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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

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Topic 2 spatial representation

Topic 2 – Spatial Representation

A – Location, Shape and Scale

B – Map Projections


Location shape and scale

A

Location, Shape and Scale

  • 1. Spatial Location and Reference

  • 2. The Shape of the Earth

  • 3. Map Scale


Spatial location and reference

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.


Spatial location and reference1

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 coordinates

2

7

7

4

2

4

1

Spatial Location and Reference : Plane Coordinates

Y axis

Y axis

(7,4)

(7,4)

a

(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


Spatial location and reference global systems

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

1

Spatial Location and Reference: Global Systems


Spatial location and reference global systems2

1

Spatial Location and Reference: Global Systems


The shape of the earth

2

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.

B

A

A = B

A / B = 1


The shape of the earth1

2

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.

B

A

A > B

F = A / B = 0.9966099


Reference ellipsoids used in geodesy

2

Reference Ellipsoids used in Geodesy


The shape of the earth2

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.


International geodetic survey geoid 96

2

International Geodetic Survey, Geoid-96


The shape of the earth3

2

The Shape of the Earth

Altitude

Topography

Geoid

Sea Level

Ellipsoid

Sphere


Map scale

3

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.


  • 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


Map scale1

3

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.

10 km


Map projections

B

Map Projections

  • 1. Purpose of Using Projections

  • 2. Cylindrical Projections

  • 3. Conic Projections

  • 4. Azimutal Projections

  • 5. Other Projections


Purpose of using projections

Plane (2 dimensions)

Sphere (3 dimensions)

1

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.

Projection


Purpose of using projections1

1

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.


Cylindrical projections

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 projections1

2

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


Cylindrical projections2

2

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


Cylindrical projections3

2

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.


Mercator projection

2

Mercator Projection


Conical projections

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 projections1

3

Conical Projections

Tangent

Secant


Conical projections2

3

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.


Albers equal area conic

3

Albers Equal Area Conic


Lambert conformal conic

3

Lambert Conformal Conic


Azimuthal projections

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 projections1

4

Azimuthal Projections

Tangent

Secant


Azimuthal projection north pole

4

Azimuthal Projection, North Pole


Robinson projection

5

Robinson Projection


Hammer aitoff projection

5

Hammer Aitoff Projection


Fuller projection

5

Fuller Projection


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