slopes are $0.025005\,\mathrm{m}$ on the apron, $0.073510\,\mathrm{m}$ on the ramp, and $0.009764\,\mathrm{m}$ on the plate face.

CPU meters and the beauty shader share the compare. $\tilde d$ is the stored window $z$ of the winning fragment, decoded with the same $n_l$, $f_l$. Depth fetch is nearest-neighbor. Bias $b(n_p)$ depends on the receiver normal:

Constant sweep. With $k_s=0$, $b(n)=b$. The ladder is $b=0, 4, \ldots, 120\,\mathrm{mm}$ (31 samples). The map is built once; checksum a816326e2dbe2366 is invariant across every $b$ and after the last beauty. Bias changes only the compare, never the stored depths.

Slope-scale, one marked point. Here $b_c=0$ and $k_s=1$. With light-image basis $e_1$, $e_2$,

$\Delta_{\mathrm{tex}}$ is metres of light depth across one texel: $73.510\,\mathrm{mm}$ on the ramp, $25.005\,\mathrm{mm}$ on the apron. Neutral's display shoulder $K_s=0.76$ is unrelated to this geometric $k_s$.

Acne fraction. An $8\,\mathrm{mm}$ grid over a receiver; a sample counts only if analytic visibility toward the sun is exactly 1:

Both regions are fully analytically lit: ramp $2703/2703$, apron $3654/3654$.

Contact gap. The plate's downstream face sits at $x_c=1$ with thickness $\tau=12\,\mathrm{mm}$ toward the sun. Visible ground is majority-filtered over a $31\,\mathrm{mm}$ window for boundary finding only—never for $A$. $G$ is the start of the longest shadowed run minus $x_c$. At $b=0$, the raw compare shadows $451/451$ transect samples downstream of the plate, so a closed gap is a real contact shadow, not a missing caster.

Ideal-map oracle. For this plate and sun altitude,

The knee sits at $b=\tau/\cos\alpha$. Across five locking biases the unfiltered ideal traversal tracks $G_{\mathrm{ideal}}$ within $1.5\,\mathrm{mm}$; peak error is $0.647\,\mathrm{mm}$. The raster gap is measured, not fitted to the oracle.

Operating point. With $A_0=A_{\mathrm{ramp}}(0)$ and $G_{\mathrm{hi}}=\max(G(b_{\max}),0)$,

$b^\star=\arg\min J$ (ties take the smaller $b$). This run isolates $b^\star=36\,\mathrm{mm}$ at $J=0.202$, interior to the swept extremes. The sweep was not extrapolated past $b_{\max}$.

Photographs are scene-referred linear Rec.709 through Neutral at $K=3.40$, $e=1.00$. The sweep plot and metrics snapshot are authored sRGB, bypassing Neutral: photographs show the curb; tables and curves carry $A$ and $G$.

Discussion: what if bias is short, long, or slope-scaled?

What if you stop at the interior minimum of $J$?

The cover is that operating point: $1280\times 720$, $b^\star=36\,\mathrm{mm}$. A faint residue remains on the ramp ($A_{\mathrm{ramp}}=0.018868$); the apron is clear ($A_{\mathrm{apron}}=0$); the annotated gap is the measured $15\,\mathrm{mm}$ along the apron, not a JPEG luminance cue. Apron acne is already gone by $16\,\mathrm{mm}$ while the ramp still shows $0.283019$ with $G=0$. At $b^\star$ the gap has just opened; ramp acne nulls only at $40\,\mathrm{mm}$ with $G=20\,\mathrm{mm}$. The operating point sits where ramp acne is mostly gone and detachment has only begun.

What if $b=0$?

Same camera and the same shadow map, compare at $b=0$. Ramp and apron self-shadow as high-frequency stripes: lit receivers marked shadowed. $A_{\mathrm{ramp}}=0.509434$, $A_{\mathrm{apron}}=0.482759$, $G=0$. Contact is correct; about half of every lit grid is a false shadow.

What if $b=b_{\max}=120\,\mathrm{mm}$?

Same camera and map at the long extreme. Both acne fractions are $0$, but the plate's contact shadow has left the foot: $G=91\,\mathrm{mm}$ ($0.091\,\mathrm{m}$) along the apron. The curb's own shadow remains a separate band.

What if slope-scale marks one point instead?

Slope-scale at $k_s=1$, $b_c=0$ is one marked point in the same space—not a second ladder. It clears $A_{\mathrm{ramp}}=0$ and $A_{\mathrm{apron}}=0$ with $G=13\,\mathrm{mm}$ ($0.013000\,\mathrm{m}$), shorter than $G(b_{\mathrm{match}})=20\,\mathrm{mm}$ at matched ramp acne. The claim is that inequality, not a product ranking of filtered shadows.

The teaching pin plots $A_{\mathrm{ramp}}$, $A_{\mathrm{apron}}$, and $G$ against constant bias from 0 to 120 mm, marks $b^\star$, and callouts the slope-scale triplet. Eight contact strips sit under the plot at 0, 24, 48, 72, 96, and 120 mm, then $b^\star$, then slope-scale; the gap staircase is left visible. Selected constant-bias rows are in Appendix A; the full 31-sample ladder is what the figure maps.

Limits: what this run can and cannot claim

Can claim. On this OSMesa / llvmpipe build (core 3.3 request), one directional sun, one $512^2$ nearest-texel shadow map, and one constant-bias sweep from 0 to 120 mm produced falling acne on both receivers and a rising contact gap at a 12 mm plate. The interior minimum of the normalised sum of ramp acne and gap is $b^\star=36\,\mathrm{mm}$ ($J=0.202$). Slope-scale at $k_s=1$ clears ramp acne at a shorter gap than the matching constant sample (13 mm against 20 mm). The regime ratio places the failure in the texel-slope regime. The run prints 11 pass / 0 fail.

Cannot claim. A discrete GPU, wavefront, or frame-time budget. A filtered shadow, cascade, or soft penumbra. A measured polygon offset, normal-offset bias, or receiver-plane depth. Acne or gap read from a JPEG. A second depth codebook, fight-fraction, or reverse-Z curve. A ranking of this bias length against a product shadow stack.

Honesty notes that stay out of the main argument: the meter uses a 32-bit float color copy of winning fragment $z$ (attachment DEPTH_COMPONENT24; window depth bits 0); sixteen apron depth-texture texels match that copy; sampler2DShadow agrees on 8/8 stable points; probes at $b=0$ and $b^\star$ match the CPU test at 16/16 locations. Gate 3 apron median $d-\tilde d=-0.000350\,\mathrm{m}$ ($-0.35\,\mathrm{mm}$) sits under the unquantized floor of 0 because the 8 mm lattice is sunward of $512^2$ texel centers (projected residual $-0.349\,\mathrm{mm}$); the gate accepts any median within $-2\,\mathrm{mm}$ (an unrendered apron would be about $-3.3\,\mathrm{m}$). Specular on the lit plate face is masked by the binary shadow and does not enter $A$ or $G$. The 31 mm majority filter is boundary-only; pass requires $G(b_{\max})>G(0)$, and $G$ rises monotonically from 0 to $0.091\,\mathrm{m}$ even if individual staircase steps plateau. One map, one checksum—mismatch stops the run. The ideal-map oracle checks metric code; the raster curve checks the shadow map.

Out of scope

Percentage-closer and contact-hardening filters, cascades, atlas packing, a second light, an animated sun, and a soft penumbra. Reverse-Z, a depth-format sweep, and the z-fighting note's fight-fractions. Vendor polygon-offset scales and hierarchical-Z. Reading $A$ or $G$ from a beauty JPEG.

The cover is the operating point. The sweep is the teaching figure. Bias is a geometric length. Dense meters follow.

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Appendix A — Constant-bias sweep and lede map

Metrics are CPU double precision before Neutral. The 31-sample ladder is $b=0, 4, \ldots, 120\,\mathrm{mm}$; selected rows:

Lede abbreviations map to this run's unrounded tokens:

Contact strips on the teaching pin: 0, 24, 48, 72, 96, 120 mm, then $b^\star$, then slope-scale. Sweep plot authored sRGB of the curves and strips.

Appendix B — Full meter table

Renderer row: llvmpipe, OSMesa core 3.3 request; RGBA32F copy of winning fragment $z$; attachment DEPTH_COMPONENT24; window depth bits 0; sixteen apron texels match the color copy; hardware compare 8/8; encode $K=3.40$, Neutral $e=1.00$, sRGB OETF on the CPU; map checksum a816326e2dbe2366.

Appendix C — Assertions and formula cheat sheet

Printed: 11 pass / 0 fail. The eleven checks cover window depth, the regime ratio, the apron median, lit and covered regions, the ideal-map gap, probes and the checksum, frame projection, falling acne, a rising gap, an interior operating point, and the slope-scale gap.

Honesty checklist retained from the pre-appendix draft (items 1–7): (1) CPU depth verification; (2) Gate 3 apron median; (3) display tone-mapping constraints; (4) discreteness of $G$; (5) slope-scale is one marked point; (6) one map, one checksum; (7) rasterization is llvmpipe's.

b*           = 0.036 m (36 mm),  J = 0.202
A_ramp       = 0.509434 / 0.018868 / 0     at 0, b*, b_max
A_apron      = 0.482759 / 0
G            = 0 / 0.015 / 0.091 m
slope        ks=1, bc=0 → A=0/0, G=0.013 m
b_match      = 0.040 m, G=0.020 m
regime ratio = 54787.622775  (lede 54788)
apron median = -0.000350 m
checksum     = a816326e2dbe2366
asserts      = 11 pass / 0 fail