You can only use a planar surface so far, before the equidistance assumption creates large errors

Transcription

1 You can only use a planar surface so far, before the equidistance assumption creates large errors

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3 Distance error from Kiester to Warroad is greater than two football fields in length

4 So we assume a spherical Earth

5 P. Wormer, wikimedia commons

6 Longitudes are great circles, latitudes are small circles (except the Equator, which is a GC)

7 Spherical Geometry Longitude

8 Spherical Geometry

9 Note: The Greek letter l (lambda) is almost always used to specify longitude while f, a, w and c, and other symbols are used to specify latitude

10 Great and Small Circles Great circles splits the earth into equal halves Small circles splits the earth into unequal halves wikipedia geography.name

11 All lines of equal longitude are great circles Equator is the only line of equal latitude that s a great circle

12 Surface distance measurements should all be along a great circle Caliper Corporation

13 Note that great circle distances appear curved on projected or flat maps Smithsonian

14 In GIS Fundamentals book, Chapter 2

15 Latitude - angle to a parallel circle, a small circle parallel to the Equatorial great circle

16 Spherical Geometry Three measures: Latitude, Longitude, and Earth Radius + height above/ below the sphere (hp)

17 How do we measure latitude/longitude? Well, now, GNSS, but originally, astronomic measurements: latitude by north star or solar noon angles, at equinox

18 Longitude Measurement Method 1: The Earth rotates 360 degrees in a day, or 15 degrees per hour. If we know the time difference between 2 points, and the longitude of the first point (Greenwich), we can determine the longitude at our current location. Method 2: Create a table of moon-star distances for each day/time of the year at a reference location (Greenwich observatory) Measure the same moon-star distance somewhere else at a standard or known time. The distance will be slightly different, and we can use the difference to calculate longitude

19 Longitude by Hour Angle Clock set to Greenwich time - if accurate enough, calc longitude by time difference

20 1 hr = 15 deg longitude on Earth, and measured time to get angle from Greenwich Meridian to local point along Equator 12 h at sailor s meridian, 15h, 18m 55s at Greenwich

21 1 hr = 15 deg longitude on Earth, and measured time to get angle from Greenwich Meridian to local point along Equator 12 h at sailor s meridian, 15h, 18m 55s at Greenwich 15h 18m 55s - 12h = 3h 18min 55sec in decimal hours, 3+18/ /3600 = h so angle = * 15 = h = 49 deg 0.729*60 min = 49 deg min = 49 deg 43 min 0.75*60 sec = 49 deg 43 min 45 sec

22 Longitude with no clock is more difficult - earliest accurate method use precalculated moonstar distances Given date, and time of night (to/ from midnight) the star/moon distance depends on longitude, and can be precalculated, placed in tables

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25 Now, international services broadcast time signals over radio and other channels, so you can know the exact Greenwich time instantly all over the world.

26 VLBI - Very Long Baseline Interferometry

27 A Combination of systems, but based ultimately on astronomical measurements

28 Distances and Angles on a Sphere Given L0, what is the Azimuth and distant to L1?

29 Typically solve this problem with a spherical triangle and law of sines or cosines, with one corner of the triangle at the nearest pole Note that both angles and distances are measured in spherical units (degrees or radians) and not linear units (e.g., miles or km)

30 Distances and Angles (Azimuths) between points on a Sphere

31 Law of Sines sin(a) sin(b) = - sin(a) sin(b) sin(c) sin(b) = - sin(c) sin(b) sin(a) sin(c) = - sin(a) sin(c)

32 Law of Cosines cos(a) = cos(b)cos(c)+sin(b)sin(c)cos(a) or cyclically, cos(c) = cos(b)cos(a)+sin(b)sin(a)cos(c) Remember, A is angle, a is side

33 What is the great circle distance between St. Paul, MN ( N, W) North Pole C and a St. John s, Labrador ( N, W) We know a, b, and C, so we can use the law of cosines to solve St. Paul, MN b A c B St. John s, Labrador

34 C is difference in longitudes = = b can be calculated from latitude of St Paul = N = a can be calculated from latitude of St John s = N = N, W N, W LOC, cos(c) = cos(b)cos(a)+sin(b)sin(a)cos(c) cos(c)= cos( )cos( )+sin( )sin( )cos( ) c = distance = R * angle = 6,371km * = 3,082.6km

35 Cotangent formulas, derived from LOS, useful for calculating azimuths, distances By definition, angle A is azimuth from St Paul to St John s Tan(A) = sin(b) tan(a) sin(c) - cos(b)cos(c) pgs 37 and 38 of text, similar formula for angle B, can back calculated for Azimuth from B to A

36 Tan(A) = sin(b) tan(a) sin(c) - cos(b)cos(c) C = b = a = Tan(A) = sin( ) sin( ) tan( ) - cos( )cos( ) Tan(A) = A = ArcTan (2.745) = 70.00deg Note some online calculators use an ellipsoidal calculation, so values may differ a bit.and some are just plain wrong!

37 Note, your initial azimuth won t get you to your destination on the shortest path (on great circle) (rhumb line, follows constant azimuth)

38 Due to longitudinal convergence, all fixed azimuth paths that are not on a great circle will spiral to the nearest pole

39 Three-dimensional Earth centered coordinate system

40 Why use 3-D Cartesian? Certain common calculations are easier, and so they ve been adopted as standards by most governments It is easy to convert from spherical coordinates to 3-D coordinates and back (also true for ellipsoidal coordinates, more about those later)

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