Chapter 4: Geometry

FIGURE 4.29
The Bernoulli or logarithmic spiral (left), the Archimedes or linear spiral (middle), and the
Cornu spiral (right).
¾
¾
with the logarithmic spiral, which spirals down to the origin without reaching it. The
Cornu spiral or clothoid (Figure 4.29, right), important in optics and engineering,
has the following parametric representation in Cartesian coordinates:
´Øµ
Ø
¼
Ó×´ ½
¾ ×¾ µ ×
Ý Ë´Øµ
Ø
¼
×Ò´ ½
¾ ×¾ µ × (4.7.8)
( and Ë are the so-called Fresnel integrals; see page 547.) A curve parameterized
by arc length and such that the radius of curvature is inversely proportional to the
parameter at each point is a Cornu spiral (compare to the Bernoulli spiral).
4.7.5 THE PEANO CURVE AND FRACTAL CURVES
There are curves (in the sense of continuous maps from the real line to the plane)
that completely cover a two-dimensional region of the plane. We give a construction
of such a Peano curve, adapted from David Hilbert’s example. The construction
is inductive and is based on replacement rules. We consider building blocks of six
shapes:
, the length of the straight segments being twice
the radius of the curved ones. A sequence of these patterns, end-to-end, represents a
curve, if we disregard the gray and black half-disks. The replacement rules are the
following:
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