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CH APT E R 13

Smooth Surfaces and Their Outlines

Several chapters of this book have explored the quantitative relationship between
simple geometric figures such as points, lines, and planes and the parameters of their
image projections. In this one, we investigate instead the qualitative relationship
between three-dimensional shapes and their pictures, focusing on the outlines of
solids bounded by smooth surfaces. The outline, also named object silhouette or
image contour in this chapter, is formed by intersecting the retina with a viewing
cone (or cylinder in the case of orthographic projection) whose apex coincides with
the pinhole and whose surface grazes the object along a surface curve called the
occluding contour,or rim (Figure 13.1).
The image contour of a solid shape constrains it to lie within the associated
viewing cone, but does not reveal the depth of its occluding contour. In the case of
solids bounded by smooth surfaces, it provides additional information. In particu-
lar, the plane defined by the eye and the tangent to the image contour is tangent
to the surface. Thus, the contour orientation determines the surface orientation
along the occluding contour. In 1977, Marr argued that the silhouette does not, in
general, tell us anything else about shape, claiming for example that the inflections
of a snake’s contour, that separate its convex parts from its concave ones (see the
next section for a formal definition), in general have nothing to do with the intrin-
sic local surface shape, but correspond instead to the boundaries between far and
near parts of the snake’s body, the near regions appearing larger than the far ones
due to perspective effects (Figure 13.2, left). Although intuitively plausible, this
interpretation is incorrect. Indeed, as shown by Koenderink in a delightful 1984
article, the inflections of the contour are the projections of parabolic surface points
that separate convex parts of the surface from saddle-shaped, or hyperbolic ones
(Figure 13.2, right; we will prove a quantitative version of this result later in this
chapter). Thus, they indeed always reveal something of the intrinsic shape of the
observed object.
Koenderink’s view is that of a physicist trying to understand and model the
laws that govern the visual world, and it prevails in this chapter, where the accent
is not on applications but on a theoretical understanding of what can, or cannot
be said about the world by merely looking at it. The proper mathematical setting
for this study is differential geometry, a field of mathematics whose primary aim
is to model the shape of objects such as curves and surfaces in the small—that
is, in the immediate vicinity of a point. This local analysis is particularly fruitful
for understanding the relationship between solid shapes and their outlines. In
particular, it can be shown that the occluding contour is in general a smooth curve,
formed by fold points where the viewing ray is tangent to the surface and a discrete
set of cusp points where the ray is tangent to the occluding contour as well. The
image contour is piecewise smooth, and its only singularities are a discrete set
of cusps formed by the projection of cusp points and T-junctions formed by the

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