Topologically safe curved schematization

Traditionally schematized maps make extensive use of curves. However, automated methods for schematization are mostly restricted to straight lines. We present a generic framework for topology-preserving curved schematization that allows a choice of quality measures and curve types. Our fully-automated approach does not need critical points or salient features. We illustrate our framework with Bézier curves and circular arcs

Deconstructing Approximate Offsets

We consider the offset-deconstruction problem: Given a polygonal shape Q with n vertices, can it be expressed, up to a tolerance \eps in Hausdorff distance, as the Minkowski sum of another polygonal shape P with a disk of fixed radius? If it does, we also seek a preferably simple-looking solution P; then, P's offset constitutes an accurate, vertex-reduced, and smoothened approximation of Q. We give an O(n log n)-time exact decision algorithm that handles any polygonal shape, assuming the real-RAM model of computation. A variant of the algorithm, which we have implemented using CGAL, is based on rational arithmetic and answers the same deconstruction problem up to an uncertainty parameter \delta; its running time additionally depends on \delta. If the input shape is found to be approximable, this algorithm also computes an approximate solution for the problem. It also allows us to solve parameter-optimization problems induced by the offset-deconstruction problem. For convex shapes, the complexity of the exact decision algorithm drops to O(n), which is also the time required to compute a solution P with at most one more vertex than a vertex-minimal one.Comment: 18 pages, 11 figures, previous version accepted at SoCG 2011, submitted to DC

Computation and visualization of ideal knot shapes

We investigate numerical simulations and visualizations of the problem of tying a knot in a piece of rope. The goal is to use the least possible rope of a fixed, prescribed radius to tie a particular knot, e.g. a trefoil, a figure eight, and so on. The ropelength of the knot, the ratio to be minimized, is its length divided by its radius. An overview of existing algorithms to minimize the ropelength is given. They are based on different discretizations. Our work builds on the biarc discretization, for which we have developed an entire C++ library "libbiarc". The library contains a variety of tools to manipulate curves, knots or links. The biarc discretization is particularly well suited to evaluation of thickness. To compute ideal knot shapes we use simulated annealing software, which is also included in "libbiarc", on a biarc discretization. Simulated annealing is a stochastic optimization algorithm that randomly changes the point or tangent data. In the quest to find appropriate moves for this process we arrived upon a Fourier representation for knots, which allows global changes to the curve in the annealing process. Moreover, with the Fourier representation we can enforce symmetries that a given knot might have. To identify these symmetries we use visualization of simulations where symmetry was not enforced. Visualization of knot shapes and their properties is another important aspect in this work. It ranges from simple graphs of the curvature of a knot, through 2-dimensional plots of certain distance, circle or sphere functions, to 3-dimensional images of contact properties. Specially designed color gradients have been developed to emphasize crucial regions of the plots. We show that the contact set of ideal torus knots is a curve that is ambient isotopic to the knot itself, which is a result first suggested by visualization. A combination of numerics and visualization made us aware of a closed trajectory within the trefoil knot, a 9-billiard. Consequently the symmetries and the billiard make it possible to represent the trefoil with only two curve sub segments. We also anneal and visualize knot shapes in the unit 3-sphere or S3. In particular we present the contact set of a candidate for optimality, whose curved contact chords form Villarceau circles, which in turn span a Clifford torus embedded in the unit 3-sphere. Finally some knots and contact surfaces are constructed as physical 3D models using 3D printers

Ropelength of tight polygonal knots

A physical interpretation of the rope simulated by the SONO algorithm is presented. Properties of the tight polygonal knots delivered by the algorithm are analyzed. An algorithm for bounding the ropelength of a smooth inscribed knot is shown. Two ways of calculating the ropelength of tight polygonal knots are compared. An analytical calculation performed for a model knot shows that an appropriately weighted average should provide a good estimation of the minimum ropelength for relatively small numbers of edges.Comment: 27 pages, to appear in "Physical and Numerical Models in Knot Theory and their Application to the Life Sciences

Ideal knots and other packing problems of tubes

This thesis concerns optimal packing problems of tubes, or thick curves, where thickness is defined as follows. Three points on a closed space curve define a circle. Taking the infimum over all radii of pairwise-distinct point triples defines the thickness Δ. A closed curve with positive thickness has a self-avoiding neighbourhood that consists of a disjoint union of normal disks with radius Δ, which is a tube. The thesis has three main parts. In the first, we study the problem of finding the longest closed tube with prescribed thickness on the unit two-sphere, and show that solutions exist. Furthermore, we give explicit solutions for an infinite sequence of prescribed thicknesses Θn = sin(π/2n). Using essentially basic geometric arguments, we show that these are the only solutions for prescribed thickness Θn, and count their multiplicity using algebraic arguments involving Euler's totient function. In the second part we consider tubes on the three-sphere S3. We show that thickness defined by global radius of curvature coincides with the notion of thickness based on normal injectivity radius in S3. Then three natural, but distinct, optimisation problems for knotted, thick curves in S3 are identified, namely, to fix the length of the curve and maximise thickness, to fix a minimum thickness and minimise length, or simply to maximise thickness with length left free. We demonstrate that optimisers, or ideal shapes, within a given knot type exist for each of these three problems. Finally, we propose a simple analytic form of a strong candidate for a thickness maximising trefoil in S3 and describe its interesting properties. The third and final part discusses numerical computations and their implications for ideal knot shapes in both R3 and S3. We model a knot in R3 as a finite sequence of coefficients in a Fourier representation of the centreline. We show how certain presumed symmetries pose restrictions on the Fourier coefficients, and thus significantly reduce the number of degrees of freedom. As a consequence our numerical technique of simulated annealing can be made much faster. We then present our numeric results. First, computations approach an approximation of an ideal trefoil in S3 close to the analytic candidate mentioned above, but, supporting its ideality, are still less thick. Second, for the ideal trefoil in R3, numerics suggest the existence of a certain closed cycle of contact chords, that allows us to decompose the trefoil knot into two base curves, which once determined, and taken together with the symmetry, constitute the ideal trefoil

Exploring curved schematization

Hand-drawn schematized maps traditionally make extensive use of curves. However, there are few automated approaches for curved schematization most previous work focuses on straight lines. We present a new algorithm for area-preserving curved schematization of geographic outlines. Our algorithm converts a simple polygon into a schematic crossing-free representation using circular arcs. We use two basic operations to iteratively replace consecutive arcs until the desired complexity is reached. Our results are not restricted to arcs ending at input vertices. The method can be steered towards different degrees of 'curviness': we can encourage or discourage the use of arcs with a large central angle via a single parameter. Our method creates visually pleasing results even for very low output complexities. We conducted an online user study investigating the effectiveness of the curved schematizations compared to straight-line schematizations of equivalent complexity. While the visual complexity of the curved shapes was judged higher than those using straight lines, users generally preferred curved schematizations. We observed that curves significantly improved the ability of users to match schematized shapes of moderate complexity to their unschematized equivalents

Map schematization with circular arcs

We present an algorithm to compute schematic maps with circular arcs. Our algorithm iteratively replaces two consecutive arcs with a single arc to reduce the complexity of the output map and thus to increase its level of abstraction. Our main contribution is a method for replacing arcs that meet at high-degree vertices. This allows us to greatly reduce the output complexity, even for dense networks. We experimentally evaluate the effectiveness of our algorithm in three scenarios: territorial outlines, road networks, and metro maps. For the latter, we combine our approach with an algorithm to more evenly distribute stations. Our experiments show that our algorithm produces high-quality results for territorial outlines and metro maps. However, the lack of caricature (exaggeration of typical features) makes it less useful for road networks

Stenomaps: Shorthand for shapes

We address some of the challenges in representing spatial data with a novel form of geometric abstraction-the stenomap. The stenomap comprises a series of smoothly curving linear glyphs that each represent both the boundary and the area of a polygon. We present an efficient algorithm to automatically generate these open, C1-continuous splines from a set of input polygons. Feature points of the input polygons are detected using the medial axis to maintain important shape properties. We use dynamic programming to compute a planar non-intersecting spline representing each polygon's base shape. The results are stylised glyphs whose appearance may be parameterised and that offer new possibilities in the 'cartographic design space'. We compare our glyphs with existing forms of geometric schematisation and discuss their relative merits and shortcomings. We describe several use cases including the depiction of uncertain model data in the form of hurricane track forecasting; minimal ink thematic mapping; and the depiction of continuous statistical data