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|Title:||A variation of the opaque problem|
|Keywords:||Opaque problem;Convex set;Curve|
|Abstract:||The opaque problem on a plane is a problem which seeks, among curves (or unions of curves) that intersect all the straight lines passing through a given shape (e.g., a square, a circle, a polygon, etc.), the curve (or union of curves) with the shortest length. In this study, we simplify this problem by ﬁxing a point on the plane outside the given shape and considering only the straight lines that pass through this point. Our problem is then to ﬁnd a “minimizer” inside the given shape that intersects all those straight lines that intersect the given shape. We prove that, for most of the convex shapes on the plane, solutions to this problem always fail to exist. However, the greatest lower bound of the length of the candidates is found to be simply θ2 θ1 r dθ, where r = r(θ) is the distance, in polar form, from the ﬁxed point to a certain part of the boundary of the region in question. We also extend our study to non-convex shapes, whose boundary is a simple closed curve. We prove that the solution exists if its boundary satisﬁes certain conditions.|
|Description:||Thesis (M.Sc. (Applied Mathematics))--Mahidol University, 2009|
|Appears in Collections:||Mathematics: International Proceedings|
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