Computational Geometry in the Human Brain - Discrete and Computational Geometry and Graphs - page 150

( ( (

Fig. 7. Impossible motion “Magnet-Like Slopes” (Sugihara, 2009): (a) three-
dimensional object; (b) result of apparently impossible motion; (c) another view of
the object.

illusion was found as a byproduct of computer vision research [ 18 , 19 ]; refer to
[ 19 , 21 ] for the details of designing this class of illusion.
A remarkable aspect of this illusion is that even after we understand the
actual shape of the object, as in Fig. 7 (c), we incorrectly perceive the shape
when we return to the vantage view point shown in Fig. 7 (a). This observation
may be expressed in the following way.

Observation 3. Computations in the human brain persistently retain the
initial interpretation.

5 Robust Geometric Computations Suggested by the
Human Brain

As we have observed, computations in the human brain are imprecise, local, and
persistent. However, in spite of these disadvantages, the human brain still can
robustly and eciently process visual geometric data in our daily lives. In this
section, we consider how these remarkable characteristics of the human brain
can be used in the design of algorithms for computers.
Geometric algorithms are usually designed on the assumption that numerical
computations will be done precisely, and hence, in particular, that geometric
predicates will always be evaluated correctly. However, this is not true in real
computers, and theoretically correct algorithms sometimes fail when they are
implemented as software. This failure is common when the input is very close to
a degenerate situation.
Let us take the straight skeleton as an example. Let P be a polygon in the
plane. Suppose that from each edge of P , two copies of the edge, we will call them
the sweep lines, start moving in opposite directions away from the edge and at
the same speed, and that they continue to maintain contact with the neighboring
sweep lines at the terminal vertices. Hence, the sweep lines change their lengths
as they move. The motions of the sweep lines terminate at the points of collision
with the other sweep lines. The region swept by a sweep line is assigned to the
corresponding edge. In this way, the plane is partitioned into the regions swept