Topic 2 – Spatial Representation

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

2. A Location, Shape and Scale • 1. Spatial Location and Reference • 2. The Shape of the Earth • 3. Map Scale

3. 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.

4. 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.

5. 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

6. 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.

7. 1 Spatial Location and Reference: Global Systems

8. 1 Spatial Location and Reference: Global Systems

9. 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

10. 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

11. 2 Reference Ellipsoids used in Geodesy

12. 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.

13. 2 International Geodetic Survey, Geoid-96

14. 2 The Shape of the Earth Altitude Topography Geoid Sea Level Ellipsoid Sphere

15. 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.

16. 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

17. 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

18. B Map Projections • 1. Purpose of Using Projections • 2. Cylindrical Projections • 3. Conic Projections • 4. Azimutal Projections • 5. Other Projections

19. 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

20. 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.

21. 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.

22. 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

23. 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

24. 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.

25. 2 Mercator Projection

26. 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.

27. 3 Conical Projections Tangent Secant

28. 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.

29. 3 Albers Equal Area Conic

30. 3 Lambert Conformal Conic

31. 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.

32. 4 Azimuthal Projections Tangent Secant

33. 4 Azimuthal Projection, North Pole

34. 5 Robinson Projection

35. 5 Hammer Aitoff Projection

36. 5 Fuller Projection