# Horizontal envelopes and their limiting slope

The full editable proof is `horizontal-envelopes-and-their-limiting-slope.md`.
Its locators S1–S21 give the mathematical arguments, examples and complete
solutions. `figures/geometry.json` records the exact functions, constants,
domains and proof locators depicted in the two diagrams.

With Python, NumPy and Matplotlib available, run:

```shell
python make_figures114.py
```

This renders both PNG/SVG pairs. The sampled functions are exact examples;
the diagrams accompany the written proof rather than supply numerical evidence
for its general conclusions. The slab is the two-dimensional section with
`a=1`, `b=3`, `R=3` and `Psi(z,t)=z*z-t*t+9`.

The original material uses CC0 1.0. The DejaVu font notice is retained separately.
The classical source credit is Hörmander, *The Analysis of Linear Partial
Differential Operators II*, Lemma 16.1.6, printed page 308. The cited edition
is 1983, with a second revised printing in 1990 and a 2005 reprint.