Summary: Computational and Structural Advantages of Circular
Boundary Representation #
O. Aichholzer + , F. Aurenhammer # , T. Hackl §
B. J uttler ¶ , M. Rabl # , Z.
S r ##
Boundary approximation of planar shapes by circular arcs has quantitative and qualitative advantages
compared to using straightline segments. We demonstrate this by way of three basic and frequent compu
tations on shapes -- convex hull, decomposition, and medial axis. In particular, we propose a novel medial
axis algorithm that beats existing methods in simplicity and practicality, and at the same time guarantees
convergence to the medial axis of the original shape.
The plain majority of algorithms in computational geometry have been designed for processing linear ob
jects, like lines, planes, or polygons. On the one hand, this is certainly due to the fact that many interesting
and deep computational and combinatorial questions do arise already for inputs of this simple form. Again,
the pragmatic reason is that algorithms for linear objects are usually both easier to develop and simpler
to implement. To make things work for nonlinear objects, which arise frequently in practical settings,
such objects are usually approximated in a piecewiselinear manner and up to a tolerable error. Existing
approaches  to directly extending polygonal algorithms to curved objects are rare and, due to their
generality, of limited practical use.