Lab 4: Projections and Coordinate Systems

1 1 Lab 4: Projections and Coordinate Systems Overview: Any planar coor dinate system is based on a map projection , which is the orderly transfer of positions or places on the surface of the Earth to cor responding points on a two-dimensional surface, like a sheet of pape r. Seve ral elements go into defining a projection . To start with, Earth is nearly spherical. In fact, to better describe the shape of Earth, we use the geometrical figur e called a spheroid, which is a flattened sphere and can also be called an ellipsoid. The elliptical nature of the sphe roid is defined by two pa rameters: the semi-minor (polar) and the semi-major (equatorial) axes of the spheroid. These axes are measured from the center of the earth to eithe r the North Pole (polar) or a point along the equator (equatorial). Ove r the years and with the advancement of measurement technolog ies, the distances measured for the semi- major and semi-minor axes have changed.

2 The two most commonly us ed spheroids for North America are the Clarke spheroid of 1866 (Clarke 1866) and the Geodetic Reference system spheroid of 1980 (GRS80) . Whe n projecting from a geographic Coordinate system (latit ude, longitude) to a planar Coordinate system (UTM, state plane, etc.), there have be en seve ral standards developed that provide a frame of referenc e for measurement. These standards (da tums) define the or igin and orientation of the grid tha t make s up each particular Coordinate system . Datums that are frequently us ed within North America include: the Nor th American Datum of 1927 (NAD27) , which is defined based upon the Clarke 1866 spheroid and has its origin in Meade s Ranc h, Kansas, and North American Datum of 1983 (NAD83), which is defined on the GRS80 spheroid and uses the center of Earth s mass as its origin. Whe n specifying the pa rameters of a projection , you will typi cally specify either the spheroid or the datum but not both.

3 Finally, it must be understood that distortions of shape , area, distance and direction are inherent to all pr ojections. Projections can be conformal, equal-area, equidistant, or true-direction. Projections may minimize distor tion in various categories, but no projection can minimize distortion in all the categories. The attempt to minimize the distortions is approached by looking at various ways of projecting a spheroid onto a flat piece of paper. Projections that are cylindrical, conical, or planar in nature use va rious points of intersection or tangenc y with the spheroid upon which to minimize the distortion. In defining these points of tangenc y, or standard parallels, we are specifying the extent of our study a rea for which distor tions will be minimized. The fur ther a way you move from the se de fined areas (standard line s), the more distor tion you will enc ounter.

4 For more d etailed discus sion, see the appropriate cha pter in your textbook. Lear ning Objectives: Figure 2. Tangent interception on a conic projection . Source: Figure 1. Spheroid representation (a) semi-major axes (b) semi-minor axes. Source: 2 Lab 4: Projections and Coordinate Systems To explore the basic structure of a planar c oordinate system ; To learn to re-pr oject geographic data into various Coordinate Systems ; To underst and the distortions of shape , a rea, distanc e, a nd direction that are inherent to the various types of Projections ; To wor k with multiple data frames in a map layout To be s ubmitted in a single PDF doc ument: 1. (15 pts) A write-up (500 word maximum) answering the questions throughout the lab; inclusion of g raphics to illustrate your answers may e xtend the 500 word limit. 2. (5 pts) A single map layout with multiple data frames that depicts the United States in six different Projections as detailed in the lab.

5 Procedure: 1. Exploring a Cartesian c oordinat e system a. Start ArcCatalog and connect to your working directory b. Launch ArcMap with the a ppropriate icon from the toolba r c. Drag and drop (by hi ghlighting, holding down a l eft-click, and dragg ing to ArcMap) the triangle and grid coverages (in this order) into ArcMap f rom your working directory. Or add the coverages using the te chnique used in earlier labs. Click OK if you get the following e rror message: ** It is important to note that this error will be displayed any time a layer i s added without a defined proj ection. d. Use the identify tool to determine the coordinates of the triangle points as well as the size of the area c reated by the boundary of the triangle (think of how to do this without calculating the area manua lly) Question 1 (3 points): a. What is the differenc e be tween geographic and projected coordinates ?

6 Which utilizes a Cartesian Coordinate system ? b. In what situations might eac h Coordinate type be most appropriate? 2. Understanding distortions of p rojections u sing Tissot Indicatrix figures 3 Lab 4: Projections and Coordinate Systems A Tissot ( Tiss-oh ) Indicatrix is a figur e that shows how a projection distorts the g eometric pr operties at a point. To underst and how distortion affects a single point on a GIS l ayer, you will employ a Tissot s Indicatrix where a c ircle on a s phe re is projected onto a p lane: The resulting shape is always an ellipse. The ratio of the lengths of the semi-major axis, a, a nd the semi-minor axis, b, of this ellipse (Fig. 1) represents the maximum and minimum distortion at the point being conside red (Hrad lek and Hamilton, 1973; McDonnell, 1979). If the ellipse is circular (a = b), the projection is conformal; if the ellipse has an area equal to that of the gene rating circle, the projection is equal-area; finally, if eithe r a or b is constant, the pr ojection is equal-distant in t he corresponding direction.

7 For many common map Projections , a and b occur along the meridians and parallels of the earth making it relatively easy to calculate the scale factors associated with length, area, and angle. It should be remembered that no pr ojection of a sphere onto a plane can preserve both shape and area simultane ously. (Finlayson and Montgomery, 2002) A reference circle shows what the circle would look like if the pr ojection did not di stort geometry. The figure shows what a projected circle would look like on a map. By displaying many Tissot Indicatrix figures on a map, you can readily understand how a projection changes geometry. a. Go to om/java/maps/proj /, a java applet that demonstrates distor tions found within map Projections using the Tissot Indicatrix. Accept and allow Run. b. Scroll down to The Globe Applet c. Read the instructions.

8 D. Check the Tissot I ndicatrix box be low the map display. e. Select several different map Projections . Using t he T issot indicatrix and your own sense of proportion and direction, study how each map projection affects area, shape, a nd direction. f. Use your mouse to drag the map from left t o right and (whe n possible) f rom top to bottom and observe how cha nging the center o f the view affects the distor tion in different regions of the world. Try zooming in to areas with which you are f amiliar. Remember that although the java a pplet you are us ing allows you to dyna mically adjust the projection to best observe each region in the wor ld, if you were going to produc e a static print map you would have to settle on a single viewpoint to represent all areas of interest. 3. Displaying dat a layers of different proj ections in A rcMap 4 Lab 4: Projections and Coordinate Systems a.

9 Start a new view within ArcMap b y selecting the New Map File icon (do not save the previous view of the g rid and triangle). b. Drag and Drop (or any other method to add data) mys tery1 first. The n add mys tery2 from your working directory into the display area of A rcMap. Drop in t he mystery1 layer f irst. (If you add mystery 2 first, the d ata will be slightly of fset; this is due to an unusual discrepa ncy in converti ng be tween these two particular datums.) c. These l ayers are in two different Projections (Note the Geographic Coordinate Systems Warning ). ArcMap a utomatically re-displays all layers in the data view to the projection of the first layer added, also referred to as projecting on-the-fly. (In the future be sure to add the data with your projection of interest to the data frame f irst). d. To find out the Projections of mys tery1 and mystery2, right-click on the layer in the table o f contents (TOC) and navi gate to Prope rties then the Source t ab.

10 E. You can also find out the projection of a shape file within ArcCatal og by right-clicking on the data layer, selecting Properties , a nd clicking XY Coordinate system . The pr ojection is specified by the highlighted projected system in the list with full details in the Current Coordinate system box be low. Question 2 (2 points): What is the projection of mys tery1? Of mystery2? Make sure you underst and the different functions of on-the-fly projection (what ArcMap just did to mystery2 so that i t could be displayed with mystery1), projecting using ArcToolbox (what you are going to do next), and de fining a projection . If the p rojection is not defined to begin with, ArcMap has no idea how to re-project the d ata or project them on- the-fly. (ArcMap will display da ta with undefined Projections , but only a s data on a meaningless Coordinate system that cannot be mapped to any of the Coordinate s ystems it knows).