# When brightness becomes a landscape

## Narrow-screen variant and text equivalent

`intensity-height-narrow.svg`, `.png` and `.pdf` provide a stacked layout; the PNG is 390 × 1,280 pixels. Panel A shows the same complete intensity grid. To keep labels legible, panel B shows its exact centre row (y = 0) as two lines rather than a small perspective surface: assigned display height and true zero height. No source values change. Run `python generate-narrow.py` after the main generator; it reads the existing CSV and asserts equality of intensity and display-height samples. This is an alternative view of the same calculation, not an additional experiment.

Three exact grid positions make the conclusion available without seeing the figure. Values below are rounded to six decimal places; coordinates, intensity and heights are dimensionless normalized quantities.

| Position x (at y = 0) | Intensity | Display height | True height |
|---|---:|---:|---:|
| −1 (left edge) | 0.152244 | 0.152244 | 0 |
| 0 (centre) | 0.800000 | 0.800000 | 0 |
| +1 (right edge) | 0.152244 | 0.152244 | 0 |

The centre is brighter and is therefore drawn higher by the chosen display rule. All three positions are on the same flat object. Brightness differences alone do not establish differences in physical height.

## Ready-to-use English explanation

A bright patch can be drawn as a hill without being a hill. Here the object is perfectly flat. Its centre reflects more light than its edges under uniform lighting. The first panel shows the resulting synthetic brightness field; the second draws those same values as vertical coordinates. Nothing measured the surface's physical height.

This display can make intensity differences easier to inspect. It does not, by itself, distinguish shape from differences in reflectance or illumination. A depth estimate needs an appropriate measurement method or justified assumptions, plus checks against independently known geometry.

**Concrete use today:** imagine inspecting a printed label. A surface plot can make a faint central printing variation conspicuous. Treating that plot as measured topography would incorrectly turn a lighter ink patch into a raised feature.

**Possible next step:** compare an image-based depth method on a flat printed target and a target with known relief. Check whether changing the pattern or lighting changes the reported geometry. This is a proposed comparison, not a performed experiment or a guarantee that all reconstruction methods behave alike.

## Caption and alt text

Caption: Synthetic intensity-to-height display. A flat surface with assigned reflectance produces the brightness pattern in A under an ideal normalized imaging model. Panel B plots exactly the same intensity values as display heights; its blue outline marks the true flat surface at zero. All axes are dimensionless normalized coordinates. The display is a visualization of intensity, not recovered physical depth, and contains no measured data.

Alt text: A bright central patch on a flat surface becomes a hill when its pixel intensities are used as height coordinates. The true surface remains flat at zero height.

## Model and reproduction

Run `python generate.py` with NumPy and Matplotlib. The script deterministically exports SVG, PNG, PDF, CSV and metadata. No random numbers, external data or network access are used.

For a 65 × 65 grid over x,y in [-1,1], assign

`I(x,y) = 0.15 + 0.65 exp(-((x/0.42)^2 + (y/0.60)^2))`.

The true height is zero throughout. The displayed height equals I by definition. Intensity is an assigned normalized reflectance signal under uniform illumination and an ideal linear detector with unit gain. There is no physical length calibration. The model omits shadows, perspective, blur, noise, specular reflection and image formation on curved surfaces. It is a counterexample to automatically interpreting intensity as geometry, not a forward model of a competing raised object and not a proof that depth recovery is always impossible.

The CSV contains 4,225 rows: normalized x, y, intensity, true height and display height. The analytic maximum is 0.8 at the origin; sampled minimum is approximately 0.15013949. Script assertions verify zero true height, equality of intensity and display height, and bounded intensity. No error bars or experimental uncertainty are invented.