Homework. In GIS Fundamentals book, Chapter 2, exact formula

Transcription

1 Homework In GIS Fundamentals book, Chapter 2, exact formula

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3 Homework 2) Use the spherical triangles method to calculate the ini7al azimuth from St. Paul to: Reykjavik

4 t r o N e l o hp C a B Reykjavik N, W b= = a=25.87 = C=71.27 = b c Tan(A) = sin(c) sin(b) - cos(b)cos(c) tan(a) N, W A St.Paul sin(71.27) = sin(45.046) - cos(45.056)cos(71.27) tan(25.87) = A = Tan-1 (0.768) = deg

5 Azimuth Complications: Angles gt 90 o Tan(A) = sin(b) tan(a) sin(c) - cos(b)cos(c) St Paul to Buenos Aires A = Tan -1 ( ) GC distance calcs. wellbehaved, but angle tricky, because of inverse tangent function

6 Angle tricky, because of tangent function It repeats every 180 degrees Same tan(), different azimuth tan(f) = tan(f +180) = tan(f - 180) For angles gt 90 degrees, add or subtract 180 (or pi radians) to atan output. How do you know gt 90? From figure, or signs of sin and cos output.

7 Arctan ambiguous, e.g., returns the same value for angles of 30 or 210 degrees, or 140 and 320 degrees.which to choose? From the ATAN function alone, angles in the NE and SW quadrants are ambiguous as are angles in the NW and SE quadrants

8 To Buenos Aires, S latitude, so 90+ latitude b A = Tan -1 ( ) returns a = c S, W

9 Homework Part 3 - calculate Chord dist. vs Great Circle First calculate X Y Z coordinates, then compare to part 1

10 Plane to Ellipsoidal/Geographic Use astronomical observations to set a point Collect angle and distance measures to a new point Calculate the new locations using geodetic formulas (law of signs, great circle distance), reduced to the adopted ellipsoid. Redundantly If I change the ellipsoid, or have better or more measurements, I recalculate the coordinates

11 Plane to Ellipsoidal/Geographic Use astronomical observations to set a point Collect angle and distance measures to a new point Calculate the new locations using geodetic formulas (law of signs, great circle distance), reduced to the adopted ellipsoid. Redundantly If I change the ellipsoid, or have better or more measurements, I recalculate the coordinates

12 Plane to Ellipsoidal/Geographic Use astronomical observations to set a point Collect angle and distance measures to a new point Calculate the new locations using geodetic formulas (law of signs, great circle distance), reduced to the adopted ellipsoid. Redundantly If I change the ellipsoid, or have better or more measurements, I recalculate the coordinates

13 Plane to Ellipsoidal/Geographic Use astronomical observations to set a point Collect angle and distance measures to a new point Calculate the new locations using geodetic formulas (law of signs, great circle distance), reduced to the adopted ellipsoid. Redundantly If I change the ellipsoid, or have better or more measurements, I recalculate the coordinates

14 Plane to Ellipsoidal/Geographic Use astronomical observations to set a point Collect angle and distance measures to a new point Calculate the new locations using geodetic formulas (law of signs, great circle distance), reduced to the adopted ellipsoid. Redundantly If I change the ellipsoid, or have better or more measurements, I recalculate the coordinates

15 Plane to Ellipsoidal/Geographic Use astronomical observations to set a point Collect angle and distance measures to a new point Calculate the new locations using geodetic formulas (law of signs, great circle distance), reduced to the adopted ellipsoid. Redundantly If I change the ellipsoid, or have better or more measurements, I recalculate the coordinates

16 Plane to Ellipsoidal/Geographic Use astronomical observations to set a point Collect angle and distance measures to a new point Calculate the new locations using geodetic formulas (law of signs, great circle distance), reduced to the adopted ellipsoid. Redundantly If I change the ellipsoid or positions, or have better or more measurements, I recalculate the coordinates

17 Geodesy on an Ellipsoid

18 Geodesy on an Ellipsoid This is the most useful surface for estimating horizontal (lat/lon) positions We specify an ellipsoid - the semi-major and semi-minor axes We reduce our set of measurements to the best estimates (in a least squares sense) to their location on the ellipsoid Both the ellipsoid and the points are needed to specify a horizontal datum

19 Preliminaries for Ellipsoidal Calculations a = semi- major axis length b= semi-minor axis length e = first eccentricity =[2f - f 2 ] 1/2 e = second eccentricity = [(a 2 -b 2 )/b 2 ] 1/2 f = flattening = (a-b)/b GRS80 is the internationally recognized, globally best fitting datum

20 Survey a set of Datum Points Locations calculated using ellipsoidal geometry

21 Three Latitudes parametric latitude Geodetic Latitude is our Standard geodetic latitude geocentric latitude

22 p is the diameter of a parallel of latitude How do we calculate p if we know a, f? p = N * cos (f)

23 p is the diameter of a parallel of latitude How do we calculate p if we know a, f? p = N * cos (f) We need to calculate N

24 Remember, f is latitude

25 N is the Prime Vertical - Line containing a vertical line up from the surface, extending to polar axis At a right angle to meridian, and to a parallel at the intersection point

26 Two points on an (Earth) Ellipsoid What are the distance and direction (azimuths) between the points? 2nd Questions: If I leave a point on a given azimuth, and travel a given distance, where will I end up? Wikimedia commons

27 A Geodesic - Shortest Distance Between Points Insert Figure 5.13 here

28 Computations on an Ellipsoid Surface Bowring Equations pg 78, here Remember, e = second eccentricity = [(a 2 -b 2 )/b 2 ] 1/2 where a and b are semi-major and minor axes pg 78, E & F

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31 Two points on an (Earth) Ellipsoid What are the distance and direction (azimuths) between the points? 2nd Questions: If I leave a point on a given azimuth, and travel a given distance, where will I end up? Wikimedia commons

32 These calculations allow us to, from known starting points, survey a set points with latitudes/ longitudes on the ellipsoid surface - Benchmarks Starting points from astronomical measurements Intermediate locations determined from surveys Monumented, and entered in a database

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34 p is the diameter of a parallel of latitude How do we calculate p if we know a, f?

35 Homework - what s up with St. Paul?

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38 The Coordinates for each point depend on the Ellipsoid you choose (a and b), and the measurements for your starting points NAD83(xxxx) Datums are Offset from the WBS84/ITRF Datums, although both sets use the GRS80 Ellipsoid - different starting point/orientation

39 e.g, ArcMap Warning: Which to choose, and what are we doing?

40 We Are Applying a 3-D Cartesian Coordinate Transformation

41 The List of Datum Transformations is Going to Grow: NGS Ten Year Plan Updated Plan Four main goals One enterprise goal Replace NAVD 88 with a GPS/geoid datum Replace NAD 83 with a geocentric GPS based datum Implement New Datums in