Step 1 — pick the face normal
Each of the six face sweeps carries a fixed (sx, sy, sz) — one of the six unit axis vectors, the outward normal of the face being swept (panel 01).
Consequence: every sample on the same face shares the same normal. The shading we compute below is therefore per-face, not per-pixel — this is exactly why each face appears as one ASCII character at a time.
Step 2 — rotate the normal with M
Apply the rotation rows from Panel 03 to (sx, sy, sz) — same coefficients, but with the +1.5 translate omitted. Translation moves points; it must never move directions. The rotated normal is N̂ = (x1, y1, z1) — and since the light has no x-component, the code never computes x1 at all.
Step 3 — project onto the light
The cube's fixed light direction is L̂ = (0, 2, 1) / √5 — mostly from above, leaning into the screen. Since both N̂ and L̂ are unit vectors, their dot product is exactly the cosine of the angle between them — the textbook Lambertian shading term:
|L| = √(0² + 2² + 1²) = √5 ≈ 2.236
L̂ = (0, 2/√5, 1/√5)
luminance = N̂ · L̂
= x1·0 + y1·(2/√5) + z1·(1/√5)
= (2·y1 + z1) / 2.236 ← matches code
Step 4 — clamp; the z-buffer does the hiding
If luminance ≤ 0 the face is tilted away from the light. The cube does not skip it — the value clamps to 0, and after panel 06's inverted ramp that face fills with the densest glyph @. Hidden surfaces are removed only by the 1/z depth buffer.
Step 1 — the normal varies smoothly
Every cup sample stores its own outward normal. On the walls it is radial — N = (±cosθ, 0, ±sinθ) (outer “+”, inner “−”); on the handle it sweeps the torus — N = (cosψ·cosφ, cosψ·sinφ, sinψ).
Consequence: unlike the cube, the normal changes at every sample — so shading is a continuous gradient, not one flat value per face.
Step 2 — rotate the normal with M
Apply the cup's rotation Ry(θ)·Rx(−45°) from Panel 03 to the surface normal — identical coefficients, +1.5 translate omitted (directions never translate). The rotated normal is N̂ = (x1, y1, z1).
Step 3 — project onto the light, scale by AO
The cup's fixed light is L̂ = (0, 1, −2) / √5 — from above, toward the viewer (the cube uses a different light: (0,2,1)/√5). Each sample also carries a baked ambient-occlusion factor — inner wall 0.30–0.60, floor 0.35, rim 1.25 — darkening the cavity and popping the lip:
luminance = (N̂ · L̂) · AO
= ((ny1 - 2·nz1) / 2.236) · AO
if (luminance > 1) luminance = 1 // rim boost can overshoot
Step 4 — depth buffer resolves the hollow
Outer wall, inner wall and floor can hit the same pixel; the 1/z depth buffer keeps the nearest, so you see into the cup through the rim while the closed bottom blocks any see-through.