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 and Changing Projections

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.

Definitions 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 to spheres to complex ellipsoids. Flat earth models can be accurate for 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.

Ellipse Z b O a X   An ellipse is defined by: For the earth: F2 F1 P Focal length =  Flattening ratio: f = (a-b)/a Distance F1-P-F2 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 b F1 O a F2 X   P

Ellipsoid or Spheroid b O a Z X Z Y a b Rotational axis Rotate an ellipse around one of its axis.

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.

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

Earth Surfaces Topographic surface Ocean Ellipsoid Gravity Anomaly Geoid Topographic surface Ellipsoid Gravity Anomaly

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

Standard Vertical Geodetic Data 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 and 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 then projected onto a flat surface.

Map Projection Projection Scale Scale Fraction Representative Fraction Globe distance Earth distance Scale Fraction Map distance Globe distance (e.g. 1:24,000) (e.g. 0.9996)

Map Distortion 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 than preserving another.

Map Distortion Conformal projections: Preserves shapes. Equal area projections: Preserves area. 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. Example: 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 Mercator Transverse Oblique Tangent Secant

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 and 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 (f, l, z) for the whole earth Projected coordinates (x, y, z) on a local area of the earth’s surface

Global Cartesian Coordinates Z Greenwich Meridian O • Y X Equator

Geographic Coordinates 90 W 120 W 60 W 30 N 0 N 60 N

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

Geographic Coordinates Earth datum defines the standard values of the ellipsoid and geoid. Latitude (f) and longitude (l) are defined using an ellipsoid. Elevation (z) is defined using a geoid.

Latitude f Take a point S on the surface of the ellipsoid and define there the tangent plane mn. Define the line pq through S and normal to the tangent plane. Angle pqr is the latitude f, of point S. f S m n q p r

If Earth were a Sphere ... r D r Dl C B Df 0 N R A Length on a Meridian: AB = R Df (same for all latitudes) Length on a Parallel: CD = r Dl = R Cosf Dl (varies with latitude) r r D Dl C B Df 0 N R A

If Earth were a Sphere ... 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: p radians = 180°, so 1º = p/180° = 3.1416/180° = 0.0175 radians For the meridian: DL = R Df = 6370 Km * 0.0175 = 111 km For the parallel: DL = R Cosf Dl = 6370 * Cos30° * 0.0175 = 96.5 km Meridians converge as poles are approached

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

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 (lo). Reference Latitude (fo) is the equator. (Xshift, Yshift) = (xo,yo) = (500,000, 0). Units are meters.

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

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

Overview Geodetic Datum Map Projections Coordinate Systems Viewing, Defining and Changing Projections

Viewing Projection There are two ways to view projection details in ArcCatalog. One is by viewing the projection details in the map display. To do this, make sure that dataset is selected in table of contents and click on the Metadata tab in the map display. The first paragraph defines the projected coordinate system and the geographic coordinate system. The second paragraph defines the extent of the dataset.

Viewing Projection Another way is viewing the projection details in Spatial Reference Properties wizard. To view the projection details of a dataset, right-click on the layer and then click Properties. In the wizard that opens up, click on the Fields tab; then on geometry, under data, type “column.” Finally, in the Field properties frame, click on the button located on the Spatial Reference row.

Viewing Projection Defines the extent of the shapefile and feature class.

Defining Projection Before moving feature classes into a feature dataset, the spatial reference of the feature dataset should be defined. In the Spatial Reference Properties wizard: Select can be used to select a coordinate system available. Import can be used to define the projection using an existing projection from a feature dataset, feature class or shape file. New can be used to assign custom projection.

Projecting On-The-Fly Each data frame in an ArcMap document has its own projection. The projection of the data frame can be defined using Data Frame Properties/Coordinate System. Data (with map projection defined) added to a data frame is re-projected on-the-fly to the data frame’s projection. A layer in a data frame can be exported either with its original projection or with data frame’s projection.

Defining and Changing the Projection ArcToolBox contains tools for data conversion and data management. Projection definition and projection change can be accomplished using ArcToolBox ArcCatalog can also be used for defining the data projection.

Projecting Data ArcToolBox can be used for changing projections. To change the projection of a dataset, its original coordinate system should be known.