Related papers: Plane-filling trails
The `linear orbit' of a plane curve of degree d is its orbit in the projective space of dimension d(d+3)/2 parametrizing such curves under the natural action of PGL(3). In this paper we compute the degree of the closure of the linear orbits…
Modular forms are highly self-symmetric functions studied in number theory, with connections to several areas of mathematics. But they are rarely visualized. We discuss ongoing work to compute and visualize modular forms as 3D surfaces and…
Graphs are commonly used in mathematics to represent some relationships between items. However, as simple objects, they sometimes fail to capture all relevant aspects of real-world data. To address this problem, we generalize them and model…
Restricting path tracing to a small number of paths per pixel for performance reasons rarely achieves a satisfactory image quality for scenes of interest. However, path space filtering may dramatically improve the visual quality by sharing…
For two closed curves on a plane (discrete version) and local criteria for similarity of points on the curves one gets a potential, which describes the similarity between curve points. This is the base for a global similarity measure of…
Modern computer networks support interesting new routing models in which traffic flows from a source s to a destination t can be flexibly steered through a sequence of waypoints, such as (hardware) middleboxes or (virtualized) network…
In this paper, we describe an algorithm for fitting an analytic and bandlimited closed or open curve to interpolate an arbitrary collection of points in $\mathbb{R}^{2}$. The main idea is to smooth the parametrization of the curve by…
We study plane curves over finite fields whose tangent lines at smooth $\mathbb{F}_q$-points together cover all the points of $\mathbb{P}^2(\mathbb{F}_q)$.
Obtaining complete information about the shape of an object by looking at it from a single direction is impossible in general. In this paper, we theoretically study obtaining differential geometric information of an object from orthogonal…
The visibility graph of a simple polygon represents visibility relations between its vertices. Knowing the correct order of the vertices around the boundary of a polygon and its visibility graph, it is an open problem to locate the vertices…
Preprocessing a 2D image often produces a noisy cloud of interest points. We study the problem of counting holes in unorganized clouds in the plane. The holes in a given cloud are quantified by the topological persistence of their boundary…
The paper presents a systematic strategy for implementing Hilbert's space filling curve for use in online exploration tasks and addresses its application in scenarios wherein the space to be searched obstacles (or holes) whose locations are…
Given a finite collection of two-dimensional tile types, the field of study concerned with covering the plane with tiles of these types exclusively has a long history, having enjoyed great prominence in the last six to seven decades. Much…
The class of closed graphs by a linear ordering on their sets of vertices is investigated. A recent characterization of such a class of graphs is analyzed by using tools from the proper interval graph theory.
We provide criteria for deciding whether a given planar curve is an image of a given spatial curve, obtained by a central or a parallel projection with unknown parameters. These criteria reduce the projection problem to a certain…
The quest for regular models of arithmetic surfaces allows different viewpoints and approaches: using valuations or a covering by charts. In this article, we sketch both approaches and then show in a concrete example, how surprisingly…
We present a procedure to approximate a plane contour by piecewise polynomial functions, depending on various parameters, such as degree, number of local patches, selection of knots. This procedure aims to be adopted to study how…
Two-, three- and four-dimensional representations of Penrose tilings of the plane are described. The vertices that occur in these representations lie on lattices. Symmetries and methods of visualizing these representations are discussed.…
In this paper, we give a double twist to the problem of planning under uncertainty. State-of-the-art planners seek to minimize the localization uncertainty by only considering the geometric structure of the scene. In this paper, we argue…
Two averaging algorithms are considered which are intended for choosing an optimal plane and an optimal circle approximating a group of points in three-dimensional Euclidean space.