Map Projections And State Plane Coordinate Systems. Thomas Maxell Beer, PLS

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1 Map Projections And State Plane Coordinate Systems By Thomas Maxell Beer, PLS

2 Introduction The origin for the modern State Plane Coordinate System in the United States began with two highway engineers from North Carolina. In 1932, George Syme and O.B. Bester contacted Dr. Oscar. S. Adams of the United States Coast and Geodetic Survey, about an idea they had about initiating a plane grid system for use in North Carolina. Being highway engineers, they were well aware of the problems of using 2D plane surveying techniques over large areas of the curved earth s surface. Dr. Adams and his assistance, Charles Claire, thus began what is called the State Plane Coordinate System (SPCS). This lesson will focus on the history and rational of the use of the various Mapping Projections and origin of SPCS systems in the United States. The lesson is not intended to be used as a guide to performing all the necessary computations for use of any particular SPCS. Cartesian Coordinates The system of mathematics we call Cartesian coordinate system was derived from the mind of the French Philosopher and Mathematician Rene' Descartes (Latinized form Renatus Cartesius). Descartes lived from 1596 to He is considered to be the father of modern analytical geometry that incorporates the study of geometry with the principles of algebra. This is widely used in engineering today. Cartesian coordinates allow any point on a plane 2D surface to be located by two numbers an X or Y coordinate. They allow any point in 3D space to be located by three numbers being N X, Y and Z coordinate as illustrated below. Copyright 2014 Thomas Maxell Beer Page 2 of 27

3 Illustration of 3D Cartesian Coordinates Notice that the origin point of the three number lines X, Y, and C have a common point of origin with an intersection point of 0,0,0. The Y-axis represents a north-south direction in civil engineering and surveying, the X-axis represents an east-west direction; the Z-axis represents an elevation above a particular datum. In a 2D Cartesian system a plane is divided into four sections or quadrants. By custom, they are numbered clockwise with quadrant I being the northeast quadrant, II for the southeast quadrant, III for the southwest and IV for the northwest quadrant. In a 3D Cartesian system space is divided into eight octants. These coordinates depend on where they are positioned in relation to the point of origin otherwise 0,0,0 as follows: First octant (+, +, +), top-back-right (, +, +), top-back-left (,, +), top-front-left (+,, +), bottom-front-left (+,, ), bottom-back-left (,, ), bottom-back-right (, +, ), bottom-frontright (+, +, ) In regards to establishing point locations with SPCS, we are concerned with the northeast quadrant only so the coordinates for both X and Y are positive numbers. Euclidean distances (a straight line distance between two points) can be calculated between a set of coordinates using formulas based on the Pythagorean Theorem: The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c). Copyright 2014 Thomas Maxell Beer Page 3 of 27

4 Distance for a two-dimensional coordinates Distance for a three-dimensional coordinates Angles from a 2 dimensional plane Angles can be calculated between a set of coordinates using trigonometry rations for sin, tangent and cosine. The angle A of the hypotenuse then derived from the process of inversing using the arc functions of arc-tan, arc-sin, or arc-cos. Plane and Geodetic Surveying Plane surveying is defined as a surveying system that assumes that the surface of the earth is a perfectly flat plane. It is not concerned about making corrections for the curvature of the earth. Copyright 2014 Thomas Maxell Beer Page 4 of 27

5 Measurements are made in two dimensions. The theory assumes that in small areas any distortion due to curvature is insignificant. The Plane Surveying system relies on simple plane trigonometry. This is defined as the study of triangles in the Euclidean plane including the use of functions defined by the ratios of sides of right triangles. In contrast, geodetic surveying is highly concerned with the aspect that the earth is curved. Trigonometry deals with triangles on a sphere. Mathematically, it is a far more complex system than the simpler mathematics used in ordinary plane surveying. Triangle on Sphere Unlike the earlier example of an ordinary simple straight line calculations in Euclidean geometry, lines on a sphere are the shortest path from point A to point B are the shorter arc of a great circle passing through points on A and points B. A great circle is defined as a circle that cuts the earth into two equal halves. Any circles that are not cutting the sphere into equal halves are called a small circle. These are not the shortest distance between points. On a sphere, the sum of the angles of a triangle does not equal 180. A sphere is not a Euclidean space. It lies within a curved non Euclidean space. Therefore, geodetic surveys are typically carried with great precision and require highly trained personnel. They take into account the true shape of the Earth; they are used to determine the size, shape, and gravity fields of the earth, and are necessary to develop engineering plans for projects that cover great lengths and large areas. Copyright 2014 Thomas Maxell Beer Page 5 of 27

6 True Shape of the Earth Sometime around 600 BC, ancient Greek philosophers belonging to the Pythagorean School of Thought, followers began to postulate that the Earth was a sphere. It is a sphere, but it is not a perfect sphere. The Earth is flatter at the poles that at the equator. The reason for this is that the earth consists of a molten core and it spins on its axis. Do to this rotation, the forces of angular momentum created a bulge at the equatorial plane. These forces of angular momentum cause the polar axis to be a shorter diameter than the diameter of circle that bisects the equatorial plane. The Earth is thus referred as being an oblate spheroid. The earth has a radius of km (3,963 statute miles) along the equatorial major axis and a radius of km (3,949 statute miles) length along the polar minor axis. Radius of sphere of equal area 6,370,997 meters 3,958 statute miles Area of earth 510,900,000 sq. km 197,260,000 sq. miles Equatorial Circumference 40,075 kilometers 24,899 miles Earth Dimensions Copyright 2014 Thomas Maxell Beer Page 6 of 27

8 In 1980 The General Assembly of the International Union Of Geodesy and Geophysics (IUGG) adopted the geodetic reference system (GRS 80) that was based on a 6,378,137-meter semimajor axis and a 6,356, meter semi minor polar axis. When comparing NAD 27 coordinates to NAD 83 a difference of up to nearly 100 Meters occur in the contiguous United States. There is no simple formula to calculate these shifts since they vary widely from place to place. The Nation Geodetic Survey offers a free program (NADCON) that computes the difference in height between the two datum with reasonable results. Vertical Datum NGVD 29 and NAVD 88 A vertical datum, in regards to surveying and geodesy, is a reference surface from which measurement of elevations and depths are taken. The first attempts of true geodetic leveling in the United States was based on sea level and occurred along the eastern seaboard in The two most widely used vertical datum used in the United States are the NCVD 29 and NAVD 88 Datum. The National Geodetic Vertical Datum 29 (NGVD29) standard was adopted in It is also referred to as Sea Level Datum of 1929 or simply mean sea level on early quad maps. The NGVD29 datum established an independent mean sea level elevation observation at 26 base monuments, 21 in the United States and five along the Canadian Coastlines. By decree, it was declared that mean sea level would be 0.00 feet at each of these local stations pending tidal observations over a long period of time. That period is 19 years. The sun, earth, and moon, come back into the same relative position every 19 years. This period reflects all of the influences of gravity of the moon and the sun alignment that subsequently influence tides. The problem with a system based on mean sea level is that no such thing exists. The sea is not level; it varies from place to place. Besides gravity wind, barometric pressure, currents, the Copyright 2014 Thomas Maxell Beer Page 8 of 27

9 salinity of the water, and seabed topography are a few contributing factors that contribute to sea levels. The North American Vertical Datum (NAVD 88) was adopted in 1991 as a means of correcting the inadequacies of the NGVD 29 system. It adopted a single primary benchmark reference as its basis. This basis is known as the Great Lakes Datum of Its local mean sea level value at a station at Father Point in located Rimouske, Quebec, Canada. It is no simple matter to convert NGVD 29 elevations to NAVD 88 elevations. The conversion factor varies from place to place and can be as much as three vertical feet. The Nation Geodetic Survey offers a free program (VERTCON) that computes the difference in height between the two datums. The program uses the Latitude and Longitude of a particular point to compute reasonable results. Geographic Coordinates Latitude is based on the equator being an equal distance between the North and South Poles. It was given a value of 0 degrees. The poles are given a value of 90 degrees. Lines of the same degrees are said to be parallel to the equator. For example, the dividing line between North and South Korea is an arbitrary east west line along the 38th Parallel. Lines in the Northern Hemisphere are given positive numbers and those in the Southern Hemisphere are given negative numbers. Points are described in degrees, minutes and seconds. Instead of a positive or negative numbers the letters N or S are most often used. Sometimes the Greek letter (φ) phi is used. Longitude is based on an arbitrary line running from pole to pole. Unlike latitude, there is no point that can be naturally being assumed as a starting point. This starting point is referred to as being the prime meridian. It is only an assumed line and is arbitrary in its location. By international convention in 1884, the line used today runs through the Royal Observatory in Greenwich England. All longitudinal lines or meridians are great circles. The Prime Meridian Copyright 2014 Thomas Maxell Beer Page 9 of 27

10 was given the value of 0 degrees. From the Prime Meridian, lines are designate as being +180 eastward and 180 westward. Again the letters E and W are most often used instead of the Greek letter λ (lambda). Royal Observatory located at the Prime Meridian Copyright 2014 Thomas Maxell Beer Page 10 of 27

11 Accurate positioning on longitude is based on determining the difference from local time to that of Greenwich time. This could be accomplished using a quadrant or sextant to determine local noon based on an observation of the sun at its highest point and compare that time with Greenwich time. These instruments were used to accurately determine latitude from observing the sun or particular stars. A Quadrant Circa 14th Century The invention that made determining longitudes possible was the marine chronometer. Prior to this navigators, depended on dead reckoning to keep track of longitude. This system was inaccurate, to say the least. It depended on keeping track of a ship s speed, time and direction, and plotting position on a chart. In 1714 the Longitude Act was passed in England by Parliament. The Act provided a monetary prize for anyone who invented a simple means to accurately measure longitude. The prize was eventually awarded to John Harrison, an English self-taught clockmaker, in His devise incorporated a balance wheel attached to a spring that compensated for motion. Copyright 2014 Thomas Maxell Beer Page 11 of 27

12 Harrison s Extremely Accurate Marine Chronometer H4 Prior to this invention, clocks with pendulums were useless at sea because of a ship s movement. Complex astronomical observations of heavenly bodies were the only means of accurately determining longitude. Chronometers used during this time were set at precisely at 1:00 pm with a ball dropping at the Naval Observatory located at 0 degrees longitude. From combining both latitude and longitude there is a system that is called geographic coordinates. These coordinates designate any point on the globe as illustrated below. For example Washington DC has the coordinates of Latitude: 38 53' North, Longitude: 77 02' West. The NADCON software also allows conversion of Geographic to NAD 27 and 83 systems. Locating Geographic Coordinates on the Surface of the Globe. Map Projections Copyright 2014 Thomas Maxell Beer Page 12 of 27

13 The ideal model of a map is a globe. It truly represents a map that is three-dimensional. When trying to cast a 3D image on to a 2D plane, a distortion occurs like the case as plane surveying vs. geodetic surveying. One can envision the problem by peeling an orange and placing the remains on a flat surface. There would be no end to the differences number of outcomes based how the orange was peeled. Therefore, there are many ways to project a 3D surface on a plane surface. Each projection has particular advantages and disadvantages. There is no perfect way to project the Earth s surface other than a globe. To various degrees, map projections are selected on the following four parameters: 1) Area: If your main concern is that areas on the map are uniform and proportional in scale, one should select a projection that would produce an equal area map. Equal Area Projection Examples of equal map areas are relatively proportional. There is no perfect solution other than a globe to produce a perfect equal area map on a flat surface. Copyright 2014 Thomas Maxell Beer Page 13 of 27

14 2) Direction: A map that is used in navigation is mostly concerned with using azimuth to correctly plot a course. These maps are accurate for this purpose only. They make a poor choice for other purposes. The most famous example is the Mercator projection that originated in the 16th century and was used by early navigators. Mercator Projection Above is an example of a directional projected based on a Mercator projection? Note the gross distortion of areas of Greenland and South America. They appear to be the same size, but in reality Greenland is about a one eighth the size of South America. Mercator never intended his map to be used for any other purpose that navigation. Compare the size ratio of Greenland and South America with the equal area map shown above. Map projections have even been used for propaganda purposes. For example, the upper latitude countries of the former Soviet Union and Red China are grossly misrepresented as their true areas in a Mercator projection. By using this projection, in addition using a vivid color of red, it made these areas seem much more menacing during the Cold War. If the author of this map wanted to adhere to the true, correct projection would be equal area projections with neutral colors. 3) Distance/Scale: A map that preserves distance from the center of the projection to any other point on map is refereed to being an equal-distant projection. Copyright 2014 Thomas Maxell Beer Page 14 of 27

15 4) Shape/Conformity: This projection concerns itself with the preservation of the shape. It is used for topographic mapping and navigation. Its advantages are that every parallel line intersects with every meridian at right angles, and that for any point; scale must be the same in all directions. This projection is used primarily on smaller scaled maps. This feature is also referred to as conformity. A map cannot be both equal area and conformal. Although though there are numerous projections available for use, the State Plane Coordinate system predominately use two. They are the Transverse Mercator projection and the Lambert Conformal Conic projection. Mercator Projection The Mercator projection is a cylindrical projection of Earth's surface created in the mid 1500 s by Gerhardus Mercator, a Flemish mathematician and cartographer. Gerard Mercator's parents were Hubert and Emerentia Kremer. The name Kremer means a merchant in German. Gerhardus Mercator is the Latinized version of his name. The Mercator projection is to be considered as one of the most important nautical navigation tools of all times. It made possible many the great explorations of the past 500 years and is still used today and will be in the future. Mercator projected the world on a cylinder with the parallels and meridians intersecting each other at right angles. He stretched the distances of the parallels with the same factor as the distances of the meridians. As a result, angular measures are preserved with this projection. Copyright 2014 Thomas Maxell Beer Page 15 of 27

16 Mercator wrote on his map spread on a plane the surface of a sphere in such a way, that the positions of all places shall correspond on all sides with each other, both in so far as true direction and distance are concerned and as concerns true longitudes and latitudes. Straight lines on the Mercator projection are called rhomb or loxodrome line. They represent lines of constant compass bearings that make this projection so useful for navigation and surveying. Mercator Map of the World Copyright 2014 Thomas Maxell Beer Page 16 of 27

17 Transverse Mercator The standard Mercator projection illustrated above has a cylindrical axis that is the same as the Earth's axis. A variation of this projection has a cylindrical axis that is perpendicular to the Earth s axis. This projection is called transverse. In this projection the point of tangency between cylinder and globe is a meridian, or line of longitude. This meridian is called the central meridian. This projection is often used to map areas with larger north-south extensions than east west. For example the state of Florida has most of its landmass running from north to south. Universal Transverse Mercator Zones in the United States In the 1940 s the United States Army Corps of Engineers adopted a standard universal grid for military use throughout the world called the Universal Transverse Mercator (UTM). The sphere is divided into 60 zones running along meridians. Each zone covers a 6 area. They are numbered from zone 1, located between West longitude 180, and 174. From here the zones run eastward around the globe to zone 60. The zones for the contiguous United States are numbered from zone 10 in the west to zone 19 in the east. Lambert Conformal Conic Projection Johann Heinrich Lambert was born in 1728 in an enclave of Switzerland. He was a mathematician and cartographer. One of his achievements formulated hyperbolic functions into Copyright 2014 Thomas Maxell Beer Page 17 of 27

18 the field of trigonometry concerning triangle on concave surfaces. Lambert developed the Lambert Conformal Conic projection in This projection superimposes a cone over the sphere. The intersection line produces tworeference parallels secant to the globe and intersecting it. Instead of the references being longitudinal, as in the case of the Transverse Mercator projection, they are associated with latitudinal lines. This projection is often used to map areas with larger east-west extent than north-south extent. For example, the state of Tennessee has most of its landmass running from east to west. The Lambert Conformal Conic map projection is one of the most widely used map projections in the United States. The USGS uses this projection for many 7.5 and 15-minute topographic maps. Conic Projection State Plane Coordinate System (SPCS) State Plane Coordinate systems were developed for specifying positions of geodetic stations using plane rectangular coordinates for surveying and mapping. Zones were created based on the premise that distortion less than 1 in 10,000 would occur. In order to accomplish this ratio, the width of zone must be limited to 158 miles. There are 124 separate number zones in the Unites States. Each zone has a code number. Copyright 2014 Thomas Maxell Beer Page 18 of 27

19 The main advantages of SPCS are that all surveys and mapping are permanently tied to a single system so that no point can be lost. Simpler methods of plane geometry can be applied over large statewide areas. All SPCS systems use either the Transverse Mercator or the Lambert Conformal polyconic projections with the exception of one unique zone in Alaska that uses an oblique Mercator projection. This projection was chosen for the panhandle area because it lies at an angle. State Plane Zones United States Note SPCS zone boundaries running in from east to west like the five zones Texas are Lambert Conformal Polyconic projections. Those running north to south in extense like the two zones in Illinois are Transverse Mercator. Florida has two zones that are Transverse Mercator and one zone covering it east to west extense of the panhandle as being a Lambert Conformal. Some states like North Carolina and Tennessee have only one zone. Alaska has the most with as many as ten zones. SPCS systems employ three basic units of measurement for distances. These are the International Foot, the US Survey Foot, and the meter. It depends in which state the SPCS is located, as to whether the international foot or US Survey foot is used. For example by state statute in North Carolina the unit is the US Survey Foot. In South Carolina it is the International Foot and in Virginia it s the US Survey foot. Even counties within Virginia use use the International Foot! Coordinate systems based on the more modern NAD 83 use the meter as the primary unit of measurement. The International Foot was defined in 1959 as being meters. The US Copyright 2014 Thomas Maxell Beer Page 19 of 27

20 Survey Foot is defined as being 1200/3937 meters. Although the differences between the twofoot units are small, they can amount to a significant amount over a large statewide area. It is also important to round off numbers to at least nine significant figures to avoid large errors. For example one meter should equal feet. It is also important not to mix calculations from two different SPCS since they are not based on a common framework. The following is taken from a letter written in January 1991 by the National Geographic Survey regarding policy: In preparation for the adjustment of the North American Datum of 1983, 31 states enacted legislation for the State Plane Coordinate System of 1983 (SPCS 83). All states defined SPCS 83 with metric parameters. Within the legislation, the U.S. Survey Foot was specified in 11 states and the International Foot was specified in six states. In all other states the meter is the only referenced unit of measure according to the SPCS 83 legislation. The remaining 19 states do not yet have any legislation concerning SPCS 83. State Plane Coordinate System for North Carolina As mentioned in the beginning of this lesson, the first state responsible for the initiation a SPCS was North Carolina. It would be impracticable to examine all of the various state systems in this short lesson, so let us begin with the first state to become involved. North Carolina has most of its landmass in an east west direction. It is considerably narrow in its north south direction. Therefore, the State of North Carolina best fits using the Lambert Polyconic Conformal projection, because it is so narrow in the north south direction it only requires one zone. Referring to the map on the next page the two standard parallels or the lines where the conic projection intersects the globe are along 36 10' 00 in the north and 34 20' 00 In the south. Along these parallel lines the scale is exact. This is because they intersect with the globe along these lines. The area between the two parallels are said to have a scale projection that is too small. These area covers 2/3 the distance of the north-south projection limits. The reason is that the grid plane Copyright 2014 Thomas Maxell Beer Page 20 of 27

21 is below the curved surface of the ellipsoid. The two areas above and below the parallels each covering 1/6 of the area of the projections limits are said to be too large in scale. This is because the curved surface of the ellipsoid is below the grid plane. The base meridian for the North Carolina SPCS is 79 degrees. This system is depicted on the map on the next page. The map is from a state publication entitled DEVELOPMENT & APPLICATION OF THE STATE COORDINATE SYSTEM dated rev Copyright 2014 Thomas Maxell Beer Page 21 of 27

22 Converting Ground Distances to Grid Below is an illustration that explains terms used in converting ground distances to grid distances. Note that ground distances will be longer than grid or ellipsoid distance. The Combined Factor (CF) consists of the elevation factor and grid factor both of which are used to reduce ground distances to grid distances. Note from the illustration that the grid scale is only exact where the north latitude intersects the ellipsoid. Between these intersections, where the ellipsoid surface and ground is above the grid plane, the scale projection is too small. Outside the scale projection it is too large because the ellipsoid surface is below the grid plane. Copyright 2014 Thomas Maxell Beer Page 22 of 27

23 To obtain the combined factor, the ground horizontal distances must first be reduced to the ellipsoid. The formula for this reduction is as follows: Ellipsoid Distance = Horizontal Distance X = R TERMS - - H + Geoid Height + R - - H = elevation (NGVD29/NAVD) Geoid height = Use the average geoid height for the particular area, in this case for North Carolina -33 meters is sufficient for most surveys. R = Mean radius of the Earth in (North Carolina uses meters.) Often tables are compiled for this data using the above formula in the following format: (This is for a -33 meter geode height.) Elevation H meters Ellipsoid factor Copyright 2014 Thomas Maxell Beer Page 23 of 27

24 A more precise means would be to average the geoid heights from data sheets. Below is an example for a particular monument control station in Montgomery County, North Carolina. This data sheet example states that the GH is meters. A survey tying in two such monuments would take the mean average of the two GH factors. The next step is taking the calculated ellipsoid lengths and further reducing the original ground distance to the length they would be on the particular State Plane coordinate grid. This grid factor is a scale factor and is based on the previously mentioned scale factor can be exact, in the case of where parallels intersect the ellipsoid, or small or large depending on the latitude of the points surveyed. For example if the point falls on Lat 34 Degrees 20 minutes in North Carolina the scale factor is or unity. Refer to Map on page 23. Scale factors are obtained from data sheets, by software or by published tables. Copyright 2014 Thomas Maxell Beer Page 24 of 27

25 North Carolina publishes this table in the following format: Latitude R (Meters) Tab diff. K scale factor d 18 m d 19 m d 20 m Unity 34 d 21 m d 22 m R in above table refers to mapping radius at grid origin The above is only a portion of the table going from scale too great to unity on the parallel described above to scale factor too small. We then can determine the combine factor that allows us to project ground distances to grid distances. Combined Factor = Ellipsoid Factor X Scale Factor Grid Distance = Horizontal Ground Distance X Combined Factor Copyright 2014 Thomas Maxell Beer Page 25 of 27

26 Convergence Angles The next concern is that true north only occurs along the central meridian for each state plane coordinate system. Referring to the diagram on page 30, this meridian is 79 degrees in the case of the North Carolina Lambert projection. Any other place except the central meridian will need to address the difference between the grid and geodetic azimuths at any point. Grid meridians are parallel, Geodetic meridians converge at the geographic poles. This angle has been called the convergence angle, the mapping angle, or simply the theta (θ) angle. The angle is a positive number to the east of the Central Meridian and a negative to the west of the Central Meridian. Grid Azimuth equals Geodetic True Azimuth (-) theta (θ) + the 2 nd term. The second term (T-t) is a minor correction that is applied to surveys of great accuracies and is not normally used in ordinary surveying. They apply to directions more than five miles and in east-west directions. It is needed for surveys of great precision because measured direction between two points is a curved line and when projected on a plane appears as a curved line. The mapping angle is used only when it is necessary to convert grid azimuth to geodetic azimuth or geodetic azimuth to grid azimuth. Grid azimuth can be computed from an inverse between two points with plane coordinates. Geodetic azimuth is usually provided by NGS or from GPS data. Example: For North Carolina is the ration of theta to the longitude difference from the central meridian. Caution- this ration is for NC only. (Central meridian longitude) x = theta (θ) (79 degrees-80 degrees) x = = - 34' = theta (θ) Conclusion Copyright 2014 Thomas Maxell Beer Page 26 of 27

27 State plane Coordinate Systems have contributed greatly to Surveying and Engineering as follows: 1. Large projects, like highway projects, can be an undertaking using the methods of plane surveying. The projects need not have to resort to the more difficult methods and calculations employed with geodetic surveying. 2. Plane surveying is less expensive that Geodetic Surveying. 3. Surveyors do not have to be trained as extensively using the simpler methods of plane surveying. 4. All surveys within the SPCS have a single established framework. 5. Lost monuments can be reestablished when tied to state plane coordinates. 6. Projects surveyed in different areas can be linked together by a common framework. For further information regarding each State s Plane Coordinate Systems visit the following web site. Copyright 2014 Thomas Maxell Beer Page 27 of 27

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