A. Jon Kimerling

Geodesic Discrete Global Grid Systems

In recent years, a number of data structures for global geo-referenced data sets have been proposed based on regular, multi-resolution partitions of polyhedra. We present a survey of the most promising of such systems, which we call Geodesic Discrete Global Grid Systems (Geodesic DGGSs). We show that Geodesic DGGS alternatives can be constructed by specifying five substantially independent design choices: a base regular polyhedron, a fixed orientation of the base regular polyhedron relative to the Earth, a hierarchical spatial partitioning method defined symmetrically on a face (or set of faces) of the base regular polyhedron, a method for transforming that planar partition to the corresponding spherical/ellipsoidal surface, and a method for assigning point representations to grid cells. The majority of systems surveyed are based on the icosahedron, use an aperture 4 triangle or hexagon partition, and are either created directly on the surface of the sphere or by using an equal-area transformation. An examination of the design choice options leads us to the construction of the Icosahedral Snyder Equal Area aperture 3 Hexagon (ISEA3H) Geodesic DGGS.

Comparing area and shape distortion on polyhedral-based recursive partitions of the sphere

Regular grid sampling structures in the plane are a common spatial framework for many studies. Constructing grids with desirable properties such as equality of area and shape is more difficult on a sphere. We studied the distortion characteristics of recursive partitions of the surface of the globe starting with the octahedron and icosahedron polyhedral models. We used five different methods for mapping from the polyhedral model to the surface of the sphere: the Gnomonic projection, Fullers Dymaxion projection, Snyders equal area polyhedral projection, direct spherical subdivision, and a recursive polyhedral projection. We increased partition density using both a 4-fold and a 9-fold ratio at each level of recursive subdivision by subdividing to the 8th level with the 4-fold density ratio (65 536 cells per polyhedral face) and to the fifth level with the 9-fold density ratio (59 049 cells per polyhedral face). We measured the area and perimeter of each cell at each level of recursion for each method on e...

A Cartographic Study of Equal Value Gray Scales for Use With Screened Gray Areas

The determination of the functional relationship between the tone of a screened gray area and its perceived value is not of interest only to cartographers. This relationship has been studied by psychologists and others for over two centuries. Various experimental methods have been employed in the past to define the relationship, with each form of experiment giving differing results. The scale resulting from this authors testing program involving screened gray areas was very similar to the Munsell equal value scale. The effect of age, sex, experience, wearing of glasses, and the background on which the gray areas were viewed on the form of the gray scale was studied. The viewing background was found to be the only factor which significantly affected the form of the equal value scale.

Dotting the Dot Map, Revisited

Dot maps show the geographic distribution of a phenomenon in an area by placing dots representing a certain quantity of the phenomenon where it is most likely to occur. The fundamental steps in dot mapping are selecting the dot size, determining the dot unit value, and placing the correct number of dots in a manner that correctly reflects the geographic distribution of features. Selecting the dot size is a subjective decision, but the dot unit value has long been determined with the aid of the Mackay nomograph. Close examination of the nomograph finds it is not appropriate for determining the dot unit value when dot placement is based on computer-generated random numbers that result in overlapping dots. A new graphical aid for the determination of dot unit value was created by modeling aggregate area of dots and amount of dot overlap using a truncated form of the unification equation from probability theory. Aggregate dot areas predicted by this equation were tested against actual random dots created for several common dot sizes, and high agreement was found between measured and predicted aggregate areas. Pseudo-random dot placement with a maximum overlap constraint for dot pairs appears to mimic more closely how cartographers have traditionally placed dots. Pseudo-random dot placement can be thought of as similar to rigid random placement of circles in a square with maximum circle overlap limits from mutually exclusive to completely random dots. Thinking of dot placement in this manner allowed a general equation for aggregate dot area to be devised as a weighted combination of the mutually exclusive and completely random dot endpoint equations. Aggregate areas predicted by this general equation were found to match actual assemblages of pseudo-random dots with differing maximum dot pair overlaps closely.

A comparison of intercell metrics on discrete global grid systems

Abstract A discrete global grid system (DGGS) is a spatial data model that aids in global research by serving as a framework for environmental modeling, monitoring and sampling across the earth at multiple spatial scales. Topological and geometric criteria have been proposed to evaluate and compare DGGSs; two of which, intercell distance and the “cell wall midpoint” criterion, form the basis of this study. We propose evaluation metrics for these two criteria and present numerical results from these measures for several DGGSs. We also consider the impact of different design choices on these metrics, such as predominant tessellating shape, base modeling solid and partition density between recursive subdivisions. For the intercell distance metric, the Fuller–Gray DGGS performs best, while the Equal Angle DGGS performs substantially worse. For the cell wall midpoint metric, however, the Equal Angle DGGS has the lowest overall distortion with the Snyder and Fuller–Gray DGGSs also performing relatively well. Aggregation of triangles into hexagons has little impact on intercell distance measurements, although dual hexagon aggregation results in markedly different statistics and spatial patterns for the cell wall midpoint property. In all cases, partitions on the icosahedron outperform similar partitions on the octahedron. Partition density accounts for little variation.

Non-Photorealistic Rendering and Terrain Representation

In recent years, a branch of computer graphics termed non-photorealistic rendering (NPR) has defined its own niche in the computer graphics community. While photorealistic rendering attempts to render virtual objects into images that cannot be distinguished from a photograph, NPR looks at techniques designed to achieve other ends. Its goals can be as diverse as imitating an artistic style, mimicking a look comparable to images created with specific reproduction techniques, or adding highlights and details to images. In doing so, NPR has overlapped the study of cartography concerned with representing terrain in two ways. First, NPR has formulated several techniques that are similar or identical to antecedent terrain rendering techniques including inclined contours and hachures. Second, NPR efforts to highlight or add information in renderings often focus on the use of innovative and meaningful combinations of visual variables such as orientation and color. Such efforts are similar to recent terrain rendering research focused on methods to symbolize disparate areas of slope and aspect on shaded terrain representations. We compare these fields of study in an effort to increase awareness and foster collaboration between researchers with similar interests.

Hillshading of Terrain Using Layer Tints with Aspect-Variant Luminosity

Hillshading provides a rendering of topographic surfaces by assigning brightness to surface elements based on the orientation of these elements and a selected direction of illumination. Users easily visualize many topographic features, but some areas lack detail, as one shade of gray does not define a unique surface orientation. We clarify some of this ambiguity by varying the color of layer tints with aspect direction. We use the CIELAB color model to quantify color specifications and map variations in luminosity onto slices of the Hue-Saturation-Value (HSV) color model. Traditionally, cartographers assign an aspect-invariant color (or colors) based on H and S and vary V with the hillshading values. In our research, we assign aspect-variant H and V values in close proximity in HSV color space. We use values of luminosity and saturation from the CIELAB and HSV color models to select colors that are least saturated, most saturated, least luminous, and most luminous to represent the northwest, southeast, southwest, and northeast directions, respectively. We then vary V in the traditional manner with hillshading from the northwest. Topographic details not apparent in the original hillshaded maps are highlighted with this technique.

Modifications of Tanaka's Illuminated Contour Method

Visualization of topography can be greatly facilitated by the illuminated contour method. This method, popularized in a hand-drafted map by Tanaka, uses a gray background with black and white contours. A direction of illumination is assumed, and white contours represent illummated topography, while black contours represent non-illuminated or shaded areas. Additionally, thickness of contours varies with the cosine of the angle between the azimuth of maximum slope (i.e., aspect) and the azimuth of illumination. We modified Tanakas method by basing thickness of contour lines on twice the cosine of the angle between the surface normal and the illumination vector. The cosine of this angle is most commonly used in analytical hill shading. In addition, we present maps with changes in other visual variables and offer our evaluations. Lines with gray tones instead of black and white lines do not improve the illumination effect. We believe variations in the colors of contours and background with elevation can visually enforce information regarding topography. Our use of colors for aspect and variations in the width of contours for slope adds information to the map but does not assist with visualization of topography.

Color Specification in Cartography

If cartographers are to replicate color research by colleagues and compare different studies in a consistent manner, color definition must be standardized. Of prime importance is agreement on psychophysical color specification systems, and a common method for converting between these systems and process color printing specifications. Cartographers widely employ CIE and Munsell specification systems in research and instruction, while color charts based on tint screens are used in printing specifications. Process color tint screen percentage combinations can be converted mathematically to CIE chromaticity coordinates. Diagrams can be used to convert between Munsell hue, value, and chroma designations and CIE coordinates. Conversion equations are applied to the Defense Mapping Agency process color system to illustrate the methods in a cartographic setting.