1. Topic 2 – Spatial Representation
A – Location, Shape and Scale
B – Map Projections

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Citation
Dr. Jean-Paul Rodrigue, Dept. of Economics & Geography, Hofstra University.

3. Location, Shape and Scale
1. Spatial Location and Reference
2. The Shape of the Earth
3. Map Scale
Adapted from: Dana, P.H. (1995) Map Projection Overview, The Geographer’s Craft Project, Department of
Geography, The University of Texas at Austin.

4. Spatial Location and Reference1
Spatial Location and Reference
Precise location is very important
Provide a referencing system for spatial objects.
Distance.
Relative location.
Navigation.
Ownership.
Coordinate systems
Provide a set of coordinates identifying the location of each objects relatively to others or to an origin.
Many basic coordinate systems.
Represent points in 2-D or 3-D space.
A map cannot be produced without some implicit spatial location and referencing system.

5. Spatial Location and Reference1
Spatial Location and Reference
Cartesian system
René Descartes ( ) introduced systems of coordinates based on orthogonal (right angle) coordinates.
The origin is where the values of X and Y are equal to 0.
By tradition, the value of X is called an easting, because it measures distances east of the origin.
The value of Y is called a northing, because it measures distances north of the origin.
A computer represents vector graphics as a Cartesian system.
The earth’s surface in a GIS is “projected” in a Cartesian system.

6. Spatial Location and Reference : Plane Coordinates1
Spatial Location and Reference : Plane Coordinates
Y axis
Y axis 7 2 7
(7,4)
(7,4) a 4 4
(2,2) 2 b
X axis
X axis
Distance (a, b)= √((X2-X1)2+(Y2-Y1)2)
Distance (a, b)= √ ((2-7)2+(2-4)2)
Distance (a, b)= √ ((-5)2+(-2)2)
Distance (a, b)= √ (25+4) = 5.38

7. Spatial Location and Reference: Global Systems1
Spatial Location and Reference: Global Systems
Longitude / Latitude
Most commonly used coordinate system.
The equator and the prime meridian (Greenwich) are the reference planes for this system.
Latitude of a point:
Angle from the equatorial plane to the vertical direction of a line.
90 degrees north and 90 degrees south.
Tropic of Cancer: summer solstice = 23.5 N
Tropic of Capricorn: winter solstice = 23.5 S
Longitude of a point:
Angle between the reference plane and a plane passing through the point.
180 degrees east of Greenwich and 180 degrees west.
Both planes are perpendicular to the equatorial plane.

8. Spatial Location and Reference: Global Systems1
Spatial Location and Reference: Global Systems

9. Spatial Location and Reference: Global Systems1
Spatial Location and Reference: Global Systems

10. 2 The Shape of the Earth Datum B Possible representations A Sphere
Base elevation model for mapping.
Representation of the earth’s surface.
Using a set of control points.
Possible representations
Sphere.
Ellipse.
Geoid.
Sphere
Simplistic representation.
Assumes the same length of both its axis. B A
A = B
A / B = 1

11. 2 The Shape of the Earth Ellipse B A A > B F = A / B = 0.9966099
Assumes different lengths for each axis.
More appropriate since the earth is flatter at its poles due to its rotation speed.
Polar circumference: 39,939,593.9 meters.
Equatorial circumference: 40,075,452.7 meters.
Flattening index. B A
Source: Clarke, 2001.
A > B
F = A / B =

12. Reference Ellipsoids used in Geodesy2
Reference Ellipsoids used in Geodesy
Name of ellipsoid
Earth’s axis (m)
Datum
Geodetic Reference System 1980 (GRS80)
6,378,137
World Geodetic System 1984
World Geodetic System 1972 (WGS72)
6,378,135
World Geodetic System 1972
Geodetic Reference System 1967
6,378,160
Australian Datum 1966
South American Datum 1969
Krassovski (1942)
6,378,245
Pulkovo Datum 1942
International (Hayford 1924)
6,378,388
European Datum 1950
Clark (1866)
6,378,206
North American Datum 1927
Bessel (1841)
6,377,397
German DHDN

13. 2 The Shape of the Earth Geoid
Figure that adjusts the best ellipsoid and the variation of gravity locally.
Computationally very complex.
Most accurate, and is used more in geodesy than for GIS and cartography.

14. International Geodetic Survey, Geoid-962
International Geodetic Survey, Geoid-96

15. 2 The Shape of the Earth Altitude Topography Geoid Sea Level Ellipsoid
Sphere

16. 3 Map Scale Maps are reductions of the reality Scale
How much a reduction we need?
Proportional to the level of detail:
Low reduction - Lots of details.
High reduction - Limited details.
Scale
Refers to the amount of reduction on a map.
Ratio of the distance on the map as compared to the distance on the real world.
Knowing the scale enables to understand what is the spatial extent of a map.

17. Generalization Abstraction Displacement Simplification
Giving away details and accuracy to fit elements on a map.
Abstraction
Real world objects displayed differently as they are (e.g. a city as a point).
Displacement
The location of an object may be moved to fit on a map.
The object may be enlarged.
Simplification

18. 3 Map Scale Equivalence Scale Representational Fraction Graphic Scale
Difference of representational units.
“one centimeter equals 1,000 meters”
“one millimeter equals 5 kilometers”
Representational Fraction
The map and the ground units are the same.
Reduces confusion.
1:65,000 means that one centimeter equals 65,000 centimeters, or that one meter equals 65,000 meters.
Graphic Scale
Measured distances appear directly on the map.
10 km

19. B Map Projections 1. Purpose of Using Projections
2. Cylindrical Projections
3. Conic Projections
4. Azimutal Projections
5. Other Projections
Adapted from: Dana, P.H. (1995) Map Projection Overview, The Geographer’s Craft Project, Department of
Geography, The University of Texas at Austin.

20. Purpose of Using Projections1
Purpose of Using Projections
Purpose
Represent the earth, or a portion of earth, on a flat surface (map or computer screen).
Geometric incompatibility between a sphere (3D) and a plane (2D).
The sphere must be “projected” on the plane.
A projection cannot be done without some distortions.
Sphere (3 dimensions)
Projection
Plane (2 dimensions)

21. Purpose of Using Projections1
Purpose of Using Projections
Conformal
Preserve shape (angular conformity).
The scale of the map is the same in any direction.
Meridians (lines of longitude) and parallels (lines of latitude) intersect at right angles.
Equivalent
Equal area:
Preserves area.
Areas on the map have the same proportional relationships to the areas on the Earth (equal-area map).
Equidistant:
Preserves distance.
Compromise
No flat map can be both equivalent and conformal.
Most fall between the two as compromises.
To compare maps in a GIS, both maps MUST be in the same projection.

22. Cylindrical Projections2
Cylindrical Projections
Definition
Projection of a spherical surface onto a cylinder
Straight meridians and parallels.
Meridians are equally spaced, the parallels unequally spaced.
Normal, transverse, and oblique cylindrical equal-area projections.
Scale is true along the central line.
Shape and scale distortions increase near points 90 degrees from the central line.

23. Cylindrical Projections2
Cylindrical Projections
Tangent
Cylinder is tangent to the sphere contact is along a great circle.
Circle formed on the surface of the Earth by a plane passing through the center of the Earth.
Secant
Cylinder touches the sphere along two lines.
Both small circles.
Circle formed on the surface of the Earth by a plane not passing through the center of the Earth.
Tangent
Secant

24. Cylindrical Projections2
Cylindrical Projections
Transverse
When the cylinder upon which the sphere is projected is at right angles to the poles.
Oblique
When the cylinder is at some other, non-orthogonal, angle with respect to the poles.
Transverse

25. Cylindrical Projections2
Cylindrical Projections
Mercator projection
Mercator Map was developed in 1569 by cartographer Gerhard Kremer.
It has since been used successfully by sailors to navigate the globe since and is an appropriate map for this purpose.
Straight meridians and parallels that intersect at right angles.
Scale is true at the equator or at two standard parallels equidistant from the equator.
Often used for marine navigation because all straight lines on the map are lines of constant azimuth.

26. 2
Mercator Projection

27. 3 Conical Projections Definition
Result from projecting a spherical surface onto a cone.
When the cone is tangent to the sphere contact is along a small circle.
In the secant case, the cone touches the sphere along two lines, one a great circle, the other a small circle.
Good for continental representations.

28. 3
Conical Projections
Tangent
Secant

29. 3 Conical Projections Albers Equal Area Conic Lambert Conformal Conic
Distorts scale and distance except along standard parallels.
Areas are proportional.
Directions are true in limited areas.
Used in the United States and other large countries with a larger east-west than north-south extent.
Lambert Conformal Conic
Area, and shape are distorted away from standard parallels.
Used for maps of North America.

30. Albers Equal Area Conic3
Albers Equal Area Conic

31. Lambert Conformal Conic3
Lambert Conformal Conic

32. Azimuthal Projections4
Azimuthal Projections
Definition
Result from projecting a spherical surface onto a plane.
Tangent
Contact is at a single point on the surface of the Earth.
Secant case
Plane touches the sphere along a small circle.
Center of the earth, when it will touch along a great circle.

33. Azimuthal Projections4
Azimuthal Projections
Tangent
Secant

34. Azimuthal Projection, North Pole4
Azimuthal Projection, North Pole

35. 5
Robinson Projection

36. Hammer Aitoff Projection5
Hammer Aitoff Projection

37. 5
Fuller Projection