Lecture 2

Map Projections and GIS

Coordinate Systems

Tomislav Sapic

GIS Technologist

Faculty of Natural Resources Management

Lakehead University

• The two main reasons why map projections are needed:

Maps and GIS data are most often displayed on flat surfaces, be it a paper sheet

or a computer screen.

Spatial calculations, such as distance, can be much easier performed within a

two-dimensional coordinate system of a flat surface than a three-dimensional

coordinate system of an oval (ellipsoid) surface.

Map Projections

•Map projections are mathematical formulas used

to transfer shapes from an oval surface (the Earth

surface – more accurately, a model of it) onto a flat

surface.

• (Map) Projection: a systematic presentation of

intersecting coordinate lines on a flat surface upon

which features from the curved surface of the earth

or the celestial sphere may be mapped (Merriam

Webster Dictionary).

y

x

y

x

y

x

• Although GIS features are a model of real-world features existing on an oval surface, their

positional references, i.e., x, y, coordinates, are stored and displayed in a GIS software within a

plane coordinate system.

• The employed plane coordinate system is either projected or geographic.

• Projected coordinate systems are derived from map projections and geographic coordinate

systems from latitude and longitude degrees.

• Coordinate systems are referenced to the physical earth surface through the use of datums.

The Shape of the Earth and the Model Representing It

• The Earth is an irregular body.

• The shape of the Earth is given the form of a surface that is everywhere perpendicular to the

direction of gravity – so called equipotential surface.

• The force and direction of gravity are affected by irregularities in the density of the Earth’s

crust and mantle, therefore the Earth’s form is somewhat irregular.

• The true shape of the Earth is described as a geoid, defined as that equipotential surface that

most closely corresponds to mean sea level.

• The geoid, with its local irregularities, remains a difficult surface for spatial calculations,

which is why a more simple model was chosen to represent the Earth – the ellipsoid.

• The ellipsoid is an ellipse that has been rotated about its shortest (the minor) axis – the reason

why an ellipsoid is used as the model for the Earth is that the Earth is flattened at the poles, not

by much though, 22 km over 6378 km.

6378 km 6356 k

m

Datum

• When creating an ellipsoid model, the goal

is to get a model that best fits the Earth geoid.

• The mechanism through which an ellipsoid

model, i.e., its coordinate system, is related to

the Earth geoid is called geodetic datum, or

just datum.

• A datum is realized by establishing physical

monuments on the ground, which have x,y,z

coordinates assigned to them, forming a

geodetic coordinate reference system.

• Historically, the practice was to develop an

ellipsoid model and a datum that make a good

local fit to the Earth geoid; meaning that this

level of fit would be lost in other areas on the

Earth.

• With the advent of satellites, a global

ellipsoid has been developed (as part of GRS

1980) that has been internationally accepted

as the best fit globally and by the majority of

countries as the model used locally as well.

Local Ellipsoid

Global Ellipsoid

Source: Ilifee and

Lott (2008).

• Positions on the ellipsoid are expressed through a coordinate system.

Ellipsoidal Coordinate

System

Latitude: the angle north or

south from the equatorial plane.

Longitude: the angle east or west

from an identified meridian.

.

Source: Ilifee and

Lott (2008).

Geocentric Cartesian

Coordinate System

• The formulae involved in computations based on ellipsoidal coordinates

are complex and inappropriate when considering observation made to or

from satellites.

• More appropriately, a set of Cartesian axes is defined with its origin

at the centre of the ellipsoid.

Source: Ilifee and

Lott (2008).

• The center of the Geocentric

Cartesian Coordinate System

is at the centre of the model of

the Earth, which may not

coincide with the centre of the

Earth.

http://itrf.ensg.ign.fr/

GIS/index.php

• International Terrestrial Reference System

(http://www.iers.org/IERS/EN/DataProducts/ITRS/itrs.html) is realized through a

geocentric Cartesian coordinate reference set known as the International Terrestrial

Reference Frame.

• It doesn’t require an ellipsoid, the centre of the ITRS is at the mass centre of

gravity of the Earth.

• Because tectonic plates are constantly moving, the physical stations with ITRF

coordinates are constantly moving as well and their coordinates get updated for the

velocity of the plate (of the location on the plate).

• ITRF is used to improve the accuracy of NAD83.

Datums Commonly Used in Resource Management GIS in Canada

WGS84 (World Geodetic System of 1984)

- datum developed and used by GPS.

-WGS84 gets aligned with the ITRF, meaning that at high levels of accuracy it

departs from the datums tied to a particular plate.

- no physical monuments.

NAD83 (North American Datum of 1983)

-developed based on GRS 1980 – the ellipsoid defined through the use of

satellites.

- tied to the North American tectonic plate, meaning that over time it diverges

from WGS84 – currently by ~1.5 - 2 m.

NAD27 (North American Datum of 1927)

- developed based on the Clarke Ellipsoid of 1866.

- discontinued from use but there are still quite a few GIS datasets with it.

- its coordinate difference from NAD83 depends on the location – ~20 m

around Thunder Bay.

To be measured in a different datum, the x,y position on the Earth needs to be

transformed between the two datums. The datum transformation is especially important

between the datums that significantly differ (for particular practical purposes); it can be

very specific and needs to be paid attention to.

Map Projections - Distortions

• “Projection” refers to the notion of shining light through the earth surface and projecting latitudes,

longitudes and geographic features onto a developable surface.

• In the process of projection the earth is represented by a model – an ellipsoid. A datum, then, links the

ellipsoid to the actual surface of the earth and by doing so, links the map projection/coordinate system to the

surface of the earth.

• Because of the transfer from an oval to a flat surface a degree of distortion is inevitable.

• A developable surface is such a surface that can be unravelled without increasing distortion – a cylinder,

cone, plane.

• However, a developable surface is not a necessary intermediary step in designing map projections – some

projections are defined in pure mathematical terms.

Source: Ilifee and Lott

(2008).

Cylindrical

Conic

Plane

• Distortions do not have to happen across all spatial aspects, they can be eliminated or

drastically reduced in one aspect while recognizing their existence in other aspects.

Preserving distances along the meridians

Preserving areas Preserving shape

Source: Ilifee and

Lott (2008).

Map Projection

Category

Maintained Distorted

Equivalent (Equal-area) areal scale angle, shape, distance

Equidistant distance (over some

portions)

direction, area

Azimuthal angular relationships from a

central point

shape, distance, area

Conformal small shapes and because of

that angles as well

large area

Map Projection Categories and Resulting Distortions

• Secant projections, compared to tangent projections, result in increased low and decreased high

distortion.

Conical tangent and

secant projection

Cylindrical tangent and

secant projection

• The points (lines) where the

ellipsoid and the developable

surface are in common

become part of the map

projection’s parameters.

Source: Ilifee and

Lott (2008).

• Projected features are placed within a plane, Cartesian coordinate system grid.

The graticule represents meridians and parallels.

UTM (Universal Transverse Mercator) Projection

• Cylindrical, secant, conformal projection

• UTM projection is derived by positioning the cylinder east-west. “Transverse”

indicates that the cylinder is perpendicular to the one in the standard Mercator

projection, where it is positioned north-south .

• “Universal” points to the fact that the projection is world-wide

• UTM is made of 60 zones, 6 degrees of longitude wide (360/6 = 60), around the

world, positioned north-south.

• Small shapes and

local angles accurate.

• Area minimally

distorted within each

zone.

• 6 degrees of longitude wide ~ 672 km on the Equator, becoming narrower towards north and south.

• x and y coordinates are expressed in meters.

• The central meridian has a false easting of 500000 m to avoid negative values within a zone.

• x coordinates are expressed relative to the central meridian x coordinate; e.g., x = 400,000 in a zone means that the point is 100 km west from the zone’s central meridian, and x = 600,000 means that the point is 100 km east from the zone’s central meridian.

• The equator has a false northing of 10,000,000 m to avoid negative values in the southern hemisphere.

• Y coordinates in the northern hemisphere represent the distance from the Equator; in the southern hemisphere they represent 10,000,000 – distance in meters from the equator.

• Stretches from 84°N to 80°S.

• Scale factor = 1 along secant meridians and 0.99960 along the central meridian.

• Secant longitude 180 km from the central meridian on the Equator.

Secant

meridians

UTM Zone and Its Coordinate System

Canada stretches

across 16 zones, 7

- 22

Ontario stretches

across 4 zones, 15 –

18.

UTM in Canada

Thunder Bay is in UTM zone 16, which has meridian 87° for its central

meridian; the central meridian for UTM zone 15 is at 93° (six degrees

apart), and so on.

The central meridian

for each of the UTM

zones lies on one of

geographic meridians,

six degrees apart from

the central meridians

on each side.

UTM Projection (cont’d)

• UTM is suitable as a map projections for areas whose width is similar to the width of one UTM zone. Because many administrative areas cross UTM zone boundaries but are stored as features in a same dataset, a decision is then made in which UTM zone should the dataset be projected. • Features covering an area that is wider than two UTM zones should be projected in another, more suitable, map projection, such as Canada Lambert Conformal Conic, in Canada.

• MNR used to apply false northing (y shift) of -4 000,000 m to GIS datasets. These

datasets would also have NAD27 for their datum.

Lambert Conformal Conic Projection

• Conformal, often secant projection, with two added parallels.

• Small shapes and local angles are accurate.

• Area minimally distorted near the standard parallels.

• Correct scale along the standard parallels.

Parameters:

• Units

• 1st standard parallel

• 2nd standard parallel

• Central meridian

• Latitude of projection’s

origin

• Suitable for areas or countries extending east-west (Canada, USA), large and medium scales

(widths of two or more UTM zones).

•Extremely widely used – LCC and Transverse Mercator between them account for 90% of base

map projections world wide.

• ESRI’s Canada Lambert Conformal Conic is a customized map projection package that includes

the datum NAD 1983 and is designed with parameters that suit the geographic extent of Canada.

Web Mercator Projection

• Started by Google in 2005 and has become the

standard Web map projection, used in Google Maps,

Bing Maps, OpenStreet Map, Web map services.

• A mathematical formula that is a variant of the

Mercator, cylindrical, projection.

• Ellipsoid coordinates are transferred onto a

developable surface using the formulas of the

spherical Mercator (Stefanakis 2015).

• Advantages: north-up orientation, quicker

computation, easier tiling.

• Disadvantage: large deviations away from the

Equator - ~40 km at 70° latitude (Stefanakis 2015).

• Should be used for visualizations only (correct

spatial calculations on the Web are done on servers

by transferring the coordinates back to the ellipsoid

coordinates).

Mercator projection of the world between 82°S and 82°N

(https://en.wikipedia.org/wiki/Mercator_projection)

Google Maps

Source: (ArcGIS 10 Help 2012)

Meridians (Longitudes) and Parallels (Latitudes)

Parallels – lines of equal

latitude, running parallel to

the equator on the surface

of the ellipsoid.

Meridians – lines of equal

longitude, running pole to

pole on the surface of the

ellipsoid

Geographic Coordinate System

• Not a map projection, it is a Cartesian coordinate system composed of longitudes (meridians) and latitudes

(parallels), and expressed in decimal degrees.

• Longitudes range 180 degrees, start from Greenwich, positive to the east and negative to the west

(International Date Line is on the opposite side of Greenwich).

• Latitudes range 90 degrees, start from the Equator and end on the poles, positive to the north and negative

to the south.

• Decimal degrees (DD) are converted from degrees according to DD = deg + min/60 + sec/3600 .

0 – 180 0 – (-180)

0 –

90

0

– (-9

0)

As a standard point of geographic reference, GCS is often used in GIS datasets stored in broadly

shared databases; however, GCS should never be used for displaying or for spatial calculations.

UTM Zone 16, NAD83 GCS, WGS84

• Contrary to datums, map projection positions (GIS data) can be straightforward

converted (projected) between map projections (and back) without a loss in accuracy.

This assumes that the datum stays the same.

• If the datum stays the same, no specification of transformation is required.

• E.g., a shapefile in UTM, Zone 16, NAD83, can be projected to a new shapefile, in

the Canada Lambert Conformal Conic projection, and back, and features’ x,y

coordinates should stay the same.

• As well, speaking of projecting data, GIS data geographically laying in UTM zone 16,

for example, can be projected to any other UTM zone.

GIS Datasets and Map Projections in ArcGIS

The description of the map projection of a shapefile is stored in its .prj file.

(a) Dataset’s projection

is properly defined.

(b) Dataset’s projection

is undefined.

(c) Dataset’s projection is

improperly defined.

Y = 5363900

X = 333800

thunder_bay.shp

thunder_bay.prj

UTM, Zone 16, NAD83

Y = 5363900

X = 333800

thunder_bay.shp

?

Y = 5363900

X = 333800

thunder_bay.shp

thunder_bay.prj

UTM, Zone 15, NAD83

Fix: - Define the shapefile

with the proper map

projection.

Delete in Win. Expl. the

.prj and the .xml files

and define the shapefile

with the proper map

projection

GIS datasets can come in several different states with respect to map projections:

Important tips in dealing with coordinate systems in GIS:

- A dataset needs to have a defined coordinate system (map projection) in order to be

projected into a different map projection.

- A dataset which does not have a defined map projection (it lacks the projection, .prj,

file) has to have its map projection defined to the map projection in which its features’

coordinates are.

- A very common and grave mistake is to attempt to project a GIS file with an undefined

map projection into a new map projection by assigning through the Define function the

new map projection to the undefined file. A GIS file with an undefined map projection

first needs to be properly defined (with the map projection in which the file`s features

are), and then projected into a new map projection – i.e., into a new file that now has

features in the new map projection.

- If a mistake is made in defining the projection for a shapefile, erase the projection file

(.prj extension) and the metadata file (.xml extension) in the Microsoft file manager.

- X and y values shown in ArcCatalog can be used to get a general idea about the

projection if the definition is missing: y shift issue, UTM versus Lambert, decimal

degrees versus distance units (e.g., metres), etc.

- Try to create a habit of having all datasets with a defined projection.

Sources:

ArcGIS 10 Help. 2012. About Map Projections.

http://help.arcgis.com/en/arcgisdesktop/10.0/help/index.html#//003r000

0000q000000.htm (January 22, 2012)

Ilifee J. and R. Lott. 2008. Datums and Map Projections: For Remote Sensing,

GIS and Surveying. Whittles Publishing.

Furuti C. A. 2011. Cartographical Map Projections.

http://www.progonos.com/furuti/MapProj/Normal/TOC/cartTOC.html

(January 20, 2011)

Stefanakis, E. 2015. Web Mercator: the de facto standard, the controversy, and

the opportunity. Gogeomatics Magazine.

http://www.gogeomatics.ca/magazine/web- mercator-the-de-facto-

standard-the-controversy-and-the-opportunity.htm# .