Villanova University, Department of Mathematical Sciences

Cartographiometry (MAT 1210/ GEO 1700)

(  copyright 1996-1999, Timothy G. Feeman and Elaine F. Bosowski.)

Key words and phrases

What follows are brief descriptions or definitions of some of the important terms and concepts encountered in the Cartographiometry course. Neither this list nor the information in it is intended to be complete. You should consult the course notes for additional information.


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Antipodes Azimuthal equidistant projection Azimuthal projections
Cartesian coordinate system Cartographic Communication System Cartography
Conformal maps Conic projection Cylindrical projection
Equal area map Equatorial Equivalent map
Eratosthenes Geodesy Geoid
Gnomonic projection Great circles Latitude
Longitude Loxodrome Map attributes
Map perspective Map projection Mean time
Mercator Mercator's map Oblique
Orthographic projection Polar Polar coordinate system
Polyconic projection Rectangular grid system Rhumb line
Scale Factors Representative Fraction Bar/Graphic Scale
Solar time Spherical coordinate system Spherical geometry
Spherical triangles Spheroid Standard line
Stereographic projection T-in-O maps Time Zones


Cartography

Cartography is the art, science, and technology of making maps. Cartography employs the scientific method in the construction of its products -- maps. The fields of Geography, Geodesy, Psychology, Mathematics, and Art all contribute to the work of the cartographer.

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Cartographic Communication System

The Cartographic Communication System describes the complex of relationships that exist between the Map User, the Environment, Maps, the Map Author, and the Mental Maps of the Map User and Map Author.

Each of us has a Mental Map created through both direct and learned experience as well as our imagination. The Map Author, having a specific purpose or message in mind, creates a map that becomes part of the Map User's learned experience, thus enriching the Map User's Mental Map. The Map Author uses cartographic language, a sort of code, in order to formulate and express the map's message for the Map User.

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Eratosthenes

Eratosthenes lived, worked, and studied in Alexandria, Egypt, over two thousand years ago, from about 276 B.C. to about 196 B.C.. He was the first person, as far as we know, to scientifically calculate and record the size of a planet, the earth.

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Geoid

Geoid, which means "earthlike", defines the smooth shape that most closely approximates the true shape of the earth with all its bumps, bulges, depressions, and divisions.

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Geodesy

Geodesy is "the branch of applied mathematics that determines the exact positions of points, and the shapes and areas of large portions of the earth's surface; the shape and sixze of the earth; and the variations of earth's gravity and magnetism." (Webster's Seventh New Collegiate Dictionary, 1969.)

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Spheroid

The true shape of the earth, the geoid, can be averaged out to form a more regular, mathematical shape -- the spheroid, a sort of three-dimensional oval shape. The earth is more accurately described as an oblate spheroid, not a perfect sphere, but a spheroidal shape that is oblate, or flattened, at the poles and bulges somewhat south of the equator.

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Rhumb line, or loxodrome

A rhumb line, or loxodrome, is a path of constant compass bearing on the earth's surface (or in the air just above the surface). Compass bearings are measured as angles away from the northerly direction, so a rhumb line, or loxodrome, crosses every meridian at the same angle. The purpose of Mercator's map is to show all rhumb lines as straight lines on the map, thus facilitating navigation by compass.

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Latitude

The latitude of a point on the earth's surface is given by the point's angular distance north or south of the equator. If we slice the earth with a plane parallel to the plane of the equator, the resulting circular cross-section is called a parallel. All points lying on the same parallel share the same latitude.

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Longitude

The longitude of a point on the earth's surface is given by the point's angular distance east or west of a pre-selected prime meridian. A semi-sircular arc running from the North Pole to the South Pole is called a meridian. All points lying on the same meridian share the same longitude.

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Time and time zones

As the earth rotates east-to-west about its axis, the height of the sun in the sky changes. The position of the sun in the sky determines ones local solar time. The moment at which the sun is highest in the sky corresponds to local solar noon. All points lying on the same meridian experience local solar noon at the same moment.

An earth day is the time it takes for the earth to complete one full rotation about its axis. For several reasons, this amount of time varies (only very slightly) throughout the year. Mean timeis an averaging out of these slight variations so that each mean earth day represents the same amount of time. One hour is one-twentyfourth part of a mean earth day.

The earth rotates east-to-west about its axis once eachday. That is 360 degrees of rotation every 24 hours, or 15 degrees of rotation per hour. For this reason, the time zones that we commonly use today change roughly every 15 degrees of longitude.

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Great circles

A great circle on a sphere is a circle that is obtained by intersecting the sphere with a plane containing the center of the sphere.

Use Dan Schwalbe's java applet to "draw" arcs of great circles on the earth.

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Spherical geometry

Spherical geometry is the study of figures on the surface of a sphere in much the same way that Euclidean geometry is the study of figures in a plane.

In the geometry of the sphere, great circles play the part of straight lines. They represent the shortest distance between two points. Every great circle is determined by a plane that contains the center of the sphere. Two such planes intersect in a line that punctures the sphere at a pair of antipodal (opposite) points. In other words, two great circles always intersect. This is different from the geometry of the plane (Euclidean geometry) where there are straight lines that do not intersect (parallel lines).

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Spherical triangles

A spherical triangle is formed when arcs of three different great circles meet in pairs (just as a triangle in the plane is formed by three mutually intersecting straight line segments). The angles of a spherical triangle always add up to at least 180 degrees and never more than 540 degrees. The area of a spherical triangle depends on the angles at which its sides meet. Specifically, the area of a spherical triangle, on a sphere of radius R, is equal to (Pi*R^2)*(angle sum - 180)/180. The maximum value this area can have is 2*Pi*R^2 which occurs when the spherical triangle fills up an entire hemisphere and has an angle sum of 540 degrees.

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Cartesian coordinate system

The Cartesian coordinate system is a coordinate system in the plane based on two perpendicular axes, one horizontal and one vertical, each marked with a scale. The point where the two axes meet is called the origin of the system. Each point in the plane can then be located by an ordered pair of numbers (x, y), called its coordinates. The first coordinate tells the point's distance to the right or left of the origin while the second coordinate tells how far above or below the origin the point lies.

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Polar coordinate system

The polar coordinate system is a coordinate system in the plane based on the selection of a point, designated as the origin of the system, and a reference direction (really a half-line emanating from the selected origin). Each point in the plane can then be located by an ordered pair of numbers (r, t), called its coordinates. The first coordinate tells how far the point lies from the selected origin. The second coordinate gives the angle, measured counterclockwise, from the reference direction to the line segment connecting the point to the origin.

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Spherical coordinate system

The spherical coordinate system is a coordinate system on a given sphere based on the selection of a pair of antipodal (opposite) points, designated as the poles of the system. It is also necessary to select one particular semi-circular arc connecting the two poles to serve as the prime meridian for the system. The circle that lies halfway between the poles is called the equator. Each point on the sphere can then be located by an ordered pair of numbers (u, v), called its latitude and longitude, respectively. The first coordinate gives the angular distance of the point north or south of the equator. The second coordinate gives the angle of the point east or west of the prime meridian.

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Rectangular grid system

Developed by the military to aid in the positioning of artillery, the rectangular grid system is a coordinate system in the plane based on the Cartesian coordinate system. Beginning with a rectangular grid, label the nodes along the bottom edge of the grid with a sequence of two-digit numbers. Do the same for the left edge of the grid. Each section of the grid can be subdivided into ten equal parts each of which can be further subdivided into ten and so on. (Thus, 215 lies halfway between node 21 and node 22, for example.) Each point within the grid can be located by a string consisting of an even number of numbers. The first half of the string tells how far to the right the point lies while the second half tells how far up. For example, the point 215320 lies halfway between nodes 21 and 22 moving to the right in the grid and even with node 32 moving up in the grid.

Such a grid can be placed anywhere on a map and used to assign locations to points on the map.

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T-in-O maps

T-in-O maps, commonly used during the Medieval period in Europe, are based on the Biblical story of the division of the earth's land to Noah's sons, Shem, Ham, and Japheth, following the Great Flood. The name "T-in-O" comes from the basic form of such a map as a letter "T" inscribed within a large letter "O". The three branches of the letter "T" represent the Nile and Don Rivers and the Mediterranean Sea which together divide the three large land masses of Africa, Asia, and Europe, represented by the regions inside the letter "O" and bounded by the branches of the "T".

On most "T-in-O" maps, Jerusalem is located at or near the center while Asia (the "Orient") is located at the top, perhaps because of the notion that the Spice Islands of the Far East might be the location of "Paradise". For this reason, the term "to orient" came to mean aligning one's map to have Paradise (or Asia, at least) at the top. Hence, the two uses of the word "orient" today.

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Map attributes

Every flat map of the earth must distort some of the earth's features. Therefore, when we analyze a the properties of a given map, we focus on certain important map attributes that the map may or may not have.

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Map projection

The term map projection, or simply projection, is used by cartographers to describe any representation of the earth as a map. The origin of this terminology lies in the idea of literally "projecting" the earth's surface onto some other surface by means of an imaginary light source. For specific examples, see azimuthal projections, cylindrical projections, and conic projections.

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Azimuthal projections

An azimuthal projection is obtained by projecting the earth, either literally or figuratively, onto a plane that touches the earth at a single point, called the center of the map.. The perspective of the projection is either polar, equatorial, or obliqueaccording to whether the map's center is at a pole, on the equator, or somewhere else. As part of your laboratory work, you will construct an azimuthal equidistant projection. Other classical examples of azimuthal maps are the gnomonic projection, stereographic projection, and orthographic projection.

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Map perspective

The perspective of a world map indicates the location of the map's center. If the center is at one of the poles, the perspective is said to be polar. A map centered on a point on the equator has an equatorial perspective. The map's perspective is said to be oblique if the center is anywhere else.

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Azimuthal equidistant projection

An azimuthal equidistant projection is a map of the earth (or some part of it) obtained by choosing a central point and portraying all other points at their correct (relative) distances and directions from the central point. An azimuthal equidistant projection centered on your home would show the correct direction and distance to go to reach any other place by the shortest possible route. In your laboratory work, you will construct an azimuthal equidistant projection of one hemisphere centered on one of the poles.

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Gnomonic projection

A gnomonic projection is an azimuthal projection obtained by projecting the earth onto a plane that touches the earth at a single point, using an imaginary light source located at the earth's center.

With a gnomonic projection, points on the earth that are half-way or more around the world from the central point of the projection will not cast a shadow on the map plane. So a gnomonic projection cannot show more than a hemisphere.

Given any two points on the earth, the great circle connecting them is defined by the plane passing through those two points and the center of the earth. This plane will intersect the projecting plane of a gnomonic projection in a straight line passing through the shadows (images) of those two points on the map. Thus, gnomonic projections have the important property that shortest routes (great circle arcs) on the earth are portrayed as a straight line segments on the map.

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Stereographic projection

A stereographic projection is an azimuthal projection obtained by projecting the earth onto a plane that touches the earth at a central point using an imaginary light source located directly opposite (antipodal to) the map's center. A stereographic projection can show the whole earth, except the point where the light source is located. The principal importance of the stereographic projection is that it is conformal.

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Orthographic projection

An orthographic projection is an azimuthal projection obtained by projecting the earth onto a plane that touches the earth at a central point using an imaginary light source located "at infinity" and shining at right angles to the projecting plane. To avoid overlapping images, only the hemisphere closest to the central point of the map can be shown. The principal feature of an orthographic projection is that the earth appears in the map as it would if viewed from deep in space. A photograph of the earth from space is essentially an orthographic projection (though the camera is not really far enough away).

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Cylindrical projection

A cylindrical projection is obtained by projecting the earth onto a cylinder. The cylinder is then slit open and laid out flat to produce the map. A cylindrical projection may have either one or two standard lines depending on whether the projecting cylinder wraps around the earth, touching it along a great circle, or passes through earth, cutting the earth in two smaller circles.

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Conic and polyconic projections

A conic projection is obtained by projecting the earth onto a cone. The cone is then slit open and laid out flat to produce the map. A conic projection may have either one or two standard lines depending on whether the projecting cone is placed on the earth like a sort of hat, thus touching the earth along a circle, or passes through the earth, cutting the earth in two circles.

A polyconic projection is created by projecting different portions of the earth's surface onto portions of different cones each with its own standard lines. The United States Geological Survey frequently uses polyconic projections for its maps.

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Conformal maps

A conformal map is one that preserves angles. A conformal map also shows the true shapes (at least locally) of regions on the earth's surface.

By preserving angles, it follows that distances and areas are necessarily distorted by a conformal map. Mercator's map, the stereographic projection, and Lambert's conformal conic projection are some examples of conformal maps.

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Equal area / Equivalent maps

An equal area, or equivalent map is one that shows regions of the earth's surface with their correct proportional areas.

By portraying areas correctly, and equal area map must necessarily distort distances and shapes. The Mollweide projection, the Gall-Peters projection, and Lambert's equal area azimuthal projection are some examples of equal area, or equivalent, maps.

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Standard line

A standard line for a given map projection is a line on the map along which there is no distortion.

For a cylindrical or conic projection, a standard line will occur wherever the projecting cylinder or cone touches the sphere. An azimuthal projection generally has no standard line as the projecting plane touches the globe at only one point.

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Mercator projection

Use Dan Schwalbe's java applet to compare lines of constant bearing to shortest routes.

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Gerhard Kremer (Mercator)

Gerhard Kremer, known more commonly as Mercator (the Latin version of his name), was a sixteenth century Flemish geographer and mathematician. He presented his famous map in 1569.

Mercator's map actually requires the Calculus of integrals for its exact formulation. Designing it nearly a century before the invention of Calculus in the 1660's, Mercator's original version was an approximation to the more exact design we use today.

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Antipodes/ Antipodal points

Two points on the sphere are called antipodal points, or antipodes, if the line connecting them passes through the center of the sphere. This means that the points are opposite to each other on the surface of the sphere, like the North and South poles.

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Scale Factors

One means of indicating the scale of a map, a representative fraction (or RF) is, as its name suggests, a fraction or ratio giving the relative sizes of distances on the map and the actual distances they represent. Thus, an RF of 1:1,000,000 means that a distance of one unit on the map represents an actual distance of 1 million units on the earth. If a map is enlarged or reduced, an RF printed on it becomes invalid (why?).

A second way of indicating a map's scale is a verbal scale such "one inch to 16 miles". This means that a distance of 16 miles on the earth is represented by a length of 1 inch on the map. A verbal scale printed on a map becomes invalid if the map is enlarged or reduced.

A third means of indicating scale, and the only one that is still valid if the map is enlarged or reduced, is a bar scale (or graphic scale). A bar sale shows in a picture the length on the map that represents one mile or one kilometer on the earth. If the map is doubled, this picture is also doubled and still represents the stated distance on the earth.


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timothy.feeman@villanova.edu

10-october-2002


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