A Topological Picture Book, Rendered

Plate I. The torus.

A Topological Picture Book
Hand-hatched surfaces after the mid-century manner of Francis, Apéry, Hilbert–Cohn-Vossen. Drag to turn the model freely (shift-drag to roll it); scroll to approach.

Surface

parametric
implicit

Preset

x(u,v)
y(u,v)
z(u,v)

u from / to

v from / to

u segments
v segments

trace the double curve (self-intersection)

f(x,y,z) = 0

box half-width
grid resolution

The zero set is triangulated by marching tetrahedra; normals are taken from ∇f.

Redraw
Save engraving (PNG)

Pen & hatching
Hatching

strokes along principal curvature
stripes along parameter lines (u,v)
stripes from plane sections of space
stripes in the picture plane (screen)

spacing

pen width (px)

hand wobble

broad nib

ink weight

edge filter (Sobel)

cross-hatch the shadows
fine rendering (2× supersample)
Stroke spacing re-traces the hatching when you release the slider.

Contours
draw silhouettes, folds & boundaries

contour weight (px)

hidden lines

dashed
faint
omitted

haloes where lines cross

Ink & lettering

overshoot at ends

ink pooling

bleed into paper

contour smoothing

place
clear

Labels are hand-lettered, pinned to the surface, drag to move them, double-click to remove; they dim when their point is hidden.

Light & tone

light bearing

light height

contrast

ambient

silhouette shading

ink
paper

Sources & method

The drawing is made of strokes, not pixels. Silhouettes are the zero set of n·v on the
mesh (found with interpolated normals, so they chain into smooth curves), boundaries and the
double curve of an immersion are added as further chains, and all of them are drawn as tapered
ribbons with a broad-nib pen model, weight that grows on the shadow side and toward the viewer,
and coherent hand wobble. Visibility is settled on the GPU: a hidden-line pass draws the occluded
parts dashed, and a one-sided paper halo under each near contour cuts the lines behind it.
Hatching is a set of streamlines of the principal-curvature line field, traced at build time in
three nested densities (the second along the other principal direction, for cross-hatching);
tone selects which family is inked and where each stroke feathers out. Highlights stay bare paper.
The principal directions are ordered by signed curvature, not magnitude, so the two families stay
continuous across the loci where κ₁ = −κ₂.
Contour chains are lightly smoothed before inking. Strokes overshoot their ends a little in the
manner of sketchy line rendering; the ink pooling at stroke starts and the ragged bleed into the
paper grain 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
with a leader line, and dimmed when their point is hidden.

G. K. Francis, A Topological Picture Book, Springer, 1987 — the style target: contour drawing with cusps, double curves, hidden lines, and sparing hatched bands.
P. Bénard, A. Hertzmann, “Line Drawings from 3D Models: A Tutorial,” Foundations and Trends in Computer Graphics and Vision 11(1–2), 2019 — the contour pipeline: smooth silhouettes as 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 on n·v).
A. Hertzmann, D. Zorin, “Illustrating Smooth Surfaces,” SIGGRAPH 2000, pp. 517–526 — hatching along principal curvature directions, cross-hatching only in dark regions, blank highlights, undercuts.
B. Jobard, W. Lefer, “Creating Evenly-Spaced Streamlines of Arbitrary Density,” Visualization in Scientific Computing, 1997 — the separation-distance rule used to trace the hatch streamlines.
A. Appel, F. J. Rohlf, A. J. Stein, “The Haloed Line Effect for Hidden Line Elimination,” SIGGRAPH 1979 — the paper haloes at line crossings.
J. D. Northrup, L. Markosian, “Artistic Silhouettes: A Hybrid Approach,” NPAR 2000 — chaining silhouette segments and rendering them as stylized strokes with 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 line rendering: lines that overshoot their endpoints and wiggle, drawn with a pen model whose width varies along the stroke.
M. P. Salisbury, S. E. Anderson, R. Barzel, D. H. Salesin, “Interactive Pen-and-Ink Illustration,” SIGGRAPH 1994, pp. 101–108 — stroke textures and the placement of hand-character strokes to reach a target tone.
E. Praun, H. Hoppe, M. Webb, A. Finkelstein, “Real-Time Hatching,” SIGGRAPH 2001 — nested tone levels of hatching; here realized with object-space strokes so the hatching never swims.
G. Winkenbach, D. H. Salesin, “Computer-Generated Pen-and-Ink Illustration,” SIGGRAPH 1994, pp. 91–100 — tone by stroke density and thickness; stroke textures.
G. Elber, “Line Art Rendering via a Coverage of Isoparametric Curves,” IEEE TVCG 1(3), 1995 — hatching along isoparametric curves (the parameter-line stripes mode).
T. Saito, T. Takahashi, “Comprehensible Rendering of 3-D Shapes,” SIGGRAPH 1990, pp. 197–206 — edge extraction from normal and depth buffers (the optional Sobel edge 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 test behind the double-curve computation.
R. Kusner, “Conformal Geometry and Complete Minimal Surfaces,” Bull. Amer. Math. Soc. 17(2), 1987 — source of the Bryant–Kusner parametrization used for Boy’s surface (checked here numerically: antipodal boundary gluing and threefold symmetry hold to machine precision). The general-p form used by kusner() has denominator w2p + κpwp − 1 with κp = 2√(2p−1)/(p−1) and prefactor p/(p−1); the constant was fixed by checking numerically that the pre-inversion surface is minimal (a circulating p = 2 version with √3 in place of 2√3 is not). See also F. Apéry, Models of the Real Projective Plane, Vieweg, 1987, whose Cartesian family (as tabulated on R. Ferréol’s mathcurve.com, “Morin surface”) gives the Morin preset and, with n = 3, a second model 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.

Notation

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

The process of creating a topological picture book involves translating three-dimensional geometric properties into specific artistic rendering techniques, utilizing principles from differential geometry and computer graphics. The initial step involves defining surfaces, such as the torus, using parametric or implicit representations, where coordinates are defined by functions $x(u,v)$, $y(u,v)$, and $z(u,v)$. The zero set of a function $f(x,y,z) = 0$ defines these surfaces, which are then processed using methods like marching tetrahedra to determine surface normals, necessary for tracing silhouettes.

The contour pipeline focuses on defining boundaries, folds, and hidden lines through sophisticated line drawing techniques. Silhouettes are derived from the zero set of $n \cdot v$ on the mesh, utilizing interpolated normals to ensure smooth chaining of segments. Visibility is handled by a hidden-line pass that draws occluded areas with dashed lines, supplemented by paper haloes around near contours to manage hidden line elimination, consistent with work done by Appel et al. regarding line effects.

Texturing and hatching are achieved by tracing streamlines along the principal curvature directions. These directions are ordered based on signed curvature rather than magnitude, ensuring continuity where the principal curvatures are equal ($\kappa_1 = -\kappa_2$). Cross-hatching is introduced by tracing lines in the direction of the other principal curvature to create textural variation dependent on localized density, informed by findings related to hatching along isoparametric curves.

The final appearance involves complex effects related to ink and line rendering. Strokes are rendered as tapered ribbons with varying width that grow toward the viewer, incorporating hand-tuned effects such as overshoot at endpoints and ink pooling, which act as stylistic elements rather than derived from pixel properties. The process incorporates stochastic elements like hand wobble to mimic sketching and edge filters, such as Sobel, to define boundaries. Labels are added as hand-lettered elements pinned to the surface, dimming when their corresponding point is occluded.

The overall aesthetic relies on layering light and tone effects, including ambient lighting, silhouette shading, and contrast, which interact with the striated texture created by hatching. The methodology draws heavily on concepts developed in 3D non-photorealistic rendering, such as methods for extracting silhouettes from interpolated vertex normals and techniques for rendering surfaces based on isoparametric curve coverage, incorporating mathematical structures like Apéry’s Cartesian family and the Kusner–Bryant parametrization to define the immersion of surfaces. The outcome is a style characterized by contour drawing emphasizing cusps, double curves, hidden lines, and sparse hatched bands.