Maa-6.3284 Map Projections

1. Maa-6.3284 Map Projections Lecture I Autumn 2010 Period I Mauri Väisänen

2. Short history the science of map projections was born more than two thousand years ago Greek scientists started to make maps of the Earth and celestial sky with meridians and parallels Anaximander Hipparchus Apollonius Eratosthenes

3. History AD 150 Claudius Ptolemy: GEOGRAPHY descriptions and methods for map design determination of the Earth dimensions map projections

4. History Renaissance: intensive development of cartography epoch of great discoveries development of trade and navigation military maps

5. History Abraham Ortelius and Gerardus Mercator late sixteenth century: ATLAS maps for navigation Mercator projection

6. History during eighteenth century: ellipsoidal shape of the Earth regular topographic surveys of large regions Bonne, Lagrange,Euler, De l'Isle logarithms and differential calculus Gauss: conformal transformations of one surface onto another Tissot: indicatrix

7. About Map Projections The physical body of the Earth is very complicated That is why we need some simplifications to describe or make a two dimensional map from three dimensional world The Earth can be described a sphere in such works, where the accuracy is not (so) important

8. About Map Projections • Also when the scale is small, the Earth can be a sphere • More accurate works, the Earth is described a rotation ellipsoid • There is also a third way to describe an Earth: • Physical body is called a geoid • The surface is difficult to handle

9. About map projections • But usually we want a two dimensional map from the world or from certain area • A map projection is any method of representing the surface of a sphere or an ellipsoid • Map projections are necessary for creating maps

10. How • There is no perfect solution to this 3d->2d problem • It is convenient to define a map projection as a systematic arrangement of intersecting lines on a plane that represent and have a one-to-one correspondence to the meridians and parallels on the datum surface

11. How • who creates a map projection needs to find relation between a sphere (ellipsoid) and plane • differential geometry is needed for creating formulaes • boundaries • Gaussian fundamental quantities give answer to some basic features of creating a map

12. Distortions • All map projections distort the surface in some way • distortion is a false presentation of • angles • shapes • distances • areas

13. Distortion • it depends on the purpose, which distortions are acceptaple and which are not • the same data can be stretched, compressed, twisted and otherwise distorted in different ways

14. Properties of a projection • A map of the Earth is a representation of a curved surface on a plane • Many properties can be measured on the Earth's surface independently of its geography • Some of these properties are: • Area • Shape • Direction • Bearing • Distance • scale

15. Properties of a projection • Map projection can be constructed to preserve one or more of these properties • At the same time not all • Each projection preserves or compromises or approximates basic metric properties in different ways

16. Scale • In the theory of map projections, concepts and formulas for both area and linear scale are considered • The nominal linear scale of a map must be distinguished from the local scale

17. Nominal linear scale • The nominal linear scale shows the general reduction of either the whole ellipsoid (or sphere) or some part of it to represent the surface being mapped • Scale is placed on the map but it is preserved only at some points or along some lines of the map • Changing the nominal scale does not affect the projection that is used

18. Local linear scale • The local linear scale of the reprentation of a given point and along a given direction is the ratio of the length of an infinitesimal segment on the projection to that of the corresponding infinitesimal segment on the surface of the ellipsoid (or sphere) • The ratio of this local linear scale to the nominal linear scale is called the linear scale factor µ: µ =ds'/ds

19. Equidistancy • if the graticule of the equidistant projection is orthogonal, then the base directions coinside with meridians and parallels and these projections are equidistant along meridians or along parallels. • the term equidistant projection usually applies to equidistance along meridians or verticals

20. Equidistancy • Describing a map projection that preserves scale • between certain points • There is no map that can show scale correctly throughout the entire map but some can show true scale between one or more lines • Most projections have one or more lines which • (at map scale) are the same on the earth → true distance, standard lines

21. Equidistant • A map is equidistant when there is a correct representation of a distance between two points on the datum surface and the corresponding points on the projection surface • scale is maintained along the lines connecting a pair of points • It is not possible to preserve all distances • For example central meridian has true distance

22. Equivalency (equality of areas) • Equivalence of areas means that areas of figures represented are retained but the expense of shapes and angles which are in such case deformed

23. Conformality • Strictly speaking only in small areas • When the scale of a map at any point on the map is the same in any direction, the projection is conformal. Meridians (lines of longitude) and parallels (lines of latitude) intersect at right angles. • Shape is preserved locally on conformal maps

24. Developable surface • a surface that can be unfolded or unrolled into a plane or sheet without stretching, tearing or shrinking is called a developable surface → plane, cone and cylinder

25. Aspects of the shape • The aspect describes how the developable surface is placed relative to the globe • it can be normal (also regular, direct,conventional are used) • the surfaces axis of symmetry coincides with the Earth's axis

26. Normal/Direct/Regular aspect

27. Aspects • It can be transverse -it means at right angles of the Earth's axis

28. Aspects • Or it can be oblique

29. Aspects • Projection can be tangential, when it touches plane in one point or • in regular (normal) cylindrical projection it can touch the whole equator or transverse case the whole meridian • it can touch along two lines parallel to central line (normal) • in oblique case it can touch along great circle

30. Regular / Direct / Normalcylinder projection

31. Regular conic projection

32. Regular conic projection • Conic can has one standard parallel or two standard parallels • When there are two standard parallels, distortion is smaller • Parallels are arcs of concentric cirles • Meridians are straight lines • Can be conformal, equal area or arbitrary in distortion • Arbitrary: equidistant along meridians and/or along parallels

33. Transverse cylinder

34. Transverse plane

35. Transverse conic

36. Oblique projection • Points of tangency are neither on poles nor on equator • Cylinder or conic has tangency points along a great circle

37. Examples of projections

38. Lambert's cylindrical equal-area projection

39. Azimuthal Equidistant Projection

40. Transverse Mercator projection

41. UTM projectionhttp://en.wikipedia.org/wiki/File:Utm-zones.jpg

42. The choice of a projection • It cannot be said which projection is the best • It depends on purpose • It depends on scale • It depends on geographical situation on the globe • Think about USA, Russia, Chile, Estonia, Greenland, New Zealand, poles; • It depends on media which can be used

43. The choice of a projection • Large scale mapping conformal projections are needed In Finland we have used Gauss-Krüger projection with one zone (YKJ) or in four zones (KKJ) (zones 0 and 5 not used) but now changing into new system • New system recommendation: it is also possible to take the best meridian between 19-31 degrees from Greenwich (Gauss-Krüger) • For example Lahti has chosen 26°

44. The choice of projection • Also Lambert conformal conic is very often used projection in large scale mapping • (usually mid latitude countries and countries which are situated in west-east direction)

45. Navigation If you navigate, then directions should be mapped right -Mercator -Normal conformal cylinder projection -Meridians are straight lines, parallels at right angles toward them

46. Navigation • -Gnomonic projection • -Arcs of great circles are straight lines • -meridians are straight, in equatorial case parallels but not at equal distance • -parallels concentric circles • -it is useful in aviation • -perspective azimuthal projection

47. Mercator's projection

48. Choice of Projection Geographical situation, shape and size must be taken into account For example Finland is long in north-south direction → Gauss-Krüger is a natural choice Estonia is broad in west-east direction → Lambert conformal conic is then a natural choice in large scale mapping

49. Mapping poles -For mapping polar regions plane projection is valid, for example stereographic projection -Cylindrical projections like Mercator or Gauss-Krüger are not possible

50. More about projections • http://www.colorado.edu/geography/gcraft/notes/mapproj/mapproj.html • http://www.progonos.com/furuti/MapProj/Normal/TOC/cartTOC.html • http://mathworld.wolfram.com/MapProjection.html • http://www.geog.ubc.ca/courses/klink/gis.notes/ncgia/u27.html#SEC27.4.5 • http://en.wikipedia.org/wiki/Map_projections • http://www.geography.wisc.edu/maplib/rob_proj.html