Progressive Simplification of Polygonal Curves

Simplifying polygonal curves at different levels of detail is an important problem with many applications. Existing geometric optimization algorithms are only capable of minimizing the complexity of a simplified curve for a single level of detail. We present an $O(n^3m)$-time algorithm that takes a polygonal curve of n vertices and produces a set of consistent simplifications for m scales while minimizing the cumulative simplification complexity. This algorithm is compatible with distance measures such as the Hausdorff, the Fr\'echet and area-based distances, and enables simplification for continuous scaling in $O(n^5)$ time. To speed up this algorithm in practice, we present new techniques for constructing and representing so-called shortcut graphs. Experimental evaluation of these techniques on trajectory data reveals a significant improvement of using shortcut graphs for progressive and non-progressive curve simplification, both in terms of running time and memory usage.Comment: 20 pages, 20 figure

Electrical networks and Stephenson's conjecture

In this paper, we consider a planar annulus, i.e., a bounded, two-connected, Jordan domain, endowed with a sequence of triangulations exhausting it. We then construct a corresponding sequence of maps which converge uniformly on compact subsets of the domain, to a conformal homeomorphism onto the interior of a Euclidean annulus bounded by two concentric circles. As an application, we will affirm a conjecture raised by Ken Stephenson in the 90's which predicts that the Riemann mapping can be approximated by a sequence of electrical networks.Comment: Comments are welcome

Area preservation in computational fluid dynamics

Incompressible two-dimensional flows such as the advection (Liouville) equation and the Euler equations have a large family of conservation laws related to conservation of area. We present two Eulerian numerical methods which preserve a discrete analog of area. The first is a fully discrete model based on a rearrangement of cells; the second is more conventional, but still preserves the area within each contour of the vorticity field. Initial tests indicate that both methods suppress the formation of spurious oscillations in the field.Comment: 14 pages incl. 3 figure

Lagrangian Relations and Linear Point Billiards

Motivated by the high-energy limit of the $N$-body problem we construct non-deterministic billiard process. The billiard table is the complement of a finite collection of linear subspaces within a Euclidean vector space. A trajectory is a constant speed polygonal curve with vertices on the subspaces and change of direction upon hitting a subspace governed by `conservation of momentum' (mirror reflection). The itinerary of a trajectory is the list of subspaces it hits, in order. Two basic questions are: (A) Are itineraries finite? (B) What is the structure of the space of all trajectories having a fixed itinerary? In a beautiful series of papers Burago-Ferleger-Kononenko [BFK] answered (A) affirmatively by using non-smooth metric geometry ideas and the notion of a Hadamard space. We answer (B) by proving that this space of trajectories is diffeomorphic to a Lagrangian relation on the space of lines in the Euclidean space. Our methods combine those of BFK with the notion of a generating family for a Lagrangian relation.Comment: 29 pages, 4 figure

Error bounded approximate reparametrization of NURBS curves

Journal ArticleThis paper reports research on solutions to the following reparametrization problem: approximate c(r(t)) by a NURBS where c is a NURBS curve and r may, or may not, be a NURBS function. There are many practical applications of this problem including establishing and exploring correspondence in geometry, creating related speed profiles along motion curves for animation, specifying speeds along tool paths, and identifying geometrically equivalent, or nearly equivalent, curve mappings. A framework for the approximation problem is described using two related algorithmic schemes. One constrains the shape of the approximation to be identical to the original curve c. The other relaxes this constraint. New algorithms for important cases of curve reparametrization are developed from within this framework. They produce results with bounded error and address approximate arc length parametrizations of curves, approximate inverses of NURBS functions, and reparametrizations that establish user specified tolerances as bounds on the Frechet distance between parametric curves