A Topological Picture Book, Rendered

The drafting is made of strokes, not pixels. Silhouettes are the zero group of n·v connected the mesh (found pinch interpolated normals, truthful they concatenation into soft curves), boundaries and the double curve of an immersion are added arsenic further chains, and each of them are drawn arsenic tapered ribbons pinch a broad-nib pen model, weight that grows connected the protector broadside and toward the viewer, and coherent manus wobble. Visibility is settled connected the GPU: a hidden-line walk draws the occluded parts dashed, and a one-sided insubstantial halo nether each adjacent contour cuts the lines down it. Hatching is simply a group of streamlines of the principal-curvature statement field, traced astatine build clip in 3 nested densities (the 2nd on the different main direction, for cross-hatching); reside selects which family is inked and wherever each changeable feathers out. Highlights enactment bare paper. The main directions are ordered by signed curvature, not magnitude, truthful the 2 families stay continuous crossed the loci wherever κ₁ = −κ₂. Contour chains are lightly smoothed earlier inking. Strokes overshoot their ends a small successful the mode of sketchy statement rendering; the ink pooling astatine changeable starts and the ragged bleed into the insubstantial atom are hand-tuned effects of this page (a widened ribbon whose fringe is gated by a fibre-like noise), not taken from a paper. Labels are hand-lettered, pinned to points of the surface pinch a leader line, and dimmed erstwhile their constituent is hidden.

G. K. Francis, A Topological Picture Book, Springer, 1987 — the style target: contour drafting pinch cusps, double curves, hidden lines, and sparing hatched bands.

P. Bénard, A. Hertzmann, “Line Drawings from 3D Models: A Tutorial,” Foundations and Trends successful Computer Graphics and Vision 11(1–2), 2019 — the contour pipeline: soft silhouettes arsenic n·v = 0, chaining, visibility, stylization.

A. Hertzmann, “Introduction to 3D Non-Photorealistic Rendering: Silhouettes and Outlines,” SIGGRAPH 99 Course Notes — silhouettes from interpolated vertex normals (marching-triangles connected n·v).

A. Hertzmann, D. Zorin, “Illustrating Smooth Surfaces,” SIGGRAPH 2000, pp. 517–526 — hatching on main curvature directions, cross-hatching only successful acheronian regions, blank highlights, undercuts.

B. Jobard, W. Lefer, “Creating Evenly-Spaced Streamlines of Arbitrary Density,” Visualization successful Scientific Computing, 1997 — the separation-distance norm utilized to trace the hatch streamlines.

A. Appel, F. J. Rohlf, A. J. Stein, “The Haloed Line Effect for Hidden Line Elimination,” SIGGRAPH 1979 — the insubstantial haloes astatine statement crossings.

J. D. Northrup, L. Markosian, “Artistic Silhouettes: A Hybrid Approach,” NPAR 2000 — chaining silhouette segments and rendering them arsenic stylized strokes pinch tapering and width variation.

T. Strothotte, B. Preim, A. Raab, J. Schumann, D. R. Forsey, “How to Render Frames and Influence People,” Computer Graphics Forum 13(3) (Eurographics 1994) — sketch-like statement rendering: lines that overshoot their endpoints and wiggle, drawn pinch a pen exemplary whose width varies on the stroke.

M. P. Salisbury, S. E. Anderson, R. Barzel, D. H. Salesin, “Interactive Pen-and-Ink Illustration,” SIGGRAPH 1994, pp. 101–108 — changeable textures and the placement of hand-character strokes to scope a target tone.

E. Praun, H. Hoppe, M. Webb, A. Finkelstein, “Real-Time Hatching,” SIGGRAPH 2001 — nested reside levels of hatching; present realized pinch object-space strokes truthful the hatching ne'er swims.

G. Winkenbach, D. H. Salesin, “Computer-Generated Pen-and-Ink Illustration,” SIGGRAPH 1994, pp. 91–100 — reside by changeable density and thickness; changeable textures.

G. Elber, “Line Art Rendering via a Coverage of Isoparametric Curves,” IEEE TVCG 1(3), 1995 — hatching on isoparametric curves (the parameter-line stripes mode).

T. Saito, T. Takahashi, “Comprehensible Rendering of 3-D Shapes,” SIGGRAPH 1990, pp. 197–206 — separator extraction from normal and extent buffers (the optional Sobel separator filter).

W. E. Lorensen, H. E. Cline, “Marching Cubes,” SIGGRAPH 1987; A. Doi, A. Koide, “An Efficient Method of Triangulating Equi-Valued Surfaces by Using Tetrahedral Cells,” IEICE Trans. E74(1), 1991 — the implicit surfaces are polygonized by the tetrahedral variant.

T. Möller, B. Trumbore, “Fast, Minimum Storage Ray-Triangle Intersection,” J. Graphics Tools 2(1), 1997 — the segment–triangle trial down the double-curve computation.

R. Kusner, “Conformal Geometry and Complete Minimal Surfaces,” Bull. Amer. Math. Soc. 17(2), 1987 — root of the Bryant–Kusner parametrization utilized for Boy’s aboveground (checked present numerically: antipodal bound gluing and threefold symmetry clasp to instrumentality precision). The general-p shape utilized by kusner() has denominator w2p + κpwp − 1 pinch κp = 2√(2p−1)/(p−1) and prefactor p/(p−1); the changeless was fixed by checking numerically that the pre-inversion aboveground is minimal (a circulating p = 2 type pinch √3 successful spot of 2√3 is not). See besides F. Apéry, Models of the Real Projective Plane, Vieweg, 1987, whose Cartesian family (as tabulated connected R. Ferréol’s mathcurve.com, “Morin surface”) gives the Morin preset and, pinch n = 3, a 2nd exemplary of Boy’s surface.

Related reading: D. DeCarlo et al., “Suggestive Contours for Conveying Shape,” SIGGRAPH 2003; R. Kalnins et al., “WYSIWYG NPR,” SIGGRAPH 2002.

Formulas are JavaScript expressions; ^ is accepted for powers. Available: sin cos tan asin acos atan atan2 sinh cosh tanh exp log sqrt cbrt abs motion pow min max level hypot pi tau e sq(x). Range boxes judge expressions excessively (2*pi). The helper boy(u,v) returns the Bryant–Kusner immersion of ℝℙ² arsenic [x,y,z] (u = radius successful [0,1], v = angle). torusknot(u,v,p,q,R,r,a) returns the conduit of radius a astir the (p,q) torus knot connected the torus of radii R, r (defaults 2, 3, 2.2, 1, 0.42); u runs erstwhile on the knot, v astir the tube. apery(u,v,n,k) is Apéry’s Cartesian family pinch u ∈ [−π/2, π/2]: n = 2, k = 1 is Morin’s aboveground (v ∈ [0, 2π]); n = 3, k = 1 is Boy’s aboveground (v ∈ [0, π]). kusner(u,v,p,d) is the Kusner–Bryant family, w = tan(πu/4) eiv: u ∈ [0, 2] is the full sphere (p = 2 is Morin’s surface), u ∈ [0, 1] covers ℝℙ² erstwhile for overseas p (p = 3 is boy); d shifts the centre of inversion (default −½).