The geometry of the earth has been discussed, studied, and imagined forever. The ancient Greek philosophers tried to picture a pure geometrical model. But it was largely Sir Issac Newton in the early 18th century who brought the shape of the earth into public awareness: he concluded that the earth was an oblate ellipsoid of revolution - an ellipse wider at the equator than at the poles. This theory was widely investigated. Enlightenment scientists set to the monumental task of measuring the earth.

Newton postulated that planets would be perfect spheres only if they didn't rotate. Rotating planets "flatten out" slightly, but not much. At the equator the diameter of the earth is 7927 miles; at the poles, it is very close to 7900 miles. 27 miles, a difference of 0.0034 is just enough to give mapmakers a headache.

Geodesists study the shape of the earth. Over time, they have created many different models of the earth's shape including both ellipsoid and spherical forms. These forms are perfect geometric constructs, unlike the earth, which is in a state of continual change, but these models have allowed cartographers to examine the world on a human scale.

2000 years ago, Greek scientists trying to understand our universe began to theorize about the shape of the earth and the placement of the stars. They created some of the first Geographic Coordinate Systems (GCS) - ways to specifically identify locations on the earth or in the sky. We're most familiar with the GCS based on the earth as a sphere and the sky as the inside surface of a sphere.

The sky is a sphere? We now know that the sky is limitless space but for purposes of astronomical mapping, the inside of a ball painted with stars is a convenient abstraction. Even the nearest star is so far away that our yearly trip around the sun doesn't change its locations relative to other stars. The planets change positions within a narrow belt of the celestial sphere about 20° above the horizon but only five planets (Mercury, Venus, Mars, Jupiter and Saturn) are visible to ordinary telescopes and their positions are easily plotted on the E/W, N/S grid of the dark celestial sphere.

We delineate the earth and sky with meridians and parallels using angular units of measure to define longitude and latitude. Scientists use several geometrical methods to measure angle - radians and x/y coordinates included - but the most accepted measure is the degree, which divides a full circle into 360 parts. From ancient times the number 360 was considered mystically potent, but it was also practical, since it could be divided by so many numbers into equal parts. A 360° circle can be divided into 2, 3, 4, 5, 6, 8, 9, 10, 12, 18, 24, 36, 45, 90, and 180 equal segments. Fractional locations can be expressed as decimal degrees, as in longitude 73.46° They can also be expressed in the older "circle" subcategories of minutes (minuta, meaning "small part") at 60 per degree, or seconds (secunda, the "second part" or next smallest, the same terms we use on a circular clock) at 60 per minute, as in 73° 27' 36", the same longitude.

Latitude is the angle measured from the earth's center north or south of a given point on the earth's surface - defined by geometric center of the earth's spin axis, the equator, lying at 000° latitude. "Horizontal" east-west lines of latitude are termed parallels, since each line each line of latitude is geometrically parallel to the next. Gainesville's latitude is about 29.7° North, while Durban, South Africa is about 29.9° South. No latitude has a value larger than 90°.

Longitude is the east or west location of a point, measured as an angle from the earth's center east or west of a prime meridian, a given point on the earth's surface at 0 degrees longitude. Over time this point has been identified as Jerusalem, Rome, Paris, Washington, DC, and other significant places, but because of the British Royal Navy's virtual mastery of the world's oceans between the 16th and the 20th centuries, and because of the British Admiralty's exhaustively accurate nautical charts, we measure from the Royal Navy's principal observatory on the Thames River, just below London at Greenwich, 000° longitude. Any line of longitude can be called a meridian because it extends from the geometric center of the earth. All longitudes are east or west of Greenwich, so the longitude of Gainesville is about 84° West, and the longitude of Khatmandu is about 85° East. No longitude has a value greater than 180°. The pure ideas of latitude and longitude are ideal mathematical overlays. Neither latitude nor longitude is an absolutely uniform unit of measure on the changing, impurely geometric earth. But their combination as a gridded network called a graticule is enormously useful. Picture a perfectly spherical grid generally holding a rough, imperfect ball; sometimes the perfect grid lies outside the ball, sometimes under its surface. The graticule has its origin at 000° latitude, 000° longitude - at the intersection of the equator and the prime meridian. We measure other celestial bodies - the moon, planets and even the sun - with a similar graticule. The shape and size of a graticule is defined by the sphere or ellipsoid upon which it is based. Recently, the use of satellites has led to more accurate measurements of the earth's shape and more accurate ellipsoids (there is a tiny thickening of the earth below the equator, a matter of yards and not miles). The most recently developed and widely used is the World Geodetic System of 1984 (WGS84). Another independent measurement commonly used is the Geodetic Reference System of 1980 (GRS80). These spheroids differ slightly but can be assumed to be identical for almost all mapping purposes.

An important element of a GCS is the datum, a base-point for the geometrical system. Since we use a sphere or ellipsoid to approximate the shape of the earth, we need a location point for its center. A datum defines the position of that spheroid relative to the center of the earth. The datum determines the placement of the coordinate system upon the ellipsoid. The datum defines the origin and orientation of lines of latitude and longitude, and is a "starting point" of reference for measuring locations on the surface of the earth. There are two kinds of datum - a geocentric datum is centered on the earth's center of mass, and a local datum is slightly offset to a convenient location in order to accommodate a particular region of study. The North American Datum of 1927 is a local datum used for many US maps. The North American Datum of 1983, however, is a geocentric datum, based on the most up to date measurements of the shape of the earth (WGS84 or GRS80). The High Accuracy Reference Network (HARN) or High Precision Geodetic Network (HPGN) are both names for the same project that is focused on readjusting the NAD83 datum to a higher level of accuracy state by state. This is an even more accurate form of the North American Datum. If you change the datum of map data, the coordinate values of the data also change.

During the time that Sir Issac Newton (and others) brought to the shape of the earth into public light in the early 18th century, topographic surveys of large regions were carried out. Newton's time also gave us calculus and logarithmic math, two crucial mathematical tools. The combination of new mathematical tools, topographic data, and earth measurements led to a cartographic revolution.

Think of all of the purposes that maps serve. They have helped us understand our place in the universe, on the earth, and in our own home towns. They help us understand physical, social and trading relations between different regions. They have been used as tools of politics for governing nations, pressing military campaigns and effecting trade development. They are used to navigate the oceans, skies, and even space. Maps have different scales, show many kinds of topical content and have different purposes. Maps are powerful tools suited for many applications, and focused on many different places.

With any map there is a basic difficulty: Round earth, flat paper. Long ago the Greek scientists of geometry struggled with the problem of depicting the lumpy earth and the celestial sphere on flat surfaces. How could they translate their surroundings into an understandable and useful picture? Imagine that you had to solve a comparable problem: how could you make the skin of a grapefruit lay flat while keeping the surface of the skin intact? There is no perfect solution, only partial answers that benefit special circumstances. Those answers are found in geometry and creative mathematics; this is the art of mathematical cartography.

Knowing that there are many ways to peel a grapefruit, and that there are different ways to serve grapefruit, you most likely realize that there are many different ways to make a map.

Over time, many formulas for depicting three-dimensional ellipsoid earth on flat paper have been created. Each geometric approach is called a map projection, since it geometrically "projects" points from a three-dimensional model onto a flat surface. The objective of a map projection is to express a physical surface in mathematical terms, using geometry and equations.

Think of the blueprint of a building: each three-dimensional point - the corner of a window or the position of a beam's end - is projected onto a flat map of the building. Since most buildings are rectilinear and built on the level, the projection is fairly simple. A mathematical formula is used to determine the mathematics of the distances between the points; on a blueprint this is the scale. 1/4" = 1'-0" is a common scale for room layouts, though the overall site plan may be drawn at 1/50" = 1'-0" or less, and window details may be drawn as large as 3" = 1"-0". Different scales for different purposes. On building blueprints, horizontal points are projected vertically as a plan, and vertical points are projected horizontally as an elevation. We look at the layout of the building from above (plan), and we look at the sides of a building from the side (elevations). No one blueprint will tell us everything about the building. Each blueprint "map" will show us one floor or one wall. The use of the map determines the projection.

Each method of map projections are based on different equations and different spheres or spheroids, in order to serve different purposes. Every map projection is a distortion of the earth in several ways. There is no "perfect" projection. For a particular map or application, the cartographer or data analyst must choose to highlight one set of characteristics at the expense of other characteristics.

To understand these projections a little more, we'll look at the way that they are constructed. Some projections use projective geometry, and are drawn on a flat surface with a t-square and a triangle. These projections are termed "developable." They consist of three basic developable surfaces: a plane above or below the spherical surface; a cone tangent to the spherical surface, and a cylinder wrapped around the spherical surface. Because a plane is a conveniently basic way to depict a compass direction, often called an azimuth, this projection family is called azimuthal (az i MYOOTH ul). The other surfaces give us cylindrical and conic projection families.

There are other, more mathematically inventive projections that cannot be developed by projective geometry. They are classified as azimuthal, cylindrical, or conic on the basis of their appearance or sometimes by the geometrician who devised them. Some complex projections that resemble the developable families but are different enough to be distinguished from them are classed as pseudo-azimuthal, pseudo-cylindrical, and pseudo-conic. They are less complicated to use than to make.

The figure below (coming soon) shows how a map surface is projected from an azimuthal, cylindrical and conic perspective. The plane or azimuthal class is often used for depicting the poles. The cylindrical class is often seen in atlases and other maps portraying the whole world. The conic class is used for mapping earth areas having a greater east-west extent than a north-south extent, like the United States. One important form in the conic class is the polyconic form, which creates a projection using many cones. Tough to imagine but they can do it.

Other special categories of projections include equal-area, conformal, and equidistant projections. Equal area projections are useful when looking at small features; on a map created with this projection each unit of area (a square meter or a square kilometer) at one location of covers a square meter or a square kilometer on every other map location. This sounds like the perfect map, but this type of projection distorts shapes, angles and scales. In conformal projections, large land masses are distorted while small shapes, local scales and relative angles in the large land mass are preserved. Equidistant maps show true scale between one or two points and every other point. Different maps for different purposes.

There are map "fashions." Historically, several different projections have been used to depict the United States. In the early nineteenth century, a simple polyconic projection was used first to map the US coast. Later, the USGS used the polyconic projection to create thousands of topographic quadrangles. This projection was used by the USGS until the 1950's. Other agencies began to use the Mercator cylindrical projection in the early twentieth century. Later in the century, some agencies converted to the Lambert conformal conic projection for maps of the entire country. After the 1950's, USGS quadrangles were based on the conformal State Plane Coordinate System (SPCS), which is not a projection but is based on two different projections depending on the shape of the region it describes. This system is broken down into grid zones. For states extending predominantly east-west, this system uses the Lambert conformal conic projection. For states extending predominantly north-south, the transverse cylindrical Mercator projection was adopted. Another familiar grid system came into use during the same period - the Universal Transverse Mercator (UTM) grid system which extends across the entire world in zones 6 degrees longitude wide. This system, like SPCS, is not a true projection but is based solely on the transverse cylindrical Mercator projection. In 1970, the Albers equal-area conic projection came into use for maps of the entire country.

To add to the complexity of these projections of the earth's surface, one must remember that there are mathematical models of the earth's shape - different spheres and ellipsoids - on which these projections can be based. Many of the earlier US projections were based on the Clarke 1866 ellipsoid. More recently, WGS84 and GRS80 have been used interchangeably. Changing the spheroid of a coordinate system changes all previously measured values. Due to the amount of work it takes to remedy the replacement of an out-of-date spheroid, many maps have not been converted.

As if this weren't enough, projected coordinate systems must have defined projection parameters. Parameters specify the data-point origins and customize projections to the area of interest. You will be delighted to hear that there are two categories of parameters. Linear parameters use projected coordinate system units like false eastings, false northings, and scale factors. Angular parameters use geographic coordinate system units like azimuths, central meridians, central parallels, longitudes and latitudes of origin, center, and standard parallels 1 and 2. These parameters sound confusing but they are the fine-tuning "fudge factors" that smooth out difficulties in a coordinate system. They allow the mapmaker to manipulate projections and their distortions.

What is the best way to depict a round world on a flat map? Now that you have a small basis for understanding how maps are created, you know that this is a trick question. The real question is: What do I want to learn from this mathematical distortion of a roundish, lumpy, untidy world depicted on a clean, flat sheet of paper? Maps for different purposes use different projections. You might even be able to guess, by now, how important it is to define the projection, datum, parameters and ellipsoid of your data.