Equal-area Projection Research Articles

SOME NEW EQUAL AREA MAP PROJECTIONS

AbstractThe investigation which follows was undertaken in order to clarify some confusion regarding the elliptical equal-area projection attributed to Hammer. In the course of this investigation a number of new equal-area projections of the sphere were discovered. These projections are all bounded by a circle or an ellipse. The resulting maps are of some theoretical interest and are attractive in appearance, though the angular distortion (as measured by Tissot's indicatrix) exceeds that of already available projections.

SOME NEW EQUAL AREA MAP PROJECTIONS

The investigation which follows was undertaken in order to clarify some confusion regarding the elliptical equal-area projection attributed to Hammer. In the course of this investigation a number of new equal-area projections of the sphere were discovered. These projections are all bounded by a circle or an ellipse. The resulting maps are of some theoretical interest and are attractive in appearance, though the angular distortion (as measured by Tissot's indicatrix) exceeds that of already available projections.

Geochemical and Structural Studies in Batholithic Rocks of Southern California: Part 1, Structural Geology of Rattlesnake Mountain Pluton

The Rattlesnake Mountain pluton, a funnel-shaped body of coarse-grained porphyritic quartz monzonite, is an offshoot from a batholithic mass of biotite-quartz monzonite forming the central part of the San Bernardino Mountains in Southern California. The pluton intrudes a structurally complex region of metamorphic and igneous rocks that borders the batholith on the north side of the mountains. The metamorphic rocks consist of marbles, calc-silicate hornfels, schists, quartzites, amphibolites, gneisses, granulites, and migmatites. The degree of metamorphism is in the higher grades of the almandine-amphibolite facies and of the granulite facies. Igneous rocks of the pluton range from quartz monzonite through granodiorite to quartz diorite and were emplaced at the same time as the main part of the batholith. The intrusion of the pluton was postkinematic and disrupted the regional structure. The most distinctive features of the Rattlesnake Mountain pluton are its shape and internal structure. The upper part of the funnel flares outward and is divided into several lobes and sheets separated by screens of mafic rocks. The sheets follow the funnel pattern, lying concentrically within one another and dipping inward toward a central point. The lobes are tongue-shaped and extend from the upper edge of two of the sheets. One of the lobes extends outward for more than 5 miles from the funnel. An unusual abundance of inclusions, schlieren, and dikes are present. These structures are in many cases oblique to one another; schlieren commonly cut across or are truncated by other schlieren, and inclusions lie in various orientations. Structural data were plotted on equal-area projections; in any one area, normals to the flattened surfaces of inclusions and schlieren form a pattern that commonly consists of an incomplete great circle containing a central maximum.

A CLASSIFICATION OF MAP PROJECTIONS1

THE desire for a classification of map projections stems from the fact that an infinite number of distinct projections are possible. Hence, the fundamental problem in classifying map projections is the partitioning of this infinite set into a comprehensible and useful finite number of all-inclusive and preferably non-overlapping classes. Several classifications of map projections are to be found in the cartographic literature. The advantages and disadvantages of each, of course, depend on the purpose, just as the properties by which classes are to be distinguished depend on the purpose. The classification based on geometric models (perspectivities) separating projections into conic, cylindric, planar, polyconic, polycylindric, etc., is convenient and is often used. The major shortcoming of this system is that it is not allinclusive. Another most important method of classification, based on the preservation of certain geometric properties of the surface of the referent object, separates the conformal projections from the equal-area projections, leaving a third class which is neither. Preservation of additional properties yields further classes; e.g., geodesic maps (for the sphere there is the gnomonic projection and all linear transforms thereof), azimuthal projections, equidistant projections (including polar isometries such as the azimuthal equidistant), and many more. In fact, Felix Klein, the German mathematician, defined a geometry as the study of properties preserved by a particular group of transformations. From this point of view, a classification according to properties preserved is the very essence of geometry. Thus one has projective transformations and projective geometry, affine transformations and affine geometry, Euclidean

Structural features of the Alpine Schists of the Franz Josef — Fox Glacier Region

Summary Fold axes in the steeply dipping greywackes and schists to the west of the Alpine divide may be located by determinations of the superposition of beds made from observations of: (1) graded bedding; (2) intraformational structures; (3) contemporaneous deformation structures; (4) crossbedding; (5) slip folds; (6) drag folds and minor folds: (7) axial plane cleavages, joints and quartz veins; and (R) feather ioints. Both joints and fault patterns have been demonstrated to be statistically related to the fold axes by means of equal area projections. The fold axes trend between 40° and 50° and locally diverge southward from the Alpine fault whose strike is 58°. Petrofabric study of calcite optic axes from marble shows a preferral orientation nearly normal to the Alpine fault. The direction of maximum principal stress during deformation inferred from fold axes and Joints is about 310° , while that inferred from the attitude of the Alpine Fault and the calcite orientation is about 327°, but it is not thought that the difference is significant,

Crystal Orientation in Glacier and in Experimentally Deformed Ice

AbstractMore than 8,000 ice crystals have been oriented and measured for crystal fabric studies from widely separated temperate and polar glaciers, using a large universal stage and thin-section techniques. Very strong fabrics have been found and a number of laboratory experiments on deformation and recrystallization of ice were conducted in an attempt to solve some of the perplexing problems raised concerning glacier flow.In polar glaciers the c or optic axes of the ice crystals tend to be perpendicular to the foliation plane (alternating planar structures of bubbly and clear ice). In areas of high shearing stress the preferred orientation of the axes reached 39 per cent in 1 per cent of the area when plotted on a Schmidt equal-area projection. In temperate glaciers the optic axes tend to form three or four strong maxima which also appear related to the foliation.Patterns from ice deformed in the laboratory resemble some of the fabric patterns found in polar glaciers. During deformation of laboratory specimens, large crystals have been observed recrystallizing into many smaller ones, while fine-grained ice, after completion of deformation (both glacier ice and laboratory deformed ice), has been annealed at melting temperature into a few large crystals with different orientations from the original pattern.

Crystal Orientation in Glacier and in Experimentally Deformed Ice

Abstract More than 8,000 ice crystals have been oriented and measured for crystal fabric studies from widely separated temperate and polar glaciers, using a large universal stage and thin-section techniques. Very strong fabrics have been found and a number of laboratory experiments on deformation and recrystallization of ice were conducted in an attempt to solve some of the perplexing problems raised concerning glacier flow. In polar glaciers the c or optic axes of the ice crystals tend to be perpendicular to the foliation plane (alternating planar structures of bubbly and clear ice). In areas of high shearing stress the preferred orientation of the axes reached 39 per cent in 1 per cent of the area when plotted on a Schmidt equal-area projection. In temperate glaciers the optic axes tend to form three or four strong maxima which also appear related to the foliation. Patterns from ice deformed in the laboratory resemble some of the fabric patterns found in polar glaciers. During deformation of laboratory specimens, large crystals have been observed recrystallizing into many smaller ones, while fine-grained ice, after completion of deformation (both glacier ice and laboratory deformed ice), has been annealed at melting temperature into a few large crystals with different orientations from the original pattern.

GRAPHICAL STATISTICAL ANALYSIS OF FRACTURE PATTERNS IN ROCK ENCOUNTERED IN ENGINEERING PROJECTS

The directional competence of badly fractured rock in engineering foundations can be analyzed by plotting the various types of fractures on a Schmidt equal-area projection and combining the data suitably. Interpretation of the results may increase safety factors and efficiency in grouting, strengthening devices, design, and tunnelling.

新らしい合成正積図法

新らしい合成正積図法

“Map: World Distribution of Spirochaetal Diseases. 3. Leptospiroses

The Briesemeister Elliptical Equal-Area Projection adopted may be esthetically satisfying to cartographers but is markedly inferior to the distorted but useful Mercator projection in avoiding meaningless clutter of superimposed symbols. The references listed on the reverse of the map are incomplete and nondiscriminating, e.g., omission of “Leptospirosen” by Gsell and Wiesmann (1952) and “The Laboratory Diagnosis of Leptospirosis” by Wolff (1954), and failure to include primary references in the United States. The section entitled Epidemiology is a hodge-podge of assumptions, mis-statements and mistakes. It is believed that omission of this plate from the Atlas would materially enhance its value.

A New Oblique Equal-Area Projection

OST published equal-area maps of the world show the major land masses with excessive shape distortion in certain areas or with a break in continuity. L VI v Notably, the relationship between Asia and the Americas is usually unsatisfactory, because of a lack of continuity in the Arctic and sub-Arctic regions. This is the case in such frequently used equal-area projections of the world as the Sinusoidal and the normal forms of the Mollweide and Hammer-Aitoff. Oblique forms of these projections, especially with the center at 450 N. and about 100 E., are preferable, because the Arctic Sea is shown without interruption and the land masses usually fall within areas of small shape distortion. The American Geographical Society many years ago started experimenting with the oblique Mollweide so centered, but examples of maps drawn on this projection were not published until World War II.U More recently it occurred to the writer that a somewhat better result would be obtained by utilizing an oblique version of the HammerAitoff. Such a projection was constructed and was used to illustrate H. Duncan Hall's article Zones of the International Frontier in the Geographical Reviewt.2 In both the Mollweide and the Hammer-Aitoff the major axis of the bounding ellipse is twice the length of the minor axis. In the oblique forms of these projections the parallels of latitude close to the North Pole are shown as ovals. It would be more desirable, however, to show these parallels as near circles; this would preserve closely the true relationship between the northern continents. To this end, the writer developed a projection with a bounding ellipse whose major axis is I.75 the minor axis, and constructed it graphically from the Lambert Azimuthal Equal-Area by a procedure similar to that used in constructing the Hammer-Aitoff. Such a map projection is particularly useful for the plotting of world distributions. The projection has been used for several maps in the Society's Atlas of Diseases, on various scales. In response to a number of requests, outline maps on two convenient scales have now been prepared and are available.3

Crystal Fabric Studies on Emmons Glacier Mount Rainier, Washington

Optical orientations of 1,725 ice crystals were determined at eleven locations on the Emmons Glacier by means of a 6-inch universal stage mounted between crossed polaroid sheets. The crystals were 0.2–6 inches across and from three to forty were included in each 4½ × 6-inch thin-section. Down-valley glacier movements of 3.4–9.2 inches per day were recorded at four of the higher locations, and another three were in ice believed to be essentially stagnant. The optic axes at most locations, when plotted on a Schmidt equal-area projection, form consistent patterns featuring four strong maxima at the corners of diamond-shaped quadrangles. Concentrations as high as 26 per cent in 1 per cent of the area were recorded. The angular separation of opposite corners of the diamond-shaped quadrangles average 84° and 52°, respectively. The pattern in stagnant ice is the same and as strong as the pattern in moving ice. Poles to shear planes and clear ice bands in the glacier at the locations studied fall near the centers of the diamonds. If this pattern is controlled by gliding within the ice crystals, it suggests the possibility of glide planes in ice crystals other than the well-known basal glide plane.

Equal-Area Projection on the Icosahedron

The base of each of the polar triangles is divided into equal intervals of longitude; hence areas of equal difference of longitude are mapped by triangles that have equal bases and the same altitude. The parallels are mapped by nonequidistant parallel lines. To determine the correct intervals between parallels, observe that the area of a zone between the North Pole and a parallel of latitude, L, having a colatitude 0 is

Equal-Area Projections and the Azimuthal Equidistant Projection in Maps of Disease

Equal-Area Projections and the Azimuthal Equidistant Projection in Maps of Disease Affiliation Saul Jarcho CopyRight https://doi.org/10.2105/AJPH.35.10.1005 Published Online: August 29, 2011

Four World Maps for Use in Biology and Geophysics

Even nowadays one frequently sees distributional data plotted on world maps on the Mercator projection, the only advantage of which is that loxodromes (rhumb lines) appear as straight lines, but which is grotesquely misleading for comparison of area and distance. Equal-area projections are, however, more and more used, the uncertainty mainly being which of the numerous kinds should be the most suitable one.

A COMBINED PROJECTION

The projection in question is a mean between the Cylindrical Equivalent (Equal-Area) Projection and its Transverse Projection. The position of any point on the earth's surface is defined by the mean x and the mean y of the two constituent projections.

A COMBINED PROJECTION

The projection in question is a mean between the Cylindrical Equivalent (Equal-Area) Projection and its Transverse Projection. The position of any point on the earth's surface is defined by the mean x and the mean y of the two constituent projections.

THE SCALE ERRORS OF AIRY'S PROJECTION BY BALANCE OF ERRORS, WITH SPECIAL REFERENCE TO THE BRITISH ISLES

AbstractTHE formula for the projection is based upon the spherical assumption. To calculate it for the spheroid might be very complicated and would not be worth while. The projection is suitable for very large areas as a compromise between the Zenithal Equal-area projection on the one hand and the Zenithal Equidistant or Zenithal Orthomorphic on the other. Its application to an area as small as the British Isles would not serve any useful purpose. An analysis of its errors in the general case reveals some unexpected simplicities. This analysis is given below, followed by its application to the particular case of the British Isles on the ten-mile scale. This is done merely to find out what changes would have occurred if the supposed drawing of that map on Airy's projection had been real.

An Introduction to the Mathematics of Map Projections

Preface 1. Introductory 2. The simple conic 3. Equal area projections 4. Orthomorphic projections 5. Perspective projections 6. Miscellaneous projections 7. The deformation of projections 8. Finite measurements 9. The best projection for a given country Index.

Base Maps of the World

Research Article| December 31, 1929 Base Maps of the World CHESTER A. REEDS CHESTER A. REEDS Search for other works by this author on: GSW Google Scholar GSA Bulletin (1929) 40 (4): 761–770. https://doi.org/10.1130/GSAB-40-761 Article history received: 21 Dec 1929 first online: 02 Mar 2017 Cite View This Citation Add to Citation Manager Share Icon Share Twitter LinkedIn Tools Icon Tools Get Permissions Search Site Citation CHESTER A. REEDS; Base Maps of the World. GSA Bulletin 1929;; 40 (4): 761–770. doi: https://doi.org/10.1130/GSAB-40-761 Download citation file: Ris (Zotero) Refmanager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search nav search search input Search input auto suggest search filter All ContentBy SocietyGSA Bulletin Search Advanced Search Abstract IntroductionSeven base maps of the world have recently been prepared and published by the American Museum of Natural History. These maps are primarily intended for plotting geographic and biologic data, both on the land areas of the globe and in the oceans. For this purpose several types of equal-area projections have been selected and an arrangement of continents sought, so that any undue distortion of form on one map is compensated on another.Map-makers are always confronted with a problem impossible of solution when they are asked to represent the surface of a sphere on a flat sheet of paper. However, many devices have been invented for representing the surface of the globe on a plane. They are known as map projections. There are many world-map projections and all have serious defects. The shapes of large areas are distorted and the distances are shown at different scales on various . . . This content is PDF only. Please click on the PDF icon to access. First Page Preview Close Modal You do not currently have access to this article.