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

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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.

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 Reference 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 1 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

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 Systems 1 Spatial Location and Reference: Global Systems

Spatial Location and Reference: Global Systems 1 Spatial Location and Reference: Global Systems

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

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 = 0.9966099

Reference Ellipsoids used in Geodesy 2 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

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

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

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.

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

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

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.

Purpose of Using Projections 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. Sphere (3 dimensions) Projection Plane (2 dimensions)

Purpose of Using Projections 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 Projections 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 Projections 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 Projections 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.

2 Mercator Projection

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.

3 Conical Projections Tangent Secant

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.

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 Projections 4 Azimuthal Projections Tangent Secant

Azimuthal Projection, North Pole 4 Azimuthal Projection, North Pole

5 Robinson Projection

Hammer Aitoff Projection 5 Hammer Aitoff Projection

5 Fuller Projection