Map Projections. Chapter 3. Emmanuel Stefanakis

Transcription

1 Stefanakis, E., Geographic Databases and Information Systems. CreateSpace Independent Publ. [In English], pp.386. Get a copy from Amazon Chapter 3 Map Projections Emmanuel Stefanakis

2 Map Projections A function f φ λ (Χ,Υ) ƒ(φ, λ) Y X 2

3 Errors in Maps The Earth (Mercator Projection) In reality Africa is 14 times China is 4 times larger than Greenland 3

4 Errors in Maps The Earth (Mercator Projection) In reality countries in the South are double in area than countries in the North 4

5 Which is the right shape? 5

6 Which is the right shape? 6

7 From the Sphere to the Plane Earth is almost spherical How? The map is planar 7

8 From the Sphere to the Plane Earth is almost spherical How? It is not possible to make the sphere planar! The map is planar 8

9 From the Sphere to the Plane If the earth was a cylinder 9

10 From the Sphere to the Plane If the earth was a cylinder How? It is possible to make the cylinder planar! 10

11 From the Sphere to the Plane How about the sphere? 11

12 From the Sphere to the Plane Use an intermediate surface (projective surface) 12

13 Projection Surfaces Tangent cases 13

14 Projection Surfaces 14

15 A Simulation of the process 15

16 A Cylindrical Projection 16

17 An Azimuthal Projection 17

18 A Conical Projection 18

19 The distribution of Errors 19

20 Distortions a circle on sphere equal area equal shape equal distance the projected image 20

21 Another configuration Secant cases 21

22 Another configuration 22

23 Another configuration orthogonal transversal oblique 23

24 Select the right projection 24

25 Select the right projection What about France Canada Turkey Chile 25

26 Map Projections The notion of scale 1:K 26

27 Common Projections Gnomonic Projection meridians parallels R tan(90 - φ ) parallel R 90 - φ φ projection center 27

28 Common Projections Gnomonic Projection meridians parallels parallel parallel radius Scale distortions along the meridians latitude Scale factor on meridian projection center Scale distortions along the parallels latitude 75 Scale factor on parallel

29 Common Projections Stereographic Projection meridians parallels parallel equator projection center parallel radius Shifts the projection center to the opposite pole (further) to minimize the distortions. 29

30 Common Projections Stereographic Projection meridians parallels parallel radius 2R tan[(90 - φ )/2] parallel parallel R 90 - φ φ equator [90 - φ]/2 projection center projection center 30

31 Common Projections Gnomonic (vs) Stereographic Projection R tan(90 - φ ) 2R tan[(90 - φ )/2] parallel R 90 - φ φ parallel R 90 - φ φ projection center [90 - φ]/2 projection center 31

32 Common Projections Gnomonic Projection Great Circles Straight Lines (!) projection of the great circle projection plane (map) great circle projection center 32

33 Common Projections Gnomonic Projection Great Circles Straight Lines (!)» Shortest path Florida Bristol 33

34 Common Projections Conical Projections r r 2π - α meridian parallel 34

35 Common Projections Conical Projections α = 2π sinφ r = R cotφ φ r 2π-α R φ 35

36 Common Projections Equidistant Conical Projections r s north pole radius s s North pole meridian parallel 36

37 Common Projections Equidistant Conical Projections 37

38 Common Projections Cylindrical Projections North pole equator South pole 38

39 Common Projections Mercator Projection 39

40 Common Projections Mercator Projection Distortion of Areas in Mercator Projection for a small area in latitude factor of area distortion

41 Cylindrical Projection 41

42 Cylindrical Projection 2πRcos(φ) < 2πR R cos(φ) φ 2πR φ R λ 42

43 Cylindrical Projection 2πRcos(φ) < 2πR 2πR φ 2πR 2πR λ Y X 43

44 Common Projections Mercator Projection 2πR 2πR 44

45 Mercator s goal: preserve shapes Scale factor on horizontal (along parallels at φ): 2πRcos(φ) 2πR 45

46 Mercator s goal: preserve shapes Scale factor on horizontal (along parallels at φ): 2πRcos(φ) 2πR Scale factor on vertical (along meridians at φ): 2πRcos(φ) 2πR 46

47 Common Projections Mercator Projection Preserves the directions All parallels have the same length = equator (2πR) At latitude φ the scale factor on the parallel is equal to: 2π R 2π Rcosφ secφ 47

48 Common Projections Mercator Projection Preserves the directions (shapes) Mercator applied the same factor on the meridians The actual distance between two parallels at latitude φ is equal to: dy = R dφ He made it equal to: dy = R secφ dφ 48

49 Common Projections Mercator Projection Preserves the directions (shapes) By integrating the last formula we get: φ o φ Y R secφ dφ Rlntan North pole Υ φ equator South pole 2π R 49

50 Loxodrome ΒΠ N Α θ θ θ θ θ Β Κ ισημερινός equator S

51 Common Projections Mercator Projection The loxodromes are straight lines(!) The shortest paths are curves though 51

52 Stefanakis, E., Geographic Databases and Information Systems. CreateSpace Independent Publ. [In English], pp.386. Get a copy from Amazon Chapter 3 Map Projections Emmanuel Stefanakis