We consider incidences among colored sets of lines in $mathbb{R}^d$ and examine whether the existence of certain concurrences between lines of $k$ colors force the existence of at least one concurrence between lines of $k+1$ colors. This question is relevant for problems in 3D reconstruction in computer vision.
Given a colored point set in the plane, a perfect rainbow polygon is a simple polygon that contains exactly one point of each color, either in its interior or on its boundary. Let $operatorname{rb-index}(S)$ denote the smallest size of a perfect rain
bow polygon for a colored point set $S$, and let $operatorname{rb-index}(k)$ be the maximum of $operatorname{rb-index}(S)$ over all $k$-colored point sets in general position; that is, every $k$-colored point set $S$ has a perfect rainbow polygon with at most $operatorname{rb-index}(k)$ vertices. In this paper, we determine the values of $operatorname{rb-index}(k)$ up to $k=7$, which is the first case where $operatorname{rb-index}(k) eq k$, and we prove that for $kge 5$, [ frac{40lfloor (k-1)/2 rfloor -8}{19} %Birgit: leqoperatorname{rb-index}(k)leq 10 bigglfloorfrac{k}{7}biggrfloor + 11. ] Furthermore, for a $k$-colored set of $n$ points in the plane in general position, a perfect rainbow polygon with at most $10 lfloorfrac{k}{7}rfloor + 11$ vertices can be computed in $O(nlog n)$ time.
Based on the concepts of isovists and medial axes, we developed a set of algorithms that can automatically generate axial lines for representing individual linearly stretched parts of open space of an urban environment. Open space is the space betwee
n buildings, where people can freely move around. The generation of the axial lines has been a key aspect of space syntax research, conventionally relying on hand-drawn axial lines of an urban environment, often called axial map, for urban morphological analysis. Although various attempts have been made towards an automatic solution, few of them can produce the axial map that consists of the least number of longest visibility lines, and none of them really works for different urban environments. Our algorithms provide a better solution than existing ones. Throughout this paper, we have also argued and demonstrated that the axial lines constitute a true skeleton, superior to medial axes, in capturing what we perceive about the urban environment. Keywords: Visibility, space syntax, topological analysis, medial axes, axial lines, isovists
In standard rounding, we want to map each value $X$ in a large continuous space (e.g., $R$) to a nearby point $P$ from a discrete subset (e.g., $Z$). This process seems to be inherently discontinuous in the sense that two consecutive noisy measuremen
ts $X_1$ and $X_2$ of the same value may be extremely close to each other and yet they can be rounded to different points $P_1 e P_2$, which is undesirable in many applications. In this paper we show how to make the rounding process perfectly continuous in the sense that it maps any pair of sufficiently close measurements to the same point. We call such a process consistent rounding, and make it possible by allowing a small amount of information about the first measurement $X_1$ to be unidirectionally communicated to and used by the rounding process of $X_2$. The fault tolerance of a consistent rounding scheme is defined by the maximum distance between pairs of measurements which guarantees that they are always rounded to the same point, and our goal is to study the possible tradeoffs between the amount of information provided and the achievable fault tolerance for various types of spaces. When the measurements $X_i$ are arbitrary vectors in $R^d$, we show that communicating $log_2(d+1)$ bits of information is both sufficient and necessary (in the worst case) in order to achieve consistent rounding for some positive fault tolerance, and when d=3 we obtain a tight upper and lower asymptotic bound of $(0.561+o(1))k^{1/3}$ on the achievable fault tolerance when we reveal $log_2(k)$ bits of information about how $X_1$ was rounded. We analyze the problem by considering the possible colored tilings of the space with $k$ available colors, and obtain our upper and lower bounds with a variety of mathematical techniques including isoperimetric inequalities, the Brunn-Minkowski theorem, sphere packing bounds, and v{C}ech cohomology.
We will prove that there exists a model of ZFC+``c= omega_2 in which every M subseteq R of cardinality less than continuum c is meager, and such that for every X subseteq R of cardinality c there exists a continuous function f:R-> R with f[X]=[0,1].
In particular in this model there is no magic set, i.e., a set M subseteq R such that the equation f[M]=g[M] implies f=g for every continuous nowhere constant functions f,g:R-> R .
Given a finite point set $P$ in the plane, a subset $S subseteq P$ is called an island in $P$ if $conv(S) cap P = S$. We say that $Ssubset P$ is a visible island if the points in $S$ are pairwise visible and $S$ is an island in $P$. The famous Big-li
ne Big-clique Conjecture states that for any $k geq 3$ and $ell geq 4$, there is an integer $n = n(k,ell)$, such that every finite set of at least $n$ points in the plane contains $ell$ collinear points or $k$ pairwise visible points. In this paper, we show that this conjecture is false for visible islands, by constructing arbitrarily large finite point sets in the plane with no 4 collinear members and no visible island of size $2^{42}$.