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Concept# Space-filling curve

Summary

In mathematical analysis, a space-filling curve is a curve whose range reaches every point in a higher dimensional region, typically the unit square (or more generally an n-dimensional unit hypercube). Because Giuseppe Peano (1858–1932) was the first to discover one, space-filling curves in the 2-dimensional plane are sometimes called Peano curves, but that phrase also refers to the Peano curve, the specific example of a space-filling curve found by Peano.
The closely related FASS curves (approximately space-Filling, self-Avoiding, Simple, and Self-similar curves)
can be thought of as finite approximations of a certain type of space-filling curves.
Definition
Intuitively, a curve in two or three (or higher) dimensions can be thought of as the path of a continuously moving point. To eliminate the inherent vagueness of this notion, Jordan in 1887 introduced the following rigorous definition, which has since been adopted as the precise description of the notion of a curve:

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The distance from self-intersection of a (smooth and either closed or infinite) curve q in three dimensions can be characterised via the global radius of curvature at q(s), which is defined as the smallest possible radius amongst all circles passing through the given point and any two other points on the curve. The minimum value of the global radius of curvature along the curve gives a convenient measure of curve thickness or normal injectivity radius. Given the utility of the construction inherent to global curvature, it is natural to consider variants defined in related ways. The first part of the thesis considers all possible circular and spherical distance functions and the associated, single argument, global radius of curvature functions that are constructed by minimisation over all but one argument. It is shown that among all possible global radius of curvature functions there are only five independent ones. And amongst these five there are two particularly useful ones for characterising thickness of a curve. We investigate the geometry of how these two functions, ρpt and ρtp, can be achieved. Properties and interrelations of the divers global radius of curvature functions are illustrated with the simple examples of ellipses and helices. It is known that any Lipschitz continuous curve with positive thickness actually has C1,1-regularity. Accordingly, C1,1 is the natural space in which to carry out computations involving self-avoiding curves. The second part of the thesis develops the mathematical theory of biarcs, which are a geometrically elegant way of discretizing C1,1 space curves. A biarc is a pair of circular arcs joined in a C1 fashion according to certain matching rules. We establish a self-contained theory of the geometry of biarc interpolation of point-tangent data sampled from an underlying base curve, and demonstrate that such biarc curves have attractive convergence properties in both a pointwise and function-space sense, e.g. the two arcs of the biarc interpolating a coalescent point-tangent data pair on a C2-curve approach the osculating circle of the curve at the limit of the data points, and for a C1,1-base curve and a sequence of (possibly non-uniform) meshes, the interpolating biarc curves approach the base curve in the C1-norm. For smoother base curves, stronger convergence can be obtained, e.g. interpolating biarc curves approach a C2 base curve in the C1,1-norm. The third part of the thesis concerns the practical utility of biarcs in computation. It is shown that both the global radius of curvature function ρpt and thickness can be evaluated efficiently (and to an arbitrarily small, prescribed precision) on biarc curves. Moreover, both the notion of a contact set, i.e. the set of points realising thickness, and an approximate contact set can be defined rigorously. The theory is then illustrated with an application to the computation of ideal shapes of knots. Informally ideal knot shapes can be described as the configuration allowing a given knot to be tied with the shortest possible piece of rope of prescribed thickness. The biarc discretization is combined with a simulated annealing code to obtain approximate ideal shapes. These shapes provide rigorous upper bounds for rope length of ideal knots. The approximate contact set and the function ρpt evaluated on the computed shapes allow us to assess closeness of the computations to ideality. The high accuracy of the computations reveal various, previously unrecognized, features of ideal knot shapes.

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We study the structure of planar point sets that determine a small number of distinct distances. Specifically, we show that if a set of n points determines o(n) distinct distances, then no line contains Omega(n (7/8)) points of and no circle contains Omega(n (5/6)) points of . We rely on the partial variant of the Elekes-Sharir framework that was introduced by Sharir, Sheffer, and Solymosi in [19] for bipartite distinct distance problems. To prove our bound for the case of lines we combine this framework with a theorem from additive combinatorics, and for our bound for the case of circles we combine it with some basic algebraic geometry and a recent incidence bound for plane algebraic curves by Wang, Yang, and Zhang [20]. A significant difference between our approach and that of [19] (and of other related results) is that instead of dealing with distances between two point sets that are restricted to one-dimensional curves, we consider distances between one set that is restricted to a curve and one set with no restrictions on it.

A long-standing conjecture of Richter and Thomassen states that the total number of intersection points between any n simple closed Jordan curves in the plane, so that any pair of them intersect and no three curves pass through the same point, is at least (1 - o(1)) n(2). We confirm the above conjecture in several important cases, including the case (1) when all curves are convex, and (2) when the family of curves can be partitioned into two equal classes such that each curve from the first class touches every curve from the second class. (Two closed or open curves are said to be touching if they have precisely one point in common and at this point the two curves do not properly cross.) An important ingredient of our proofs is the following statement. Let S be a family of n open curves in R-2, so that each curve is the graph of a continuous real function defined on R, and no three of them pass through the same point. If there are nt pairs of touching curves in S, then the number of crossing points is Omega(nt root logt/log log t).