EAN: 9783030125509

SPRINGERBRIEFS IN COMPUTER SCIENCE FITTING SP...

This Brief investigates the intersections that occur between three different areas of study
that normally would not touch each other: ODF  spline theory  and topology. The Least Squares
Orthogonal Distance Fitting (ODF) method has become the standard technique used to develop
mathematical models of the physical shapes of objects  due to the fact that it produces a
fitted result that is invariant with respect to the size and orientation of the object. It is
normally used to produce a single optimum fit to a specific object  this work focuses instead
on the issue of whether the fit responds continuously as the shape of the object changes. The
theory of splines develops user-friendly ways of manipulating six different splines to fit the
shape of a simple family of epiTrochoid curves: two types of Bézier curve  two uniform
B-splines  and two Beta-splines. This work will focus on issues that arise when mathematically
optimizing the fit. There are typically multiple solutions to the ODF method  and the number of
solutions can often change as the object changes shape  so two topological questions
immediately arise: are there rules that can be applied concerning the relative number of local
minima and saddle points  and are there different mechanisms available by which solutions can
either merge and disappear  or cross over each other and interchange roles. The author proposes
some simple rules which can be used to determine if a given set of solutions is internally
consistent in the sense that it has the appropriate number of each type of solution.

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