Coordinate Systems and Map Projections Cartography Coordinate Systems and Map Projections

Properties of a Globe Parallels are always parallel to each other and always evenly spaced along meridians and decrease in length poleward Meridians converge at both poles and are evenly spaced along parallels Distance between meridians decrease poleward Parallels and meridians always cross at right angles.

Space View

Earth’s Dimensions

Latitude

Longitude

Prime Meridian

Earth’s Circles

Spherical Coordinates

Geographic Coordinate Systems

Projections and Coordinate Systems Geographical coordinate system – are locations on the map or lat/long Projected coordinate system – locations that have been changed from spherical coordinates to planar (Cartesian) coordinates

Projections

Map Projection How to represent a curved globe on a flat surface? The globe is the only true map, but scale problems force us to use projections for larger scales Unfortunately, this can lead to distortion - such as misrepresenting the Great Arc

Why project at all? We have to. Flat paper maps Flat computer screen Measuring distances and areas in degrees is not a desirable thing to do.

From Round to Flat It is necessary to use a Projection Two primary questions to decide which one: Is the intended use to preserve equivalence (Equal Area) or its true shape (Conformal) Only one can be preferred and the other will suffer or lose its relationship.

Flattening Earth

Nature of Projections Modern Cartography and most GIS programs compensate mathematically for these distortions (Equivalence and Conformality). Projection is used because originally these maps were a globe projected and traced on a surface. Or wires where used to represent latitude (parallels) and longitude (meridians)

Projection Surfaces – “developable”

Projection Family Family refers to the developable surface Plane = Cylinder = Cone =

Figure 3.1

Figure 3.2

Figure 3.3

Figure 3.4

Cylindrical Projections

Cylindrical Cylindrical - whole earth – true shape – from Gerardus Mercator (AD 1569), distortion in size towards poles, but useful in navigation because true bearings are given, but distance is skewed

Cylindrical projections Transverse cylindrical Cylindrical

Developable Surface – Cylinder

Figure 3.7

Projections Mathematical projection of lines on a sphere to a flat piece of paper Cylindrical projection Courtesy of ESRI, Inc.

Cylindrical Projection

Cylindrical

Miller Cylindrical

World Time Zones

Planar projection Planar - used for state or smaller sized areas, where curvature of the earth can be ignored – usually circumpolar maps.

Orthographic Projections Polar Oblique

Figure 3.6

Planar Projection

Gnomonic Illustrating the difference between Mercator and Gnomic Great Circles – the most direct route between two points Rhumb Line – lines of constant direction

Rhumb and Circle

Rhumb and Circle

Conic Projections

Conic Projection Conic - used for continent sized maps, the distortion is less that with plane or cylindrical

Conic projections Standard parallels Tangent conic Secant conic

Developable Surface- Cone

Conic Projection

Mid Latitude Projections Albers equal-area Conic, equal area with two standard parallels Areas are proportional and directions are true in limited areas Used in the United States and other large countries with a larger east-west than north-south extent

Special Projections Mercator Straight meridians and parallels that intersect at right angles Scale is true at the equator or at two standard parallels equidistant from the equator Navigation – all straight lines on the map are lines of constant azimuth

Special Projections Gall-Peters Who’s the creator? Reaction to the Mercator projection, which showed European dominance Mercator distorts area Peters argues that his portrays distances correctly, which it does not De-emphasizes area distortions at polar areas

Gall-Peters

Figure 3.20

Figure 3.21

Figure 3.22

Figure 3.23

Figure 3.24

Figure 3.25

Datums and Coordinate Systems

Geodesy and Datums Shape of the Earth

Figure 2.2

Figure 2.3

Table 2.1

Table 2.2

Figure 2.4

National or Global Datums Surface against (or with) which horizontal or vertical positions may be defined A datum is a reference surface -

Horizontal Survey Benchmarks

1981 Survey Network

Earth’s Dimensions

There are two axes are called the Isaac Newton (1670) suggested the earth would be flattened at the poles, due to centrifugal force   Thus, the radius along the axis of rotation would be less than through the equator There are two axes are called the a) semi-major axis (through the equator) b) semi-minor axis (through the poles)

Earth’s is Flattened - an Ellipsoid Two radii: r1, along semi-major (through Equator) r2, along semi-minor (through poles)

The Earth is NOT an Ellipsoid (only very close in shape) The Earth has irregularities in it - These deviations are due to differences in the gravitational pull of the Earth

Geoidal Earth Surface where the strength of gravity equals that at mean sea level Highest point on geoid -- 75 m above elliposid (New Guinea) Lowest point on the geoid -- 104 m below ellipsoid (south of India)

Geoid earth The Geoid is a measured surface (not mathematically defined) Found via surface instruments (gravimeters) towed behind boats, planes, or in autos Or, from measurements of satellite paths May be thought of as an approximation of mean sea level

Example of Ellipsoid/Geoid Relationship Earth’s surface Ellipsoid Geoid

Spheroids and Datums The earth is not a sphere so a spheroid more accurately matches the Earth Spheroids parameterize a coordinate system in a datum Datum = a set of control points NAD27 NAD83 WGS1984

Datums Common Datums North American 1983 or NAD83 Sphere Spheroid Geoid Datum Common Datums North American 1983 or NAD83 North American 1927 or NAD27 World Geodetic System 1984 or WGS84 Global Reference System 1980 or GRS80

p3 p1 p2 h1 h2 h3 Datum height

Figure 2.5

Typical datum offset Roads in NAD83 Photo in NAD27 DOBA?

Specifying Projections The type of developable surface (e.g., cone) The size/shape of the Earth (ellipsoid, datum), and size of the surface Where the surface intersects the ellipsoid The location of the map projection origin on the surface, and the coordinate system units

Projection Case Case refers to the number of tangents, or standard parallels tangent secant azimuthal cylindrical conic

Transverse or Equatorial Projection Aspect Aspect refers to the location of tangency Normal or Polar Oblique Transverse or Equatorial

Defining a Projection – LCC (Lambert Conformal Conic) The LCC requires we specify an upper and lower parallel A spheroid A central meridian A projection origin origin central meridian

Projections – developable surfaces Cylindrical Conic

Projections in ArcMap

Projected Coordinate Systems Projections are a systematic rendering of geographic coordinates Geometric – planar/ cylindrical/ conic Gnomonic Orthographic Stereographic Mathematical – defined by formulas Distortion patterns – tangent and secant

Projection parameters False easting False northing Latitude of origin 0,0 Central meridian

Properties of a coordinate system GCS/Datum/Spheroid Map units Central Meridian False easting False northing Reference latitude Standard parallels Different projections may have different types of parameters used.

Distortion All map projections introduce distortion Type and degree of distortion varies with map projection When using a projection, one must take care to choose one with suitable properties

Types of distortion Area Shape Distance Direction

Projection distortions Mercator Equidistant Conic Distorts distance and area Preserves direction and shape Distorts direction and shape Preserves distance and ~area

Compromise projections Robinson Distorts all four properties a little

Table 3.1

Table 3.2

Projection Surface Rays “traced” from a source, through a globe to the projection surface All map projections cause distortion We control distortion by choosing the surface, and projection center

Distortion Varies Across Map

Projections in ArcMap Data may be projected or unprojected Converts all data on the fly to the coordinate system chosen for the data frame Relies on correct coordinate system definitions for each layer

Coordinate Systems Defined for a data layer, including Geographic coordinate system Spheroid/ellipsoid on which the lat/long values are based The datum on which the GCS is based Projection parameters (if any)

Coordinate systems Unprojected Projected Has GCS and datum defined Map units are decimal degrees Has no projection in CS definition Projected Has GCS and datum defined Map units usually are feet or meters Has projection in CS definition GCS_WGS_1984 World_Robinson

Extents The extent of a spatial data set indicates the range of x-y values present in the data Stored map units

Values in this range indicate units of degrees and thus a GCS CS Definitions Values in this range indicate units of degrees and thus a GCS Unprojected coordinate system

Large values usually indicate a projected coordinate system CS Definitions Large values usually indicate a projected coordinate system Projected coordinate system

Note on terminology “Projection” has two definitions The mathematical transformation applied to convert a spherical coordinate system to a planar coordinate system (used in this text) The complete coordinate system definition of a GIS data set, including the geographic coordinate system, datum, and projection (used by some, including older ESRI materials)

On the fly projection in ArcMap Source Layers Data frame World in GCS_WGS84 Robinson_WGS_84 Latlon in GCS_WGS84 Input layers have any CS Set data frame to desired CS

Map units Source Layers Data frame Decimal degrees Meters Decimal degrees On-the-fly projection is a temporary change in the CS for display only. Stored x-y values and units are unchanged.

Setting the data frame coordinate system

Modifying the properties of the selected data frame coordinate system Set: Central Meridian False easting False northing Map units Spheroid Datum

Understanding units Be careful not to confuse The stored map units Property of the file’s CS X-Y values are saved in the spatial data file Can’t be changed The data frame map units Determined by the data frame CS Changed by changing the data frame CS properties The display units Set by the user to any desired units Affects only the x-y location readout and measurements

Stored map units 1. Add file to ArcMap 2. Set data frame CS

Managing coordinate systems Three typical operations Examine a layer’s CS information Define a CS for a layer with unknown or incorrect CS Convert a layer to a different CS

Examine the CS (ArcMap) Right-click layer and open properties The extent is given in map units, not the units of the data frame CS. Click Source tab to read CS information

Examine the CS (ArcCatalog) Click Shape in Fields tab Read coordinate system definition Right-click layer to open properties Examine details or define CS

Defining the CS Places a label on the data set, specifying the CS of the coordinates already stored inside. It is imperative that the definition match the stored units. Knowing the correct CS may require contacting the person who created the data. GCS_NAD1983 ROADS -103.567,44.628 -103.678,44.653 -103.765,44.73

Two ways to define a CS Set the CS in the Properties in ArcCatalog Use the Define Projection tool in ArcToolbox

Define CS in Properties Click Shape in Fields tab Read coordinate system definition Right-click layer to open properties Examine details or define CS

Define the CS in ArcToolbox

Convert a layer to a new CS Creates a new spatial data file Does not destroy or change the original file Recalculates each x-y point into the new coordinate system and units Labels the new file with the new CS Wells.shp -103.567,44.628 -103.678,44.653 -103.765,44.732 … Wells.shp 445678,654321 445021,650001 444823,649200 … Project Units in decimal degrees Units in meters

Two ways to project Use the Project tool in ArcToolbox Export the data in ArcMap

Using a tool

Project tool

Export in ArcMap Set the data frame coordinate system to the desired output CS Export and choose to use the data frame CS TIP: You can project a coverage without an ArcInfo license this way!

CAUTION! DO NOT confuse This is a very common mistake Projecting Defining a projection This is a very common mistake It is also a dangerous one At best you won’t be able to display the data At worst you may corrupt a data file forever

Define Projection only changes the label of the coordinate system Project changes the label AND the xy coordinate values UTMZone13_ NAD1983 GCS_NAD1983 Roads.shp -103.567,44.628 -103.678,44.653 -103.765,44.732 … Roads.shp 445678,654321 445021,650001 444823,649200 … Project

Faulty reasoning I want to display ROADS and STATE together in UTM UTM Zone13 STATE 445678,654321 445021,650001 444823,649200 I want to display ROADS and STATE together in UTM ROADS is in GCS I need to open ArcCatalog and change the ROADS to UTM GCS_NAD1983 ROADS -103.567,44.628 -103.678,44.653 -103.765,44.732 Result: A data set with a mislabeled CS.

Result of faulty reasoning ArcMap sees label and says “I don’t have to reproject it to UTM on-the-fly” UTM Zone13 STATE 445678,654321 445021,650001 444823,649200 UTM Zone 13 ROADS -103.567,44.628 -103.678,44.653 -103.765,44.732 y Data frame CS: UTM Zone 13 . x 0,0

Correct reasoning I want to display roads and state together in UTM GCS_NAD1983 I want to display roads and state together in UTM Just let ArcMap project on-the-fly OR use the Project tool to convert roads to UTM, creating a new file. ROADS -103.567,44.628 -103.678,44.653 -103.765,44.732 UTM Zone13_ NAD83 STATE 445678,654321 445021,650001 444823,649200

Employment of Map Projections Carefully consider: Projection properties – is it suited (Equivalence? Conformal?) Deformational properties – acceptable? Benefit the shape? Projection center – can it be centered? Familiarity – for users? for designers? Cost – economical? available?

Base Map Compilation Manual base map rules (works with GIS) Map purpose Use objective evaluation Avoid personal bias Determine what elements typify the character of the area Strive for uniformity (if you generalize, make sure it evenly done)

Accuracy and Reliability Map accuracy = the degree of conformance to an established standard Is your source map a: Basic map – from an original survey Derived map – compiled from basic maps The further you get from the basic map the better the chance of error Map copyright – check for and honor copyright to avoid trouble