ASCII Donut (Rotating Torus)

Loading donut model...

Presets

Torus and motion

Light and view

Strawberry glaze Sprinkles ASCII inset (exact 2D algorithm) Rest on the plate (display mode)

Live readouts

Graphs

Lambert light on the tube: max(0, n.L) vs tube angle theta (current orientation)

ASCII brightness histogram (12-level ramp, live)

Lit fraction of the picture, last 12 s (periodic with the spin)

Frame time (ms), green line = 16.7 ms

Numerical checks

running...

How to use

Drag to orbit the camera, wheel or pinch to zoom. Change R1 (tube) and R2 (ring) to turn the donut into a thin hoop or a fat bagel; the baked textures follow because they live in the (theta, phi) parameter space. Switch the View to see what the maths sees: pure Lambert light, the 12-level ASCII ramp, the z-buffer depth, surface normals or the parameter grid. The inset runs the original ASCII algorithm with the same angles, so both versions always agree.

The science

A torus is the surface swept by a circle of radius R1 whose centre travels around a circle of radius R2 in the xy-plane:

P(theta,phi) = ((R2 + R1 cos theta) cos phi, (R2 + R1 cos theta) sin phi, R1 sin theta) n(theta,phi) = (cos theta cos phi, cos theta sin phi, sin theta)

Area A = 4 pi^2 R1 R2 and volume V = 2 pi^2 R1^2 R2 (Pappus). The Gaussian curvature K = cos theta / (R1 (R2 + R1 cos theta)) is positive on the outer half, negative on the inner half and integrates to zero over the whole surface (Gauss-Bonnet, Euler characteristic 0). With a point at distance d along the view axis the perspective projection is x' = K1 x / (K2 + z); the ASCII version keeps only the nearest sample per cell (z-buffer) and shades with the Lambert term max(0, n.L), L = (0, 1, 1)/sqrt 2 by default.

Here the same surface is a real triangle mesh modelled in Blender (256 x 128 quads, UV = (phi, theta)). The mesh, the baked albedo and normal maps, the glaze mask, the plate and the wood table come from Blender 5.2 (Cycles bake, Draco GLB); Three.js recomputes the vertex positions and analytic normals from the UVs every time R1 or R2 changes, so the textures stay glued to the surface. Light is a directional key light plus an image-based environment; the glaze is a clear-coated physical material.

Simplifications: the donut is a rigid, perfectly symmetric torus (no sagging, no oil film, no real subsurface scattering); sprinkles are decorative; when R1 > R2 the surface self-intersects (a spindle torus) and the volume/area formulas count the overlapping sheets. The Lambert mean of 1/4 assumes a convex-like uniform orientation average and ignores self-shadowing.

What the checks mean

Area sums the triangle areas of the loaded mesh and compares with 4 pi^2 R1 R2. Volume uses the divergence theorem on the same mesh. Normals compares analytic normals with face normals from cross products. Gauss-Bonnet adds K dA over the mesh and expects 0. Template compares the original Blender vertex positions with the positions rebuilt from the UVs, which proves that textures and geometry line up. Lambert mean averages max(0, n.L) over random orientations of random surface points and expects 1/4. The silhouette experiment counts covered cells of an orthographic projection and compares with 4 pi R1 R2 (face-on) and 4 R1 R2 + pi R1^2 (edge-on).

Keyboard and accessibility

Space pauses, R resets, C re-runs the checks (while the page has focus).
All controls are native inputs with labels; the stage is described by the canvas label.