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**Map Projections Do you know where you are?**

Mike Horsfall, Bentley

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**What do we know so far? Is the world spherical?**

What is the circumference of the earth? How fast are we spinning? What is the difference between a latitude and a longitude? Why do we have seasons? What is GPS?

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**It’s not flat, it’s not round, it is an Oblate Spheroid!**

Is the world spherical? It’s not flat, it’s not round, it is an Oblate Spheroid! The image on the left is a painting called “I Told You So” by Ed Miracle. His website says that “It’s been used to dramatize the dangers of ignoring over population and environmental issues, to illustrate social and economic trends, to teach history, science and economics, to inspire those too close to the edge…and to get a laugh.” The Earth's shape is very close to an oblate spheroid which is an rounded shape with a bulge around the equator. This bulge is caused by the rotation of the earth.

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**What is the circumference of the earth?**

The circumference of the earth at the equator is 24, miles / 40, kilometers Circumference of a circle = 2*pi*radius Clarke 1866 radius is 6,378,206.4 meters While the sphere is a close approximation of the true figure of the Earth and satisfactory for many purposes, to geodesists interested in the measurement of long distances. (ie, those that span continents and oceans), a more exact figure is necessary. Since the earth is flattened at the poles and buldging at the equator, the geometrical figure used in geodesy is an oblate spheroid. The spheroid that’s used to describe the figure of the Earth is called a reference ellipsoid. Because of their relative simplicity, reference ellipsoids are a preferred surface on which geodetic network computations are performed and point coordinates such as latitude, longitude and elevation are defined One such ellipsoid model is Clarke this historical reference ellipsoid was defined in 1866 and is named for the English mathematician Colonal Alexander Ross Clarke who derived the model. Using the Clark 1866 ellipsoid, the equatorial radius of the earth is calculated at 6,378,206.4 meters.

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**How fast are we spinning?**

At the equator 24, 901 miles around / 24 hrs = ~1000 mph One revolution every 24 hrs = 1 earth day days to revolve around the sun Are we spinning at the poles? It should be noted that the earth speed is not the same all over the globe; in fact the earth's speed can vary depending on the distance from the equator to the poles. For instance, the earth's speed is fastest at the equator- where the circumference has the most distance to travel around the axis. At the equator, the earth spins at 1,038 mph however if you are directly on the North or South Pole, the distance for the earth to revolve around the axis is practically zero meaning that the earth's speed is extremely slow. In fact, the earth's speed at the North or South Pole is about one centimeter per 24 hour period. If you live midway between the poles and the equator, the earth still spins fast, but not as fast as it does at the equator. Speeds for these locations are approximately 700 to 900 mph.

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**What is the difference between a latitude and longitude?**

Latitudes are parallel to equator, Longitude run pole to pole Prime meridian = Greenwich Meridian ~ GMT

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**Summer Northern Hemisphere Winter Northern Hemisphere**

Why do we have seasons? The earth is tilted at 23.5 degrees, to celestial horizon As the earth spins on its axis, producing night and day, it also moves about the sun in an elliptical (elongated circle) orbit that requires about 365 1/4 days to complete. The earth's spin axis is tilted 23.5 degrees with respect to its orbital plane, therefore when the Earth's axis points towards the Sun, it’s summer for that hemisphere. And when the Earth's axis points away, winter can be expected. Since the tilt of the axis is 23 1/2 degrees, the north pole never points directly at the Sun - on the summer solstice it points as close as it can, and on the winter solstice as far as it can. Summer Northern Hemisphere Winter Northern Hemisphere

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**Is the North Pole really north on a compass?**

Compass points to magnetic north Magnetic North wanders Grid north points to North Pole on the Central Meridian The Earth’s North Magnetic Pole is a wandering point on the Earth’s surface where the Earth’s magnetic field points vertically downwards. In 2001, the North Magnetic Pole was determined by the Geological Survey of Canada to lie near Ellesmer Island in northern Canada

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**What is GPS? Conceived by DOD in 1973**

Now 24 satellites, 20,000 km above earth, 12hr orbit Broadcasts two signals, one open for public, one coded for military purposes GPS stands for Global Positioning System and is a global navigation satellite system developed by the United States Department of Defense.

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What is a Datum? A map datum is a mathematical model that describes the shape of an ellipsoid (the earth). Some examples: Clark 1880, 1886 NAD27, NAD83, WGS84 World Geodetic System of 1984 (WGS84) based on the GRS ellipsoid – virtually identical with the NAD83. WGS84’s previous version is WGS72. Prior to NAD83, the most popular datum in U.S. was NAD27, based on parameters determined by Clarke in 1866 NAD27 from observation on earth, NAD83 from Satellites A geodetic datum is a set of reference points, or mathematical model, on the earth’s surface from which position measurements are made. Since reference datums can have different radii and different centre points, a specific point on the earth can have substantially different coordinates depending on the datum used to make the measurement. There are hundreds of locally-developed reference datums around the world, usually referenced to some convenient local reference point. Contemporary datums, based on increasingly accurate measurements of the shape of the earth, are intended to cover larger areas. The most common reference Datums used in North America are NAD27, NAD83, and WGS84

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**The basics of a projection**

The mathematical model to transfer a spherical object onto a flat surface. A map projection is any method of representing the surface of a sphere on a two dimensional plane. Map projections are necessary for creating maps and all map projections distort the surface in some fashion. Depending on the purpose of the map, some distortions are acceptable and others are not; therefore different map projections exist in order to preserve some properties of the sphere-like body at the expense of other properties. There is no limit to the number of possible map projections.

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Map Projections It can roughly be imagined that the shaded cylinder or cone wrap the Earth. While they slowly rotate around the Earth’s axis, a fixed source shoots light rays along a single meridian, projecting "shadows" of spherical features onto the surface. After a complete revolution, the tube is cut along a line parallel to its axis and unrolled. Just by changing the source position and tube's diameter, different maps result. (a) Azimuthal (b) Cylindrical (c) Conic. Light source positions, also called perspective positions, play an especially important role in planar projections. Different aspects, such as polar, equatorial, or oblique, will also affect the map projection.

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**Projection Classifications**

Spherical (Latitude, Longitude) Cylindrical (Mercator or Transverse Mercator) Conic (Lambert, Albers) Azimuthal (Stereographic) A fundamental projection classification is based on the type of projection surface onto which the globe is conceptually projected. The projections are described in terms of placing a gigantic surface in contact with the earth, followed by an implied scaling operation. These surfaces are Cylindrical (Mercator, or Transverse Mercator), Conic (Albers), or Azimuthal (Stereographic). Many mathematical projections, however, do not neatly fit into any of these three conceptual projection methods. Hence other categories have been described in the literature, such as pseudoconic, pseudocylindrica, pseudoazimuthal, retroazimuthal, and polyconic. Another way to classify projections is according to properties of the model they preserve. Some of the more common categories are: Preserving direction (azmithal) Preserving shape locally (conformal) Preserving area (equal-area or equiareal or equivalent or authalic) Preserving distance (equidistant) Preserving shortest route Because the sphere is not a developable surface, it is impossible to construct a map projection that is both equal-area and conformal.

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Mercator The Mercator projection is a cylindrical map projection created in 1569 by a Flemish geographer and cartographer Gerardus Mercator. Like all map projections that attempt to fit a curved surface onto a flat sheet, the shape of the map is a distortion of the true layout of the Earth's surface. The Mercator projection exaggerates the size of areas far from the equator. For example: Greenland is presented as having roughly as much land area as Africa, when in fact Africa's area is approximately 14 times that of Greenland. Alaska is presented as having similar or even slightly more land area than Brazil, when Brazil's area is actually more than 5 times that of Alaska. Finland appears with a greater north-south extent than India, although India's is the greater.

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Transverse Mercator The Transverse Mercator map projection was developed and presented by Johann Lambert in 1772. Lambert rotated the Mercator cylindrical projection 90 degrees, making the tangent line a line of longitude instead of the equator. Feature distortion increases proportionally as the distance from the central meridian (the red vertical line) increases. Only the central meridian and the equator of the projection are straight lines. All other latitude lines and longitude lines are complex curves. Unlike the Mercator projection, the Transverse Mercator projection is not used on a global scale but is applied to regions that have a general north-south orientation such as North America.

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Conic A Conical projection a map projection of the globe where a cone is placed on the Earth and is orientated so that it’s point is over one of the poles A point source of light at the center of the globe projects the surface features onto the cone. Two common conical projections are: Lamber Conformal Conic Albers conic

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Conic The Lambert Conformal Conic projection, developed by Johann Lambert around 1772, preserves shape but wasn't appreciated for nearly a century after its invention. Properties of Lambert Conformal Conic Projection: Latitude lines are unequally spaced arcs that are portions of concentric circles. Longitude lines are actually radii of the same circles that define the latitude lines. Poles are represented as single points. It produces extreme distortion of features farthest away from North America. One of the best projections for middle latitudes with an east-west orientation. It portrays shape more accurately than area and is common in many maps and geographic databases for North America.

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Azimuthal Among the oldest projections, Azimuthal projection was mentioned by the Greeks in the 2nd century B.C., but probably known earlier. This projection is mainly used for illustration purposes, since it cleary shows the Earth as seen from space infinitely far away.

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**Why we have different projections**

The conversion of geographic locations from a spherical coordinate system to a flat surface causes distortion. The projection process will distort one or more of the four spatial properties listed below. Distortion of these spatial properties is inherent in any map. Shape Area Distance Direction Shape refers to a map projection's ability to preserve the shape of geographic features. Conformality is a term often used to describe map projections that maintain the shapes of small geographic features even while the general shapes of large features are distorted. Conformal maps also preserve constant scale and angles locally. In a stricter sense, conformality means that the map projection maintains the same scale in any direction. Area refers to the ability of a map projection to maintain equal area for geographic features. That is, features are represented with the correct area relative to one another. Map projections that maintain this property are often called equal area map projections. Equal area projections usually sacrifice most other properties in favor of area. No map projection can preserve both conformality and equal area. Distance refers to a map projection's ability to maintain true distance. Accurate distance is maintained for only certain parallels or meridians unless the map is very localized. Generally a map is said to be equidistant if the distance from the map projection's center to all points is accurate. Direction refers to a map projection's ability to maintain true direction between geographic locations. Map projections that maintain direction are often called azimuthal map projections. Map projections of this type maintain true directions with respect to the map projection's center. Some azimuthal map projections maintain direction between any two points. In a map of this kind, the angle of a line drawn between any two locations on the projection gives the correct direction with respect to true north.

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**Projection benefits Definitions: GCS – Geographic Coordinate System**

UTM – Universal Transverse Mercator SPCS – State Plane Coorindate System

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**The Universal Transverse Mercator Projection**

UTM projection has been used by several coordinate systems: Part of the civilian UTM system – USGS, 1977 Part of the state plane system (SPC) And the military grid It has been used for mapping: Most of United States Many other countries The planet Mars The advantage of geographic coordinates in a GIS is that all maps can be transformed into a projection in the same way, which will allow us to merge and overlay maps for different purposes. However, to be able to show spatial data in 2D maps, we still have to covert geographic coordinates into a 2D Coordinate System. The most commonly used coordinate systems in GIS is UTM which has been included since the late 1950s on most USGS (United States Geological Survey) topographic maps.

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UTM Zones The UTM system divides the surface of the Earth between 80° S latitude and 84° N latitude into 60 zones, each 6° of longitude in width and centered over a meridian of longitude. Zones are numbered from 1 to 60 and increase in an eastery direction. Zone 1 is bounded by longitude 180° to 174° W and is centered on the 177th West meridian. Each of the 60 longitude zones in the UTM system is based on a transverse mercator projection, which is capable of mapping a region of large north-south extent with a low amount of distortion.

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UTM has two primary ordinate starting points, one at the equator and the other one hundred million meters (100,000,000) south of the equator and located at latitude 80 degrees south. Eastings are measured from the central meridian - with a 500,000 false easting to insure positive coordinates. Northings are measured from the equator - with a 10,000,000 false northing for positions south of the equator.

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**UTM – Scale Ground to Grid**

By using narrow zones of 6° (up to 800 km) in width, and reducing the scale factor along the central meridian by only (to , a reduction of 1:2500) the amount of distortion is held below 1 part in 1,000 inside each zone. Distortion of scale increases to at the outer zone boundaries along the equator. The secant projection in each zone creates two standard standard, or lines of true scale, located approximately 180 km on either side of, and approximately parallel to, the central meridian. The scale factor is less than 1 inside these lines and greater than 1 outside of these lines, but the overall distortion of scale inside the entire zone is minimized.

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UTM – coordinate range Northings Eastings

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**UTM in Alberta UTM Zones in Alberta**

In Alberta, the UTM mapping projection is typically used for all municipalities and non-urban areas and is referred to as rural cadastral map areas. The UTM mapping plane has central meridians of 111 degres for Zone 12 and 117 degrees for Zone 11. The scale factor at the central meridian is There is a 500,000 false easting and no false northing. Mapping Alberta in a UTM projection presents unique challenges since the province spans two UTM zones. It’s for this reason, that Albertans use a 3TM as well as a 10TM projection. UTM Zones in Alberta

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**10TM Projection 3TM Zones in Alberta**

The 3TM mapping plane has a zone width of 3-degrees with central meridians in Alberta of 111, 114, 117, and 120. The scale factor at the central meridian is and there is no false easting or false northing The 3TM mapping plane is typically used for all municipalities and is referred to as the urban cadastral map areas. The 10TM projection is unique to the province of Alberta. This 10 degree zone provides full coverage with a central meridian of 115 degrees. 3TM Zones in Alberta 10TM Projection

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**UTM example – Maine Zone 19**

Compare to Alberta, the state of Maine is relativly small and fits nicely withing UTM Zone 19 that’s bounded by 66 degrees and 72 degrees west. Likewise, Main lies in the center of the zone which minimizes distortion that’s found as you move towards the edges of the zone. 69 degrees W

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**State Plane Coordinate System**

The US State Plane coordinate system (SPC) was devised by the US Coast and Geodetic Survey ) in 1930s and uses either a transverse Mercator or Lambert’s conformal conic projection. This system is a set of 126 geographic zones, or coordinate systems, that are each designed for specific regions in the United States. Each state can have more than one state plane zone and the boundaries often follow county lines. Unfortunalty there is a lack of coordination between state borders and object locations on each side of a zone boundary are often to be misaligned. The most significant problem with each zone is that they each have their own coordinate syste. Typically they have an arbitrarily determined origin that is usually some given number of feet west and south of the south-western-most point on the map.

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**State Plane Projections – NAD 27**

While the state plane coordinate system is based on transverse mercator, the panhandle of Alaska, uses the Oblique Mercator projection since that region lies on a diagonal. UTM, Oblique Mercator, or Lambert Projections

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**State Plane Example LADWP**

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**Los Angeles Department of Water and Power**

Existing mapping is NAD27 State Plane, Zone VII feet Want to upgrade to NAD83 State Plane, Zone V feet, Compatible with City As with many clients who started their drawings in pre-V8, the Los Angeles Department of Water and Power is now faced with upgrading their maps. Their V7 design files were based upon NAD27 State Plane Zone 7 and in feet and they are wanting to standardize on NAD83 State Plane Zone V. This would result in design files that are compatible with the City.

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**State Plane Example LADWP**

V7 design files have 4 million false Northings and Eastings Could have set Global Origin to “shift” design plane to fit coordinates. Coordinates should be: X= , Y= Due to the limitations of the V7 design file, they lopped 4 million off both the Northings and Eastings. (We typically see a 5 million false Northing in old MicroStation files for mapping in Alberta.) Unfortunatly, both Albertans and Los Angelites could simply have adjusted the Global Origin to accommodate the large mapping coordinates. This would have bee preferrable rather than chopping off numbers to force the data to fit inside a design plane that’ s in the wrong location. To compound LADWP issues, the global origin is not consistent between all files.

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**State Plane Example LADWP**

Created user coordinate system (copy existing NAD27, Zone VII foot, subtracted 4 million off false easting and northing) New Projection LADWP-27 Cut 4 million off False Easting and False Northing To create a valid coordinate system in Bentley Map, they created a user-defined coordinate system. This was done by copying the delivered and accurate NAD27 Zone VII Foot system and subtracting 4 million off both the easting and northing. Now, even though the coordinates were wrong, they were right.

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**State Plane Example LADWP**

Create KMZ file overlays with Google Earth Once a user defined coordinate system was established, they can utilize it for interacting with Google Earth or by referencing to adjoing maps that use a different coordinate system.

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Thank You

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