 Welcome to Caltrans LSIT LS exam preparation course. One aid in your preparation for California licensure examinations. A word of caution. Don't use this course as your only preparation. Devise and follow a regular schedule of study which begins months before the test. Work many problems in each area, not just those in this course's workbook, but problems from other sources as well. This course is funded by Caltrans, but you and I owe a profound thanks to others, the courses instructors, from the academic community, the private sector, other public agencies, and from Caltrans as well. We wish you well in your study toward becoming a member of California's professional land surveying community. Welcome to today's discussion of the state playing coordinate system. A 1986 change to the public resources code of the state of California created the California coordinate system of 1983, noted as CCS83. The implementation of the system is to be phased in by January 1, 1995, when state playing work must refer to it in most cases. It replaces the previous system of 1927. The old North American datum of 1927, developed by the US Coast and Geodetic Survey, was the basis for CCS27. It used an approximation of the Earth's surface in North America named Clark's spheroid of 1866 as its geodetic reference surface. The North American datum of 1983, that's NAD83, developed by the National Geodetic Survey, underlies CCS83. A new ellipsoid, the geodetic reference system of 1980, also called GRS80, is the basis of the new system. This ellipsoid is geocentric and approximates the surface of the entire Earth, not just North America. Because the geodetic reference surfaces are different, the NAD27 latitude and longitude of a station differ from the NAD83 values for the same station. Overlying those differences are the factors that the adjustment incorporated new observations. The adjustment used was a least squares, simultaneous adjustment of the entire system, and that blunders could be isolated and culled. In fact, the variation between the two coordinate systems vary inconsistently, making only approximate transforms possible. At present, rigorous computation of coordinates is only possible by returning to the original field observations, readjusting and recomputing positions from them. The designers of the coordinate system limited the north-south extent of each Lambert grid to meet certain predetermined criteria. Therefore, six zones comprise CCS83. Zone one is at the north end of the state. The zone numbers increased southerly until zone six extends to the Mexican border. Each zone is based upon a cone whose axis is coincident with the GRS80 spheroid axis of rotation. The cone is a secant cone because it intersects the surface of the ellipsoid at two standard parallels rather than simply resting on it. This minimizes distortion while projecting data from the ellipsoid onto the cone. As enacted by the state legislature, the new coordinate systems linear units can either be meters or US survey feet. To date, NGS has implemented NAD83 as a metric system. It has published all CCS83 station data in meters. State plane coordinates expressed in one unit can be converted to the other unit simply by multiplying the north end and east end by the foot meter equivalency. This will be described later in this presentation. One may work the California coordinate system in either feet or in meters, but one must be consistent. Not only must all distances be uniformly in feet or in meters, but tabled constants must also correspond consistently with the linear unit used. This discussion is presented in terms of meters, but the accompanying written material is presented both in feet and in meters. The NGS recommends using polynomial coefficients and algebraic equations to simplify calculations for state plane coordinates. Polynomial coefficients are easy to use. Therefore, they are appropriate for both manual and programmed calculation applications. Through their use, projection tables become unnecessary. They can produce millimeter accuracy using hand-held calculators carrying only 10 significant digits. The NGS developed the coefficients by polynomial curve fitting of an array of data points for each zone. For those wishing to use projection tables, the California Land Surveyors Association has published a book of projection tables as special publication number 55 slash 88. NGS is now publishing a similar table. Please be aware, if you are preparing for the Land Surveyors examination, the new coordinate system will be phased in through the beginning of 1995. However, nothing prevents the state board from asking questions about the older system on future examinations. You need to prepare for both. While this presentation concentrates on the California coordinate system of 1983, it is broadly applicable to the system of 1927. Calculations for the older system are not as sophisticated as those used now. Therefore, if you understand the system of 1983, it is much easier to master the older system. State plane coordinate systems are intended to permit the calculation of positions on a grid using measurements made on the curving surface of the earth and at different elevations. If those measurements are accurately made and extended over a long distance, plane geometry calculations will yield inconsistent values for the position of the stations. The state plane coordinate systems were devised to provide a calculation technique that produces coordinate values that correctly relate to each other even over long distances and compare favorably with those calculated by geometric means. First, we will look at a series of pictures to help us visualize the coordinate system. The first set of illustrations is intended to acquaint us with the relationship between the ellipsoid and the cone that is the basis of the system. We now see depicted the geodetic reference system of 1980, also referred to as the GRS-80 system ellipsoid. It approximates the shape of the planet earth. Like the earth, it is slightly flattened. The polar radius is about 13 miles less than the equatorial radius. The GRS-80 ellipsoid underlies the California coordinate system of 1983 or CCS-83. The National Geodetic Survey developed the recently revised coordinate system. The system uses information from the North American datum of 1983, NAD-83, adjusted during the period 1975 through 1986. We also see depicted the axis of rotation of the ellipsoid that is the same as our planet earth's axis. We now see the addition of the equator. The equator is formed by the intersection of the surface of the ellipsoid with a plane that is at right angle to the axis of rotation and located midway between the poles of the ellipsoid. Next, two latitudes are presented north of the equator. These two latitudes are defining elements for each portion of the California coordinate system. Each selected latitude is called an exact scale or standard parallel. Latitude is measured by the angle formed at the intersection of the equatorial plane with a line normal to the surface of the ellipsoid. The two latitudes are called the north and south exact parallels. And note it as B sub n and B sub s. Now we see a cone generated through the two exact scale parallels. This means that it is a secant cone. In other words, the surface of the cone is within the surface of the ellipsoid between the standard parallels and outside it towards its apex and towards its skirt. The axis of the cone is coincident with the axis of rotation of the ellipsoid. For each unique pair of standard parallels, a unique cone will be created. Another parallel of latitude occurs between the two standard parallels where the separation between the ellipsoid surface and the cone is greatest. This latitude is nearly midway between the standard parallels, and therefore its name is the central parallel. Its notation is B sub o representing the latitude of the origin of the projection. Each zone has its own central parallel, a tabled constant for that zone. For each zone, the geometry of the cone fixes the distance from the apex to the central parallel. The notation for that distance is r sub o standing for the mapping radius of the projection origin. The angle between the axis of the cone and its surface will have a value that is equal to the latitude of the projection origin. Next, we see a geodetic meridian of longitude projected onto the cone. That is a true north-south line. The designer of the zone has selected a particular meridian to fulfill best the needs of those who will use the zone. It will form the center of the east-west extent of the zone. Therefore, it is called the central meridian and is a tabled constant. At the intersection of the central meridian with the central parallel is the projection origin. Now we see the addition of another parallel of latitude. It is the north geodetic latitude of the grid origin. At its intersection with the central meridian is the origin of the grid. In every zone, that point is assigned a northern value of 500,000 meters and an easting value of 2 million meters. The notation for the latitude of the grid origin is b sub b because through it will pass the baseline of the grid. Associated with grid latitude of the grid origin is a particular distance from the apex of the cone. This distance is known as the mapping radius through the grid origin, noted as r sub b. It is a tabled constant for each zone. Next, we see the customary north-south limits of the grid and the customary east-west limits of the grid illustrated. These limits are customary only when there is a good reason for doing so, the grid may be expanded beyond them. Before we continue to our next graphic, let's look at a simple model with a paper cone representing the large cone with its apex way above the north pole of the ellipsoid. On this cone, we can see that the geodetic meridians have been projected through and they form now north-south geodetic lines. You'll also see that the latitude line has been projected through so that they form what appear to be parallels on the cone's surface. Now, the interesting thing that this cone is to illustrate is that when we develop this cone, that is when we flatten it out, which we'll do right now, those lines that appear to be parallel become large semi-circles here. And the lines that used to be running to the apex of the zone now converge at a point so the geodetic meridians converge while the parallels of latitude become large arcs of circles. We will now see a series of pictures portraying the developed cone, that is the cone once it has opened up and flattened into a plane. On that developed cone, we will see the various lines that relate it and the grid that creates a state-plane coordinate system. This picture shows the cone for a zone developed that has opened up and flattened. The cone entirely encircled the ellipsoid. In other words, it represents a full 360 degrees of longitude, but now that it has been developed, the central angle at the apex of the cone is considerably less than 360 degrees. The central angle at the apex of the developed cone equals 360 degrees times sine B sub o. This gives us the ratio that we can use to proportion differences in longitude when calculating the corresponding central angle on the developed cone. Sine B sub o is the sine of the latitude of the projection origin, a tabled constant. The latitudinal line illustrated for our developed cone is B sub o through the projection origin. We see that the mapping radius of B sub o has been labeled as R sub o. It is the distance from the line of latitude B sub o to the apex of the cone, a tabled constant. Now added is the northern axis. The northern axis coincides with the central meridian. Therefore, its longitude is the same as that of the central meridian passing through the projection and grid origins. Next we see the addition of another latitude in the line and its mapping radius. The latitude is that of the parallel passing through the grid origin. Its latitude is noted as B sub B. It is a tabled constant for each zone. The mapping radius is labeled as R sub B. It is the mapping radius through the grid origin and is a tabled constant. Next we see the addition of the easting axis. It passes through the grid origin and is at a right angle to the central meridian. Next can be seen lines of longitude, approximately delineating the customary east-west limits of the zone. Also added is a segment of the latitude, approximately delineating the customary northerly limit of the zone. The customary southerly limit is the easting axis. Next, a meridian of longitude extends from the apex of the cone to a station, which coincidentally is on the central parallel. A longitude is a true north-south geodetic line, a geodetic meridian. The notation for the west geodetic longitude of a station is capital L. If we create a grid north line through the station, it will be parallel with the northern axis. The plane convergence angle is the angle formed on the developed cone between the central meridian and the line of longitude through the station. Lowercase gamma symbolizes the plane convergence angle. It can be seen on the screen that the difference between grid north and geodetic north at the station equals the plane convergence angle at the apex of the developed cone. The plane convergence angle is equal to the difference between the longitude of the central meridian and the longitude through the station multiplied times sine B sub o. Both the longitude of the central meridian L sub o and sine of B sub o are tabled constants for each zone. This principle imposes the same proportioning previously imposed upon the central angle for the entire cone. Let's now look at another set of illustrations dealing with the grid. It will make clear the grid's construction and its relationship to certain elements of the cone and the lipsoid. On the screen, we see the two standard parallels of geodetic latitude. Each California coordinate system zone has two of these lines of exact scale where the cone intersects the surface of the GRS-80 ellipsoid. The northern one is noted as B sub n and the southerly as B sub s. Now we see the circular arc of the central parallel added to the illustration. Noted as B sub o is a parallel of latitude that is approximately midway between the standard parallels. Along this line occurs the minimum scale within the zone. Associated with the central parallel is its mapping radius, the distance from the apex of the cone to the central parallel, a distance noted as R sub o. Both the latitude of the central parallel and its mapping radius are tabled constants. Next, we see the addition of the central meridian. It is the projection of a true north-south geodetic line noted as L sub o. The central meridian also is the northern axis of the grid. It passes through both the projection origin and the grid origin. As such, its notation is also E sub o. By definition, the central meridian has an easking value of 2 million meters in all zones. At the intersection of the central parallel and the central meridian is the origin of the projection. This location has geodetic coordinates that is latitude and longitude of B sub o by L sub o. We now see another parallel of latitude has been added. Noted as B sub b, it is the projection of the geodetic latitude passing through the grid origin and is a tabled constant. It also has a particular mapping radius associated with it, a tabled constant noted as R sub b. At the intersection of the central meridian with B sub b is the origin of the grid. Consequently, its geodetic coordinates are B sub b by L sub o. Through the grid origin, and at a right angle to the central meridian, runs the easting axis of the grid. It is the baseline of the grid noted as N sub b. A northern value of 500,000 meters is associated with the easting axis. Therefore, the origin of the grid has plane coordinates of 500,000 meters in northern by 2 million meters east. This is true for all zones by definition. Parallel with the baseline of the grid and passing through the projection origin is an east-west grid line. Its northern value is noted as N sub o, a tabled constant. Therefore, the grid coordinates of the projection origin are N sub o by E sub o. On the screen, we see the approximate customary east-west extent of the zone depicted. These limits are customary only, but it is usually not necessary to extend the coordinate zones too far into the Pacific Ocean or into adjacent states easterly of us. Next on the screen, we see the approximate north-south customary limits of the grid. The customary southerly limit is the baseline of the grid. More grid lines are added parallel with the northern axis. Each line runs grid north. Remember that geodetic meridians converge toward the apex of the cone, while grid meridians are not parallel, are all parallel. Therefore, the difference between geodetic north and grid north at any particular point on the grid depends upon the position of the point. Now grid lines paralleling the east-ing axes have been added to the display. They all run grid east and at a right angle to the north-south grid lines. Now that we understand the basic geometry of the ellipsoid cone and grid, we will view a series of pictures illustrating the geometry that allows us to convert geodetic coordinates, in other words latitude and longitude, to state plane coordinates are familiar northings and eastings. We now see on the screen a single line of latitude projected onto the developed cone surface. This line of latitude is the central parallel of the zone noted as B sub O. The central parallel is a tabled constant for each zone. We now see the central meridian, a geodetic north line added to the picture on the screen. This line is also the northern axis of the grid. All other grid north lines will be parallel with it. Its east-ing value is 2 million meters by definition. That value is noted as E sub O and is a tabled constant. The intersection of the central meridian with the central parallel is the projection origin. Now we add the east-ing axis or baseline of the grid noted as N sub B. This line runs at a right angle to the northern axis. Its northern value is a tabled constant and is 500,000 meters by definition. This northern value sometimes is called its false northern. The intersection of the northern axis with the east-ing axis is the origin of the grid. The screen now shows a dimension line extending from the apex of the cone to the grid origin. This distance is the mapping radius through the grid origin noted as R sub B. It is a tabled constant. By inspection, we can see that by adding together N sub B and R sub B, we can calculate the northern of the apex. Now added to the display is a station at which the geodetic coordinates and the grid coordinates are to be equated. We see a parallel of latitude added through the station. Its latitude is noted as capital B. Next added is a geodetic meridian of longitude extending from the apex of the cone to the station. The notation for its longitude is capital L. Now another line has been added. It is a line through the station extended at a right angle from the northern axis. The latitude through the station has a particular mapping radius associated with it. Labeled as capital R, the length of the mapping radius through the station can be measured along the meridian from the apex to the station. As shown on the screen now, the central parallel has its own particular mapping radius noted as R sub O. It is a tabled constant. The value of the mapping radius R through our station will be needed for converting geodetic coordinates to state plane coordinates and vice versa. So that we can do this, NGS devised a method to calculate the radial distance between the parallel of latitude through our station and the central parallel. It involves the use of equations employing tabled coefficients and the difference in latitude between the origin of the projection and the station. The distance obtained by this method is labeled small u on the screen. When a station is north of the central parallel, u is negative. When the station is south, it is positive. By inspection, it is apparent that the mapping radius of our station R equals the sum of the mapping radius through the central parallel, R sub O, and the radial difference, u. Remember that we previously discussed that the central angle of the developed cone could be found by multiplying 360 degrees of longitude by sine b sub O. Similarly, an angle on the grid between the central meridian and any other geodetic meridian can be found by multiplying their difference of longitude by sine b sub O. This angle is the plane convergence angle located at the apex of the cone. The plane convergence angle sometimes is called the mapping angle. Its symbol is the lowercase Greek letter gamma. We now can see that we have developed a giant right triangle, the conversion triangle. One leg of it is the mapping radius through the station. One angle of the triangle is the plane convergence through the station. Using classical Euclidean trigonometry, we can calculate the other two legs of the giant triangle. As noted on the screen, the leg along the northern axis of the grid can be found by multiplying the mapping radius through the station R by the cosine of the plane convergence angle. This yields the difference in northings between the apex of the cone and our station. Remember that we previously determined that adding n sub b with r sub b yielded the northing of the apex. Therefore, subtracting the value r cosine gamma yields the northing of the station. The northing is noted by small letter n. Next we see that the remaining leg of the giant conversion triangle, labeled r sine gamma, is shown the value found by solving this trigonometric equation is positive if the station is east of the northern axis and negative if west. We can calculate the easting of our station noted as small e by adding the result to e sub o. Thus, we have a method to convert our station's geodetic coordinates, b and l, to its equivalent grid coordinates, small n and small e. Reversing the procedure provides for the inverse computation, starting with plane coordinates and going to geodetic coordinates. We will cover three broad classes of calculations for state plane coordinate systems in this presentation. They are first, conversions of coordinates between ellipsoid and grid. Second, manipulation of observations. These are adjustments applied to lengths and asmas when converting field measurements to grid values and vice versa. Third, transformations from zone to zone. We first will consider forward or direct computation, which is the conversion from geodetic coordinates to plane coordinates. In other words, we will change latitude and longitude of a station into the corresponding northean and easting. Direct or forward mapping equations are used to compute state plane coordinates from geodetic coordinates. In the forward computation, we must find the mapping radius through the station. This may be done in two ways. The first is by using tabled polynomial coefficients, the method preferred by NGS. The coefficients are used in algebraic equations to provide the missing link in calculating the mapping radius through the station. This missing link is the radial distance, noted as small u, previously illustrated. The polynomial equations are straightforward although the numbers are long. But let us not forget that the machine does the hard work. All that we must do is press the buttons accurately. Another method for determining the mapping radius is to interpolate it from projection tables. This will be discussed in its proper place during our discussion of forward computation. We begin by calculating the difference of latitude between the station to be converted and that of the central parallel. This equation takes the north latitude of the station, noted as b, and subtracts the latitude of the projection origin on the central parallel, a tabled constant noted as b sub o. It yields the difference in latitude between them. If working on a calculator computing only 10 significant digits, such as the HP 41, it is necessary when subtracting latitudes to truncate tens of degrees. That is, if the latitude is 33 degrees, truncate it to three degrees. We can now use the difference in latitude just calculated to compute the value of u. Remember that we previously saw that u is the radial distance from the station to the central parallel. We will use L polynomial coefficients for this calculation. These coefficients can be found listed at the end of the California Coordinate System written material. Each zone has its own set. They also differ for feet or for meters. Using the latitude difference just obtained, we enter it into an either equation now displayed on the screen to calculate the value of u. Some persons feel that the second equation is easier for handheld solution because it has a nested form. Each equation takes the difference in latitude or the difference raised to a power and multiplies at times a polynomial coefficient that is tabled with the zone constants. Each coefficient used in these equations is noted with the capital L. Each has a subscripted number to indicate its proper position within the equation. Be careful to keep track of arithmetic signs. The radial difference, u, has a positive value if the station lies northerly of the central parallel and a negative value if it lies southerly. After we have calculated u, the radial distance from the station to the central parallel, we can calculate the mapping radius for our station. The equation now displays shows that by taking the length of the mapping radius of the central parallel, r sub o, and subtracting the radial distance measured from the station to the central parallel, u, we calculate the mapping radius through the station. Be careful with arithmetic signs. An alternative to using polynomial coefficients is obtaining the mapping radius from projection tables if they are available. The projection tables are entered with the latitude of the station and the mapping radius can be read in an adjacent column. Usually projection tables list latitude in whole minutes. Therefore, values of the mapping radius for a latitude line between whole minutes must be interpolated. Remember from our previous discussion that during a forward conversion, we are solving a giant right triangle. Whether obtained using polynomial coefficients or from projection tables, we now know the mapping radius of our station. In other words, we now know the length of the hypotenuse of our giant right triangle, the conversion triangle. Next, we must find the value of the angle of our conversion triangle located at the apex of the cone. As we learned earlier, this angle is the plane convergence angle. To determine the plane convergence angle, symbolized by lowercase gamma, we use the equation displayed on the screen. It determines the difference between the longitude of the central meridian, a tabled constant noted as l sub o, and the west longitude through the station, noted as l. It then proportions that difference by sine b sub o, a tabled constant. This is the same relationship that the central angle of the developed cone has to 360 degrees of longitude as previously discussed. If working on a calculator, computing only 10 significant digits, such as the HP 41, it is necessary when subtracting longitudes in this equation to truncate tens and hundreds of degrees. That is, if the longitude is 117 degrees, truncate it to seven degrees. Furthermore, it is necessary to carry all significant digits for this calculation. Therefore, it is suggested that you use register or stack arithmetic. Now that we know both the length of the hypotenuse and the angle at the apex of the cone, we may use them to solve our conversion triangle. Its solution provides the remaining elements needed to calculate the northern and east end of our station. First, we calculate the northern of our station using the equation displayed on the screen. It adds the mapping radius of the grid origin, r sub b, to the northern of the grid origin, n sub b. This gives us the northern of the apex of the cone. We then solve r cosine gamma, which is the difference in northings between the apex and the station. This is then subtracted from the northern of the apex, leaving only the northern of the station noted as small n. Finally, we calculate the east end of the station. By solving r sine gamma, we find the east end difference between the central meridian and the station. The value will be positive if the station is east of the central meridian, negative if it is west. Add the value to e sub o, the east end of the central meridian to obtain the east end of the station noted as small e. We now have finished the process known as direct or forward computation. We begin with the latitude and longitude of a station, its geodetic coordinates, and we converted them to their equivalent northern and east end, its plane coordinates. Next, we will study inverse mapping equations, which are to convert plane coordinates to geodetic latitude and longitude. The geodetic coordinates are measured on the ellipsoid of reference as latitude and longitude while the northern and east end is on the grid. Either polynomial coefficients or projection tables may be used in the process. First, we must solve for the plane convergence angle. This equation subtracts e sub o, the east end of the grid origin, from the east end of the station noted as small e. This yields the difference in east ends from the northern axis to the station. It then divides that difference by the sum of r sub b, the mapping radius of the grid base, which is a constant, and n sub b, the northern of the grid base, less the northern of the station, small n. This quantity is the difference in north ends from the apex of the cone to the station. These two quantities are the opposite and adjacent legs of the large conversion triangle. It is necessary to carry all significant digits of the plane convergence angle resulting from this calculation. Therefore, you should use register or stack arithmetic. Next, we proportion the plane convergence angle by dividing by sine b sub o. This is the reverse of the procedure that we saw earlier when learning forward computation. This equation is used to calculate the longitude through the station. Remember in our previous discussion, we found that we can proportion or scale the difference in longitudes between the central meridian and a station by the tabled constant sine b sub o when calculating the plane convergence angle. This equation takes the plane convergence angle symbolized by lowercase gamma, divides it by sine b sub o, thereby calculating the difference in longitudes. We then subtract that difference from the longitude of the central meridian, l sub o, to calculate the longitude through our station, noted as l. If working on a calculator computing only 10 significant digits, such as the HP 41, it is necessary when subtracting longitudes in this equation to truncate tens and hundreds of degrees. Furthermore, it is necessary to carry all significant digits in this calculation. Therefore, you should use register or stack arithmetic. Now that we have the longitude of our station, we are left with the problem of figuring its latitude. Similar to the direct conversion, we need to use the mapping radius through our station while calculating the latitude. This necessitates our use of the radial difference, noted as u, unless projection tables are available. We will proceed for now as if no projection tables are available. First, we will calculate the radial difference, u. In this equation, we seek the value of the radial difference between the mapping radius of the central parallel and the mapping radius through the station in question. First, the difference in estines between the station and the northern axis, e sub o, is multiplied by the tangent of half the station's plane convergence angle. We then subtract the result from the difference between the northern of the station, noted small n, and the northern of the projection origin, a tabled constant noted as n sub o, giving us u. Next, we must use the radial difference to compute the latitude of the station. G polynomial coefficients are used for inverse computation. After the radial distance has been calculated, G polynomial coefficients are used in algebraic equations to calculate the difference between the latitude of the station and that of the projection origin. The equation then adds the length to the latitude of the projection origin, and thereby we receive the latitude of the station. Either one of these equations may be used to calculate the latitude of the station using the previously calculated value for u and multiplying it by the G polynomial coefficients tabled constants for each zone. The various portions of the solution must be carefully solved and then combined, taking care with arithmetic signs and obtaining the latitude of the station. If working on the calculator, computing only 10 significant digits, such as the HP 41, it is necessary when adding latitudes to truncate tens of degrees. If projection tables are used, the latitude must be obtained by interpolation. The interpolation argues mapping radius against latitude. The mapping radius may be calculated from the following equation. You may recognize the Pythagorean theorem in this equation. This equation takes the square of the leg of the conversion triangle comprised of the difference of eastings, adds to it the square of the leg of the conversion triangle comprised of the difference of northings, and then takes its square root, yielding the hypotenuse of the giant triangle, which is the mapping radius through the station. All terms on the right side of the equation are tabled zone constants, except the northing and the easting of the station. When using projection tables, the mapping radius just calculated is used as an argument against latitude. The mapping radius is interpolated to refine the latitude of the station. We've been at this for quite a while now, and we have finished our discussion of inverse computation, so let's take a short break. Welcome back to our discussion of state plane coordinates. We will begin with adjustments to observations. First, we will deal with azimuths. On the screen, we see the central meridian or northing axis running north-south. It is labeled E sub o, the notation for its easting, which is a constant for each zone. We also see the easting axis labeled N sub b, which is the notation of its northing, also a constant. On the screen now, a station has been added. Now a geodetic meridian, that is, a geodetic north line is drawn between the station and the apex of the cone. This meridian is the line of geodetic longitude through the station. We now see a line running from our station toward another station. It is easterly of us in this case. The grid azimuth of the line to the station is desired. Now shown as an angle labeled alpha. This angle identifies the geodetic azimuth. The geodetic azimuth of the line toward the second station is the clockwise angle from the geodetic meridian around to the line to the second station. For the California Coordinate System of 1983, geodetic azimuth is reckoned from north, not south as previously done. On the screen now, a grid north line through the station has been added. Because it is grid north, it is parallel with the northern axis of the grid. Now we see the plane convergence angle labeled gamma at our station between geodetic north and grid north. In this case, convergence is negative because the station is west of the central meridian. If it had been east of the central meridian, it would have had a positive value. Be careful with arithmetic signs. Plane convergence angle is sometimes called the mapping angle. We now see the addition of an angular dimension, small t. This quantity is the grid azimuth of the line between our station and the one easterly of it. Grid azimuth is the clockwise angle at a station between the grid meridian, that is grid north, and the grid line to the other station. Next we see the equation t equals alpha minus gamma plus delta. Bearing in mind that the plane convergence angle in our drawing has a negative value, inspection of the drawing demonstrates that t is equal to geodetic azimuth minus plane convergence. The other term in the general formula is neglected here. It is variously named often arctochord correction or second term correction. It is usually minute, but may need to be considered for high quality work over long lines. This presentation neglects it. For CCS 83, both geodetic and grid azimuth are reckoned from north. Inverses between stations having state plane coordinates give grid azimuth and may be used directly for calculations on the grid. Plane convergence angles vary with longitude. Therefore, we must specify the station where the convergence was determined when quoting it on a map. Plane convergence may be determined in two ways. Where the longitude is not known without calculation, plane convergence can be calculated using the tabled constants of the zone and the plane coordinates. This is the same equation that we saw earlier in the inverse computation. E sub o, the easting of the grid origin, is subtracted from the easting of the station, small e. This yields the difference in eastings, then dividing the eastings difference by the sum of r sub b, the mapping radius of the grid base, and n sub b, the northing of the grid base, less the northing of the station. This is the other leg of the triangle. These two quantities are the opposite and adjacent legs of the large conversion triangle. Their arc tangent yields the plane convergence angle. If plane coordinates are not known, but the longitude of the station is known, the plane convergence may be calculated using the difference in longitude between the central meridian and the station, which is a constant. B sub o, to calculate the plane convergence angle, we use the equation displayed on the screen. It determines the difference between the longitude of the central meridian, a tabled constant, and the west longitude through the station. It then proportions that difference using sine b sub o, a tabled constant. This is the same relationship that the central angle of the developed cone has to 360 degrees of longitude, as mentioned previously. If using a calculator having only 10 significant digits, such as the HP 41, it is necessary when subtracting longitudes in this equation to truncate tens and hundreds of degrees. Furthermore, it is necessary to carry all significant digits for this calculation. Therefore, use register or stack arithmetic. The last element of grid azimuth is the second term or arc-decored correction. The Lambert system is called conformal because it is intended that field angles and grid angles should be identical. They nearly are. The second term correction is usually minute and could be neglected for most courses under five miles long. It is the difference between the grid azimuth and the projected geodetic azimuth. It increases directly with the change of eastings of a line and with the distance of the occupied station from the central parallel. Figuring it is neglected in this presentation. This is the equation previously seen at the end of the graphics presentation of azimuth. It states the basic relationship that grid azimuth at small t equals the geodetic azimuth alpha minus the plane convergence angle gamma plus the second term correction delta. We have now finished our treatment of azimuths. We will next consider the treatment of distances converting them from the ground surface where they were measured to their ultimate destination, the plane surface of the grid. This product takes the, this process takes the distance over several surfaces that originates at the ground surface. The process then takes it through the surface of what is called the geoid. The geoid is a gravity surface that has an equal potential. This surface is analogous to the idea of mean sea level. However, unlike the common notion of mean sea level as a uniform surface, smoothly curving like an ellipsoid, the geoid curves up and down or undulates. These changes usually occur gradually, but nonetheless are significant over long distances. The next surface involved in our process is the geodetic or ellipsoidal surface of the GRS-80 ellipsoid. The separation between the ellipsoid and the geoid is considered for the California Coordinate System of 1983. This is a new process. The older state plane coordinate system did not consider this separation because for the most part, it could not be readily quantified. The final surface that we go to is the surface of the cone, which is in fact the plane surface of the grid. On the screen, a ground surface is shown. For our present purposes, it has an elevation significantly above sea level. Now another surface is added. It is the surface of the geoid. Practically speaking, this is the surface that serves as the datum for elevations. In other words, mean sea level. The dimension between the geoid surface and our ground surface is labeled H. We commonly know this distance is elevation. Next, we see the addition of a third surface, the surface of the GRS-80 ellipsoid. In this illustration, it lies above the geoid or sea level surface. This is the case for all 48 contiguous states. The distance at a station from the geoid to the ellipsoid is called geoid separation or geoid height, and noted by capital N. For the contiguous 48 states, geoid separation has a negative value because the geoid is below the ellipsoid. Now on the screen, we see another dimension has been added. This is ellipsoidal or geodetic height, noted as small h. It is elevation using the ellipsoid for its datum. Remembering that in our illustration, capital N has a negative value, it is apparent by inspection that small h equals capital N plus large h. This relationship is now shown on the screen. Now a horizontal measured distance at ground level is added. This distance is a chord and is noted as capital D. Lines are extended normal to the surface of the ellipsoid. They converge towards the center of the ellipsoid. Now on the screen, we see that when a ground surface chord distance is reduced to a chord distance on the ellipsoid surface, the ellipsoidal chord length differs from the chord distance at ground level. The notation L sub C indicates ellipsoidal chord length. If ground length is above the ellipsoid surface, as in this case, the ellipsoidal chord length is shorter. If ground surface is below the ellipsoid, like along the Pacific shore, Death Valley and Imperial Valley, the ellipsoidal chord length is longer than the ground length. The movement of the length through the geoid is embedded in the equations used in this example and is not a separate procedure. Next, we see the addition of the geodetic surface noted S. The geodetic distance occurs on the geodetic surface, that is, the surface of the ellipsoid. It is an arc rather than a chord. It is the geodetic length that is projected onto the cone. The cone's surface is not shown in our illustration but may be located either above or below the ellipsoidal surface. This depends on whether the line is between the exact scale parallels or outside them. It also could cross one or both of them. The line is projected onto the cone's surface by lines normal to the ellipsoid. Geodetic lines between the exact scale parallels shrink when projected onto the cone while lines outside the exact scale parallels get longer. After the line has been projected onto the cone, its length is a grid length, that is, is the distance between points on the grid. We begin our conversion of measured distance to grid distance by calculating the radius of curvature of the ellipsoid. Because the ellipsoid is flattened at its poles, the radius of curvature is uniform only along the equator. Except along the equator, the radius of curvature will vary as the azimuth of the line varies and also as one travels along the line. Because it varies with azimuth, the full name is radius of curvature in the azimuth. There are complex equations for calculating precise values for this radius. An approximate radius for each zone is the geometric mean radius of curvature at the projection origin. It is close enough for all but the most precise work. It can be obtained by the following equation. This equation states that an approximate radius of curvature in the azimuth, noted as r sub alpha plus or minus, may be calculated by dividing the geometric mean radius of the ellipsoid at the projection origin scaled to the grid, a tabled constant noted small r sub o by the grid scale factor of the central parallel which is small k sub o. This calculation yields the geometric mean radius of curvature of the ellipsoid at the projection origin. Using this approximate radius of curvature of the ellipsoid, we'll next calculate the ellipsoidal reduction factor also known as the elevation factor. The value just obtained for the approximate radius of curvature is now used in the upper equation on the screen to figure an ellipsoidal reduction factor noted as r sub e. Calculate the reduction factor by dividing the approximate radius of curvature by the sum of the approximate radius, the geodetic separation noted as capital N, and the elevation to which the measured line was reduced noted by capital H. This equation in effect develops proportion between two radii. The first is the radius of the ellipsoid at the station. The other is the radius at the ground elevation of the station. This is similar to route surveys where the centerline chord lengths of a curve may be scaled to an offset by using the ratio of the centerline radius and the offset radius. The ellipsoidal chord length noted l sub c can be calculated by multiplying the ellipsoidal reduction factor times the ground level horizontal measured distance noted capital D. This is shown in the lower equation on the screen. This ellipsoidal chord length is a straight line distance between two points on the surface of the ellipsoid. It is not a distance measured on the surface of the ellipsoid which would be an arc length called a geodetic length. The elevation used for the measured line is usually one of the following. For triangulation, it is the average elevation of the baseline. For electronic distance measuring devices, it is the elevation to which the slope length was reduced. This is usually the elevation of the instrument or the elevation of the station mark. For tape lines, it is usually the average elevation of the line. Each one meter air in the geoid separation or elevation contributes 0.16 parts per million of air to the distance. Often, geoid separation can be estimated for stations in the general area based upon published values for nearby stations. The value of capital N usually does not vary greatly in a given area. Examining the variations in the proximity of your work can aid in determining if anomalies exist in the area. If precise geodetic lengths are desired, a correction from ellipsoidal chord length to geodetic length on the ellipsoid surface may be applied to lines generally greater than five miles long. Shorter lines and lines measured in segments are essentially arcs and need not be corrected. The ellipsoidal chord length must be nearly nine miles before the correction equals one one hundredth of a foot at over 19 miles before it equals one tenth of a foot. The upper equation on the screen is for calculating the chord correction for lengthening the ellipsoidal chord length to the geodetic length. Remember that arcs are always longer than their chords. You may recognize this equation from route surveying. The equation calculates the chord correction by cubing the ellipsoidal chord length and then dividing it by twenty-four times the square of the radius of curvature in the azimuth. As shown by the lower equation, after determining the chord correction, the correction is added to the ellipsoidal chord length yielding the geodetic length noted as s. Once we have calculated the geodetic length of the line, it must be projected onto the cone. Grid length is calculated by multiplying the geodetic length by a grid scale factor. Grid scale factor is an expression of the amount of distortion imposed on the length of a line from the ellipsoid as it is projected onto the grid cone. It is represented by the small letter k. Scale factor is the ratio of the length of the grid on the grid to the length on the ellipsoid. Scale factor is dependent upon latitude. It is less than unity between the standard parallels and greater than unity outside them. Scale factors can be calculated for points or four lines. For this presentation, point scale factors are used exclusively. If greater accuracy is needed for a long line, the point scale factor for the midpoint may be used. Approximate point scale factors may be interpolated from projection tables. A method precise enough for any plain coordinate work uses polynomial coefficients. We must first find the value of the radial distance discussed previously and noted as small u. If plain coordinates are known, this equation can be used to find the radial difference. This is similar to another equation we saw earlier. You may recognize the Pythagorean theorem in this equation. This equation takes the square of the leg of the conversion triangle comprised of the difference in eastings from the central meridian to the station, adds it to the square of the leg of the giant triangle comprised of difference of northings, then takes the square root of their sum yielding the hypotenuse of the conversion triangle, which is the mapping radius through the station. That mapping radius through the station is then subtracted from the mapping radius of the projection origin to obtain the radial difference. If geodetic latitude for the line is known, the following equations can be used to calculate the radial difference. We have seen them before during our discussion on forward computation. This equation takes the north latitude of the station noted as b and subtracts the latitude of the projection origin, the central parallel. A tabled constant noted as b sub o, it yields the difference in latitude. If working on a calculator computing only 10 significant digits, such as the HP 41, it is necessary when subtracting latitudes to truncate tens of degrees. Using the latitude difference just obtained, we enter it into either equation now displayed to calculate the value of u using polynomial coefficients. Some persons feel that the second equation is easier for handheld solution because it has a nested form. Be careful to keep track of arithmetic signs. The radial difference u has a positive value if the station lies northerly of the central parallel and negative value if it lies southerly. Once we have calculated the radial difference by one of the equations we have just seen, we can use it to calculate the point scale factor using polynomial coefficients. The value calculated above for the radial distance u and the tabled f polynomial coefficients can be used in the equation shown on the monitor to calculate a point scale factor noted as small k. Approximate scale factors for points may also be found in projection tables. The latitude or mapping radius through the station is argued against scale factor. Intermediate values are interpolated. Next we use the point scale factor just found to project our geodetic length onto the grid. This will give us its grid length. The grid length, l sub grid, is calculated by multiplying the point scale factor k together with the geodetic length of the line small s. If no correction from ellipsoidal chord length to geodetic length is warranted and the latitude and elevation differences are not great, measured lengths may be multiplied by a combined factor to calculate grid lengths using a value for average elevation and average latitude. The upper equation states that we can calculate a combined factor noted cf by multiplying an average ellipsoidal reduction factor r sub e by a point scale factor calculated for the mid-latitude of the survey noted small k. The lower equation shows that once we have determined a combined factor, we can multiply each ground level horizontal measured length noted capital D by it to calculate each course's grid length. Grid length may be converted to ground length by reversing the above procedures. For measurements of similar elevation, latitude and short length, ground length may be calculated by dividing a combined factor into the grid length. In dealing with areas calculated on a grid, the areas derived from state plane coordinates must be corrected if ground level areas are desired. The equation shown on the screen may be used to correct grid area to ground area. It shows that the ground level area of the land may be computed by dividing the grid area of the figure by the square of the combined scale factor. In effect, this applies the combined factor to both the length and the width of the figure. A problem arises in converting coordinates from one zone to another. To convert plane coordinates and the overlap of zones from one zone to another, convert the plane coordinates from the original zone using the constants for that zone to geodetic latitude and longitude. Then, using the coordinates for the new zone, convert the geodetic latitude and longitude to the plane coordinates in the new zone. As enacted by the California legislature, linear units may be expressed either in meters or in your US survey feet. Plane coordinates or distances expressed in meters or feet can be converted to the other unit simply by multiplying the north end and east end or the measurement by the US survey foot to international meter equivalency. Shown on the screen, the US survey foot, the linear unit of the California coordinate system, is defined as one US survey foot equals 1,200 divided by 3,937 international meters. It is the specialized foot that is used in surveying. When first using functions in a calculator or computer to convert between feet and meters, you should test to ensure that the correct conversion factor is employed. You should test it by using the machine's functions to convert exactly 3,937 feet to meters. The result should be exactly 1,200 meters. The reverse must also be true. Comparing the result while displaying the maximum decimal places should tell you whether the conversion you are using is for US survey foot. One may work the California coordinate system in either feet or in meters. However, one must be consistent. Not only must all distances and coordinates be uniformly in feet or in meters, but tabled constants must also correspond with the linear units chosen. I hope that this presentation will help you in your understanding of the state plane coordinate system. I also encourage you to study hard and prepare well for your land surveying examination. I wish you the best of luck and success. Thank you.