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Real Polygonal Covers of Digital Discs - Some Theories and Experiments

Bhowmick P., Bhattacharya Bhargab.B.

Fundamenta Informaticae

2009

Vol. 91, nr 3-4

487-505

There are several algorithms for digitization of a real disc (circle) to derive a digital disc, and also for finding the real disc corresponding to a digital disc. However, the correspondence of a digital disc with a regular polygon in the real plane is not well studied. This paper presents some theories and related experiments on setting the correspondence from a digital disc to its polygonal cover in the real plane. For an ideal regular polygon covering a digital disc, all the grid points of the digital disc should lie on and inside the polygon, and vice versa. That an ideal regular polygon corresponding to a digital disc is possible for some of the digital discs, especially for the ones having smaller radii, is shown. Further, for a disc whose ideal regular polygon is not possible, an approximate polygon, tending to the ideal one, is possible, in which the error of approximation can be controlled by the number of vertices of the approximate polygon. These (ideal or approximate) polygonal covers of digital discs have several applications in many problems of point set pattern matching. We have reported the conditions under which an ideal regular polygon always exists corresponding to a digital disc, and the conditions under which the existence of an ideal regular polygon becomes uncertain. Experimental results have been given to demonstrate the possibilities of approximation and the trade-off in terms of error versus the number of vertices in the approximate polygon.

Exceptional sets for universally polygonally approximable functions

Evans M. J., Humke P. D.

Journal of Applied Analysis

2001

Vol. 7, nr 2

175-190

In [9], the present authors and Richard O'Malley showed that in order for a function be universally polygonally approximate it is necessary that for each ε > 0, the set of points of non-quasicontinuity be σ - (1 - ε ) symmetrically porous. The question as to whether that condition is sufficient or not was left open. Here we prove that if a set, E = U∞n=1 En, such that each Ei is closed and 1-symmetrically porous, then there is a universally polygonally approximable function, f, whose set of points of non-quasicontinuity is precisely E. Although it is tempting to call this a partial converse to our earlier theorem it might be more since it is not known if these two notions of symmetric porosity differ in the class of F? sets.

Universally polygonally approximable functions

Evans. M. J., Humke P. D., O`Malley R. J.

Journal of Applied Analysis

2000

Vol. 6, nr 1

25-45

It has recently been established that any Baire class one function f : [0,1] -> R can be represented as the pointwise limit of a sequence of polygonal functions whose vertices lie on the graph of f. Here we investigate the subclass of Baire class one functions having the additional property that for every dense subset D of [0,1], the first coordinates of the vertices of the polygonal functions can be chosen from D.