Line Generalization on the Sphere

Abstract

In the literature discussing methods for processing cartographic lines, it seem!; almost axiomatic that the line data lie on a plane. For example, in Buttenfield'r; comprehensive review of line treatment, all the techniques described are planar (Buttenfield 1985). One is almost tempted to infer that geographic data become cartographic only after projection to a map plane. Given the dominance of map!; over globes for display, this is entirely understandable. There are, however, instances when it is advantageous to generalize line data on the sphere, with map projection subsequent to that processing. The intent of this note is to present arguments in favor of spherical generalization and to develop a spherical adaptation of the popular planar method due to Douglas and Ptoicker (Douglas and Peuckei-1973). We will also show that there can be a significant computational advantage in generalizing prior to projection. We do not discuss the need for generalization per se-it is assumed that line generalization is desirable for the usual reasons of visual clarity and/or economy of data.There are two areas where generalization on the sphere is appropriate and where data should be maintained in spherical coordinates. The first category is comprised of analytical applications, where the spherical line information is incorporated in other spherical processing. (This is in contrast to a display application, discussed below.) A good example of analytical use is the one that motivated this paper, namely, the production of global climate maps from point data. In smd-scale isarithmic mapping, a strong case can be made for interpolation on the sphere (see Willmott, Rowe, and Philpot 1985). In this approach isolines are created on the sphere by passing point data through a spherical interpolation and contour lacing procedure (e.g., Wahba 1981; Renka 1984; Lawson 1984). Because the line data are needed to construct the isolines (clipping contours to coastlines) they are most easily processed in the same coordinate system as the isolines themselves. If an interpolation method is thought of as a spatial model, the line data might be seen providing boundary conditions. When the interpolation model is formulated on the sphere, it is only natural that the boundary conditions should also be spherical. Interpolation is but one example of the analytical use of line data; others that come to mind are calculating areas and volume on the sphere (e.g., to compute spatial averages over continents).The second application of spherical line generalization is displaydriven, in which a line archive would be produced for repeated mapping at constant scale. This application arises because the level of generalization required is influenced more by

Robust line simplification on the surface of the sphere

“…As is stated in Burt (1989), in order to accelerate the algorithm execution in the tolerance checking, the …”

“…However, the high number of trigonometric functions involved in this computation will penalize the execution speed of the whole algorithm, so some work around in order to mitigate that this effect is mandatory. In Burt (1989), a solution based on rotations in the Euclidean 3D space is proposed.…”

“…In order to compute the spherical distance between the point P and the arc AB without any spherical triangle resolution (except for special cases, which will be explained later), and inspired in Burt (1989), the next algorithm is proposed:…”

“…In this paper, the method presented in Pallero (2013) is adapted to work on the surface of the sphere using some ideas from Burt (1989). The new algorithm is explained in detail and its implementation in C is put under a free software license.…”

“…In Burt (1989), a spherical version of the Douglas-Peucker Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/cageo method is described. By using rotations in the Euclidean 3D space, the original Douglas-Peucker algorithm is adapted to work with polylines on the surface of a sphere, so the distances are considered in the correct way as great circle arcs.…”