Map Projections Francisco Olivera, Ph.D., P.E. Srikanth Koka Department of Civil Engineering Texas A&M University

Overview Geodetic Datum Map Projections Coordinate systems Viewing,Defining,Changing Projections

Definition A geodetic datum defines the size and shape of the earth, and the origin and orientation of the axis used to define the location of points. Over time, geodetic data have evolved from simple flat surfaces and spheres to complex ellipsoids. Flat earth models can be accurate over short distances (i.e., less than 10 Km), spherical earth models for approximate global distance calculations, and ellipsoidal earth models for accurate global distance calculations.

Shape of the Earth We think of the earth as a sphere when it is actually an ellipsoid, slightly larger in radius at the equator than at the poles.

P Oa b X   Ellipse Z An ellipse is defined by: Focal length =  Flattening ratio: f = (a-b)/a Distance F 1 -P-F 2 is constant for all points P on ellipse When  = 0 then ellipse = circle For the earth: Major axis: a = 6378 km Minor axis: b = 6357 km Flattening ratio: f = 1/300 F1F1 F2F2 P

Earth Surfaces Geoid is a surface of constant gravity. Topographic surface Ellipsoid Sea surface Geoid

Standard Ellipsoids Ref: Snyder, Map Projections, A working manual, USGS Professional Paper 1395, p.12

Standard Horizontal Geodetic Data NAD27 (North American Datum of 1927) uses the Clarke (1866) ellipsoid. NAD83 (North American Datum of 1983) uses the GRS80 ellipsoid. WGS84 (World Geodetic System of 1984) uses GRS80.

Elevation P z = z p z = 0 Mean Sea level = Geoid Topographic Surface Elevation is measured from the Geoid

Standard Vertical Geodetic Datum A vertical datum defines elevation z, taking into account a map of gravity anomalies between the ellipsoid and the geoid. NGVD29 (National Geodetic Vertical Datum of 1929). NAVD88 (North American Vertical Datum of 1988).

Overview Geodetic Datum Map Projections Coordinate systems Viewing,Defining,Changing Projections

Map Projections A map projection is a mathematical algorithm to transform locations defined on the curved surface of the earth into locations defined on the flat surface of a map. The earth is first reduced to a globe and the projected onto a flat surface.

Map Projection Representative Fraction Globe distance Earth distance Scale Projection (e.g. 1:24,000) (e.g ) Scale Fraction Map distance Globe distance

Distortion Projected Maps In the process of transforming a curved surface into a flat surface, some geometric properties are modified. The geometric properties that are modified are: Area (important for mass balances) Shape Direction Length The difference between map projections has to do with which geometric properties are modified. Depending on the type of analysis, preserving one geometric property might be more important that preserving other.

Distortion Projected Maps Conformal projections: Preserves local shapes. Equal area projections: Preserves the area of displayed Equidistant projections: Preserves the distances between certain points.

Types of Projections Conic: Screen is a conic surface. Lamp at the center of the earth. Examples: Albers Equal Area, Lambert Conformal Conic. Good for East-West land areas. Cylindrical: Screen is a cylindrical surface. Lamp at the center of the earth. Examples: (Transverse Mercator). Good for North- South land areas. Azimuthal: Screen is a flat surface tangent to the earth. Lamp at the center of the earth (gnomonic), at the other side of the earth (stereographic), or far from the earth (orthographic). Examples: Lambert Azimuthal Equal Area. Good for global views.

Conic Projections Albers and Lambert

Cylindrical Projections Transverse Oblique TangentSecant Mercator

Azimuthal Lambert

Albers Equal-Area Conic

Lambert Conformal Conic

Universal Transverse Mercator

Lambert Azimuthal Equal-Area

Overview Geodetic Datum Map Projections Coordinate systems Viewing,Defining,Changing Projections

Coordinate Systems A coordinate system is used to locate a point of the surface of the earth.

Coordinate Systems Global Cartesian coordinates (x,y,z) for the whole earth. Geographic coordinates ( , , z) for the whole earth. Projected coordinates (x, y, z) on a local area of the earth’s surface. The z-coordinate in Global Cartesian and Projected coordinates is defined geometrically; and in Geographic coordinates gravitationally.

Global Cartesian Coordinates O X Z Y Greenwich Meridian Equator

Geographic Coordinates Longitude line (Meridian) N S W E Range: 180ºW - 0º - 180ºE Latitude line (Parallel) Range: 90ºS - 0º - 90ºN N S WE (0ºN, 0ºE) Equator, Prime Meridian

If the Earth were a Sphere 0 N  R r r A B C  Length on a Meridian: AB = R  (same for all latitudes) Length on a Parallel: CD = r  = R Cos  (varies with latitude) D

Example: What is the length of a 1º increment on a meridian and on a parallel at 30N, 90W? Radius of the earth R = 6370 km. Solution: A 1º angle has first to be converted to radians:  radians = 180°, so 1º =  /180° = /180° = radians For the meridian:  L = R  = 6370 Km * = 111 km For the parallel:  L = R Cos  = 6370 * Cos30° * = 96.5 km Meridians converge as poles are approached If the Earth were a Sphere

Cartesian Coordinates (  o, o ) (x o,y o ) X Y Origin A planar cartesian coordinate system is defined by a pair of orthogonal (x,y) axes drawn through an origin.

Geographic Transformations Moving data between coordinate systems may include transformation between the geographic coordinate systems. United states uses a grid-based method for conversions: NADCON NADCON: NAD 1927 and other older GCS to NAD 1983

Coordinate Systems Universal Transverse Mercator (UTM) - a global system developed by the US Military Services. State Plane - civilian system for defining legal boundaries.

Universal Transverse Mercator Uses the Transverse Mercator projection. 60 six-degree-wide zones cover the earth from East to West starting at 180° West. Each zone has a Central Meridian ( o ). Reference Latitude (  o ) is the equator. (X shift, Y shift ) = (x o,y o ) = (500,000, 0) in the Northern Hemisphere. Units are meters

UTM Zone 14 Equator -120° -90 ° -60 ° -102°-96° -99° Origin 6°

State Plane (USA only) Defined for each State in the United States. East-West States (e.g. Texas) use Lambert Conformal Conic, North-South States (e.g. California) use Transverse Mercator. Texas has five zones (North, North Central, Central, South Central, South) to give accurate representation. Greatest accuracy for local measurements

Overview Geodetic Datum Map Projections Coordinate systems Viewing,Defining,Changing Projections

Viewing the Layer Projection In ArcCatalog, right click on the layer, and then click Properties. In the window that opens up, click on the Fields tab, and then on Geometry under Data Type column. Finally, in the Field Properties frame, click on the button located on the Spatial Reference row to open the Spatial Reference Properties window. In the Spatial Reference Properties window, the layer projection is displayed in the Coordinate System tab, and the coordinate rage in the X/Y Domain tab.

Defining the Layer Projection In the Spatial Reference Properties window, one can Select, Import, create a New, Modify or Clear the layer projection. For projecting on-the-fly, the layer projection has to be defined.

Viewing/Modifying the Data Frame Projection The Data Frame projection is user- defined. All Data Frames have a projection. If not modified by the user, the projection is the one of the first layer added to it. To view/modify the Data Frame projection, in ArcMap, right-click on the Data Frame name and then click on Properties to open the Data Frame Properties window. In the Data Frame Properties window, one can select, Import or create a New projection.

Projection On-The-Fly All layers (with projection defined) are automatically projected on-the-fly to the data frame coordinate system when they added to the data frame. If the data frame projection is modified, all its layers (with projection defined) are automatically projected on-the-fly. A layer, whose projection is not defined, cannot be projected on-the-fly and is displayed according to its coordinate values. If a layer, whose projection is not defined, is added to a data frame that has the same projection, the layer will be displayed in the correct location. A layer can be exported either with its original projection or with data frame’s projection.