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the way in which human brains perform geometric processing, we may be able
to apply it to the construction of algorithms for ill-defined geometric problems.
Motivated by this observation, we used visual illusions to study the way in
which the human brain processes geometric data, in order to find an approach
for solving ill-defined problems in computational geometry.
On the other hand, our research group has long been studying an approach
to the design of robust geometric algorithms, which we call the topology-oriented
approach [ 22 , 24 , 26 ]. In this approach we start with the assumption that numeri-
cal errors cannot be avoided and moreover the amount of errors is not bounded a
priori, but still we aim at robust geometric algorithms. This task is ill-conditioned
because the correctness of the algorithms cannot be guaranteed due to numerical
errors. However, we can successfully construct stable algorithms by guaranteeing
the consistency of topological structures of geometric objects and thus circum-
vent failures.
We first applied this idea to an incremental algorithm for ordinary Voronoi
diagrams [ 25 ], and then extended to various geometric problems including Voronoi
diagrams for line segments [ 26 ] and line arrangements [ 10 ] in the plane, and convex
hull [ 15 ], Voronoi diagram [ 26 ] and polyhedra [ 20 ] in the three-dimensional space.
Therefore, we might regard the topology-oriented approach as an example of
human-like robust computation. In this paper we compare human brain process-
ing with the topology-oriented algorithms and discuss their similarities.
The structure of this paper is as follows. We first observe and discuss three
typical examples of visual illusions, the Zollner illusion [ 11 ], the Ouchi illusion
[ 14 ], and the impossible motion illusion [ 21 ], in Sects. 2 - 4 , respectively. In Sect. 5 ,
we construct a new algorithm for robust computation of the straight skeleton
as an example of a geometric problem, and discuss the similarities between the
computations in the human brain and the topology-oriented algorithms. We
present our concluding remarks in Sect. 6 .
2Zollner Illusion and Overestimation of Acute Angles
Figure 1 shows the famous Zollner illusion; the four long, straight lines are exactly
parallel and horizontal, but they look as if they are alternately slanted in opposite
directions. This optical illusion is evoked by the shorter lines crossing the longer
lines, and it is usually explained by the overestimation of the acute angles.
When two lines cross, they generate two acute angles and two obtuse angles.
It is commonly observed that the acute angles are apt to be perceived larger
than they actually are, and the obtuse angles are apt to be perceived smaller.
There are many other illusions explained in the same way, including the Hering
illusion, the Wundt illusion, and the Luckiesh illusion [ 11 ].
Various mathematical models have been proposed to explain this overesti-
mation of acute angles. A typical such model is the one by Fremuller et al. [ 9 ]. In
their theory, the retina, acting as a photo sensor, has finite resolution, and hence
images are blurred, resulting in greater rounding of acute angles than of obtuse
angles. This makes acute angles appear to be greater than the actual angles.
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