Topic 2 – Spatial Representation
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GEOG 60 – Introduction to Geographic Information System Professor: Dr. Jean-Paul Rodrigue Topic 2 – Spatial Representation A – Location, Shape and Scale B – Map Projections
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Topic 2 – Spatial Representation. A – Location, Shape and Scale B – Map Projections. A. Location, Shape and Scale. 1. Spatial Location and Reference 2. The Shape of the Earth 3. Map Scale. 1. Spatial Location and Reference. Precise location is very important - PowerPoint PPT Presentation
Transcript of Topic 2 – Spatial Representation
GEOG 60 – Introduction to Geographic Information SystemsProfessor: Dr. Jean-Paul Rodrigue
Topic 2 – Spatial Representation
A – Location, Shape and ScaleB – Map Projections
Location, Shape and Scale
■ 1. Spatial Location and Reference■ 2. The Shape of the Earth■ 3. Map Scale
A
1 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.
1 Spatial Location and Reference
■ Cartesian system• René Descartes (1596-1650) 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.
Spatial Location and Reference : Plane Coordinates1
X axis
Y axis7
4(7,4)
X axis
Y axis7
4(7,4)
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2
(2,2)
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
a
b
1 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.
Spatial Location and Reference: Global Systems1
Spatial Location and Reference: Global Systems1
The Shape of the Earth
■ Datum• 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.
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A
B
A = BA / B = 1
The Shape of the Earth
■ Ellipse• 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.
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AB
A > BF = A / B = 0.9966099
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 1966South 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
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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.
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International Geodetic Survey, Geoid-962
The Shape of the Earth2
Sphere
EllipsoidGeoid
Topography
Sea Level
Altitude
Map Scale
■ Maps are reductions of the reality• 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.
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■ Generalization• 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
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10 km
Map Scale
■ Equivalence 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.
Map Projections
■ 1. Purpose of Using Projections■ 2. Cylindrical Projections■ 3. Conic Projections■ 4. Azimutal Projections■ 5. Other Projections
B
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.
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Projection
Sphere (3 dimensions) Plane (2 dimensions)
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.
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2 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.
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
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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
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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.
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Mercator Projection
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Equator
Arctic Circle
Antarctic Circle
Tropic of Cancer
Prim
e Meridian
Tropic of Capricorn
International D
ate L
ine
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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.
Conical Projections
Tangent Secant
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Conical Projections
■ Albers Equal Area 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.• Directions are true in limited areas.• Used for maps of North America.
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Albers Equal Area Conic
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Equa
torA
ntar
ctic
Circ
le Tropic of Cap
ricorn
Trop
ic o
f Can
cer
Prime Meridian
Arc
tic C
ircle
International D
ate L ine
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Lambert Conformal Conic
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Equator
Trop
ic o
f Cap
ri cor
n
Prime Meridian
Trop
ic o
f Can
c er
Arct
ic C
ircle
Internatio
nal Date L ine
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4 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.
Azimuthal Projections
Tangent Secant
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Azimuthal Projection, North Pole
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Equator
Tropic of Cap
r icorn
Tropic of Cancer
Prim
e Meri dian
Arctic Circ le
Internationa l D
ate L
ine
Trop
ic o
f Cap
ricor
n
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Robinson Projection5
Hammer Aitoff Projection5
Fuller Projection5
