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Topic 2 – Spatial RepresentationPowerPoint Presentation

Topic 2 – Spatial Representation

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Location, Shape and Scale

- 1. Spatial Location and Reference
- 2. The Shape of the Earth
- 3. Map Scale

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 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.

7

7

4

2

4

1

Spatial Location and Reference : Plane CoordinatesY 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

- 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 Systems

Spatial Location and Reference: Global Systems

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 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

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

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 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

- 1. Purpose of Using Projections
- 2. Cylindrical Projections
- 3. Conic Projections
- 4. Azimutal Projections
- 5. Other Projections

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 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.

- Equal area:
- 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

- 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

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 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

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

- 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

Lambert Conformal Conic

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 Projection, North Pole

Robinson Projection

Hammer Aitoff Projection

Fuller Projection

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