Alan K. Zundel

Scan line display of algebraic surfaces

A robust algorithm is presented for scan line display of algebraic surfaces of arbitrary degree and topology. The algorithm correctly displays singularities of any complexity, even those missed by ray tracing or polygonization, and (for surfaces of degree less than eight) offers a significant speed improvement over ray tracing. Antialiasing can generally be accomplished very quickly. In addition to its typical function of shaded raster display, the algorithm is particularly adept at quickly plotting silhouette and intersection curves. A practical use for the algorithm is to display boolean combinations of algebraic half spaces, including blend surfaces.A new polynomial basis is introduced, referred to as the Bernstein pyramid polynomial basis, which enhances numerical stability and which simplifies several computations such as scan planesurface intersection and silhouette detection.

Fat arcs: a bounding region with cubic convergence

Abstract A new bouding region for planar Bezier curves is introduced. This bound is referred to as a fat arc , and is simply a circular arc with some finite thickness. Fat arcs are compared with three other traditional bounding regions: convex hulls, min-max ☐es and bounding strips. Based on empirical data, fat arcs demonstrate O( h 3 ) convergence, convex hulls and bounding strips O( h 2 ) convergence, and min-max ☐es converges linearly to the curve. This means that, in the limit, fat arcs for the two halves of a subdivided curve total 1/8 the area of the fat arc of the undivided curve. Operation counts are given for the four bounding regions. A curve intersection algorithm is discussed.

Pyramids that bound surface patches

This paper describes how to construct a pyramid which has the property that if its vertex is translated to any point on a given rational Bezier surface patch with positive weights, the patch will lie completely outside the pyramid. This pyramid can facilitate operations such as loop detection in surface intersection. Earlier literature asserted that the complement of a normal bounding cone or pyramid is a surface bounding cone or pyramid. This paper proves that the reverse of this statement is always true, but that the statement itself is true only under certain conditions spelled out in the paper.

Approximate parametrization of algebraic curves

This paper discusses a solution to the problem of how to approximate an algebraic curve of any degree with piecewise parametric curves. The approximation can be performed using any differentiable parametric curve form. An implementation using rational cubic Bezier curves results in a G 2 piecewise approximation which experiences better than O(h 7) convergence. If the algebraic curve happens to be a cubic curve of genus zero, a parametric curve can be found which exactly fits the given algebraic curve to within floating point precision. An error bound is developed which is much tighter than previously published bounds.

Use of image analysis to estimate average stopped delays per vehicle at signalized intersections

Average stopped delay has been used to determine the level of service (LOS) of signalized intersections. This work has mostly been done offline by using manual field observation methods, analytical models, and simulation software. Dynamic evaluation of the LOS of signalized intersections has been difficult because the use of conventional methods requires a large amount of labor and cost if such analyses were conducted periodically. However, many traffic operations centers wish to be able to coordinate surface street and freeway management when congestion occurs. It is demonstrated that image analysis can be used to dynamically determine average stopped delay. Two methods—the gap method and the motion method—were developed and compared with ITE’s field delay observation method. Traffic Software Integrated System software was used to create images of queued vehicles. To eliminate parallax problems, two simple assumptions were made: the queued vehicles were viewed from the direction normal to the pavement, and all lines of sight were parallel to the camera’s view direction. At present, the algorithms are simple, but they produced promising results. Various analysis time intervals from 1 to 20 s were used to estimate stopped delay. The gap method gave a good result when the time interval was about 15 s; the motion method produced a good result when the time interval was 1 or 2 s. The manual observation method resulted in a good estimate similar to that which the simulation software produced, especially when the time interval was about 10 s or less.

Fast algorithm for generating sorted contour strings

Abstract Automatic generation of contours for graphical display and map plotting has been studied extensively since the early days of computing. The individual segments making up a contour line are often determined by subdividing the object of interest into small triangles and computing the contours assuming a linear variation on each triangle. However, efficient storage of contour data and the need to place labels (automatically) or to smooth the contours require that the contours be generated in continuous strings of segments. A simple approach to generate such strings is to sort the randomly generated contour segments. Since sorting can be time-consuming, the majority of previous approaches are contour-tracing algorithms that traverse the surface and generate the contour in a continuous sequence of segments. In this paper, we present a new sorting algorithm. The algorithm is relatively easy to implement, can be applied to any type of surface, and works for both 2D and 3D objects. The algorithm is significantly faster than the contour tracing approach, particularly when large numbers of segments are involved.

A Real-Time System for Forecasting Extreme Waves and Vessel Motions

The University of Michigan is leading a team that includes subcontractors Ohio State University, Aquaveo, LLC, and Woods Hole Oceanographic Institute to design, implement, and test an Environmental and Ship Motion Forecasting (ESMF) system. The system has application to many challenges associated with offshore operations, including skin-to-skin transfer of cargo/personnel and extreme wave/response prediction. Briefly, the system uses a modified commercial-off-the-shelf (COTS) Doppler marine radar to determine the wave field surrounding the vessel; nonlinear wave theory to propagate the wave surface forward in time; and seakeeping theory to predict future vessel motions. A major challenge is that all computations must be done in real time. This paper will briefly describe the system and show an example application of predicting extreme waves and motions for a floating offshore type platform.Copyright

Size Function Smoothing Using an Element Area Gradient

This paper presents a method to improve element size transitions when using a size function to govern the mesh generation. The method modifies the size function to meet a user specified adjacent element area change limit. The method can be used to either refine or coarsen the resulting mesh. Two sample meshes generated using the method are presented.

Developing a Probabilistic Flood Plain Boundary using HEC-1 and HEC-RAS

Despite the advances in computer software and modeling tools for hydrologic and hydraulic analysis, the science of developing flood plain extents and depths still involves a great deal of uncertainty. While the physics of rainfall runoff modeling and water surface profile computations is well known, complexities and uncertainties in deriving accurate input parameters still exist. The result is that for any project, such as the development of a flood insurance rate map (FIRM), a number of similar solutions can be generated that can be defended using sound engineering principles. A better approach to developing a flood plain boundary is to incorporate uncertainty in the modeling parameters during model development. Using a series of stochastic simulations a number of probable results from hydrologic and hydraulic models can be computed. Computer automated flood plain delineation tools can then be used to determine flood plain boundaries and inundation depths for the series of model solutions. Finally, a resulting probabilistic flood plain map can be generated from all of the solutions. The probabilistic flood plain shows contours of percent probability that a location is flooded. For example the 100% probability flood zone is the area that is flooded as a result of each of the model runs, whereas the 50% probability flood zone is only flooded by half of the resulting model runs. Despite the sophisticated tools available for determining floodplain extents, the process is not an exact science. One of the most difficult problems in floodplain modeling is the number of input parameters required to create a viable model and the uncertainty associated with these parameters. Current methods of floodplain modeling require that a single set of input parameters be used to define the floodplain. A better method would be to use a distribution of values for each input parameter. This paper will focus on modeling and visualizing the changes in the floodplain boundary caused by random changes in model input parameters within a range of acceptable values. Hydrologic, hydraulic, and automated floodplain delineation model inputs and outputs are dynamically linked to interpret the effects of these random changes on the floodplain boundary.

Automating Average Stopped Delay Estimation Using Image Analysis of Actual Traffic Flow at Signalized Intersections

Average stopped delay for a given signalized intersection approach is the average time a vehicle waits at the traffic light and it constitutes a part of the measure of effectiveness of signalized intersections. The exploratory work done by two authors of this paper using images created by the animation feature of the CORSIM traffic simulation software indicated that such application can actually produce stopped delay estimates commensurate with the results of the ITE method. However, CORSIM simulation images do not have parallax problems and vehicle color and size and pavement color were practically the same for the entire evaluation field. The three methods (Gap, Gap -hybrid, Motion method) developed in this study are able to overcome to an acceptable level these problems associated with actual traffic flow images. They were tested with the two image data sets that were taken at two different locations with different camera angles. The performance of the t hree methods varies depending on the quality of image, camera angle, and calibrated parameter values used for each method. Nevertheless, in both cases they were able to produce average stopped delays similar to those estimated by the ITE manual method. At present the software that executes the three methods analyzes one approach lane at a time using digitized still images taken from analog video films; however, it can be expanded to analyze multiple approach lanes and digital images dynamically fed by digital traffic monitoring cameras. With this automated procedure the traffic engineer can estimate average stopped delays in his or her office, greatly saving time, money and manpower necessary for conducting field data collections.