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COORDINATE GEOMETRY

COORDINATE GEOMETRY. Except for Geodetic Control Surveys, most surveys are referenced to plane rectangular coordinate systems. Frequently State Plane Coordinate Systems are used. The advantage of referencing surveys to defined coordinate systems are:

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COORDINATE GEOMETRY

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  1. COORDINATE GEOMETRY • Except for Geodetic Control Surveys, most surveys are referenced to plane • rectangular coordinate systems. • Frequently State Plane Coordinate Systems are used. • The advantage of referencing surveys to defined coordinate systems are: • Spatial relations are uniquely defined. • Points can be easily plotted. • Coordinates provide a strong record of absolute positions of physical features and can thus be used to re-construct and physically re-position points that may have been physically destroyed or lost. • Coordinate systems facilitate efficient computations concerning spatial relationships. In many developed countries official coordinate systems are generally defined by a national network of suitably spaced control points to which virtually all surveys and maps are referenced. Such spatial reference networks form an important part of the national infrastructure. They provide a uniform standard for all positioning and mapping activities.

  2. THE TRIANGLE The geometry of triangles is extensively employed in survey calculations. B For any triangle ABC with sides a, b and c: a = b = c (LAW OF SINES) sin A sin B sin C AND a2 = b2 + c2 -2ab cosA b2 = a2 + c2 -2ac cosB (LAW OF COSINES) c2 = a2 + b2 -2ab cosC c a A C b A + B + C = 180° The solution of the quadratic equation ax2 + bx + c = 0 x = -b ±  b2 – 4acis also often used. 2a

  3. THE STRAIGHT LINE Y X 0 0 ∆XAB = XB-XA AND ∆YAB = YB-YA LAB =  ∆XAB2+ ∆YAB2 AzAB = atan(∆XAB / ∆YAB ) + C C=0° for∆XAB >0 and ∆YAB >0 C=180° for∆YAB <0 and C=360° for∆XAB <0 and ∆YAB >0 B(XB,YB) P(XP,YP) AzAB For P on line AB: YP = mXp + b where the slope m = ((∆yAB / ∆xAB ) = cot(AzAB ) AzAB = atan (1/m) + C A(XA,YA) b

  4. THE CIRCLE Y X 0 0 P(XP,YP) R R2 = ∆XOP2+ ∆YOP2 XP2+YP2 – 2XOXP – 2YOYP + f = 0 R =  XO2+ YO2- f O(XO,YO) f

  5. THE PERPENDICULAR OFFSET Y X 0 0 Given known points A,B and P, compute distance PC (LPC) C=0° for∆XAB >0 and ∆YAB >0 C=180° for∆YAB <0 and C=360° for∆XAB <0 and ∆YAB >0 P(XP,YP) B(XB,YB) AzAP = atan(∆XAP / ∆YAP ) + C LAP =  ∆XAP2+ ∆YAP2 AzAB = atan(∆XAB / ∆YAB ) + C LAB =  ∆XAB2+ ∆YAB2 AzAP C a AzAB A(XA,YA) a =AzAB – AzAP LPC = LAB sin LAC = LAB cos a b

  6. THE INTERSECTION Y X 0 0 Given A(XA,YA), B(XB,YB), AzAP and AzBP compute P(XP,YP) β = AzBA – AzBP a = AzAB – AzAP AzAB = atan(∆XAB / ∆YAB ) + C LAB =  ∆XAB2+ ∆YAB2 γ = 180° – a– β LAP= LAB (sin rule) sin β sin γ LAP = LAB (sin β/ sin γ ) XP = XA + LAP sin AzAP YP = YA + LAP cos AzAP C=0° for∆XAB >0 and ∆YAB >0 C=180° for∆YAB <0 and C=360° for∆XAB <0 and ∆YAB >0 B(XB,YB) AzBP β AzAP AzBA Similarly (as a check on the calculations): LBP= LAB (sin rule) sin a sin γ LBP = LAB (sin β/ sin γ ) XP = XA + LBP sin AzBP YP = YA + LBP cos AzBP Outside Orientation AzAB a γ A(XA,YA) P(XP,YP) WARNING: USE A THIRD KNOWN POINT TO CHECK ORIENTATIONS

  7. Y X 0 0 INTERSECTION OF A LINE WITH A CIRCLE Given A,B and C and radius R, compute P1 and P2 Note a = AzAC – AzAB and LBP1 = LBP2 = R Apply the cos rule to triangle ABP1: LBP12 = LAB2+ LAP12-2(LAB)(LAP1)cos a or LAP12 –(2LABcos a)LAP1+ (LAB2 - LBP12) = 0 which is a quadratic equation with LAP1 as unknown C(XC,YC) B(XB,YB) R Note that since LBP1 = LBP2 = R the above equation also applies to triangle ABP2. Hence the two solutions of the quadratic equation are AP1 and AP2. P2(XP2,YP2) a P1(XP1,YP1) AP=2LABcos a ± (2LABcos a)2-4(LAB2 - LBP12) 2 A(XA,YA) Now use AzAC and the solutions of AP to compute P1(XP1,YP1) and P2(XP2,YP2)

  8. INTERSECTION OF TWO CIRCLES Y X 0 0 Given A,B and C and radius R, compute P1 and P2 Compute AzAB and LAB from given coordinates of A and B Note LAP1 = LAP2 = RA and LBP1 = LBP2 = RB Apply the cos rule to triangle ABP1: a = acos ((LAB2+ LAP12-LBP12)/(2LABLAP1)) = acos ((LAB2+ RA2-RB2)/(2LABRA)) B(XB,YB) AzAP1 = AzAB – a and AzAP2 = AzAB + a P1(XP1,YP1) RB P2(XP2,YP2) a XP1 = XA + RAsin (AzAP1) YP1 = YA + RAcos(AzAP1) and XP2 = XA + RAsin (AzAP2) YP2 = YA + RAcos(AzAP2) a RA A(XA,YA) WARNING Trilatertion of a point with only two distances yields two positions!!!!!

  9. HOMEWORK: LECTURE 13 (CHAPTER 11 SEC 1-6) 11.1, 11.9, 11.13, 11.15, 11.17

  10. Types of Coordinate Systems • (1) Global Cartesian coordinates (x,y,z) for the whole earth • (2) Geographic coordinates (f, l, z) • (3) Projected coordinates (x, y, z) on a local area of the earth’s surface • The z-coordinate in (1) and (3) is defined geometrically; in (2) the z-coordinate is defined gravitationally

  11. Z Greenwich Meridian O • Y X Equator Global Cartesian Coordinates (x,y,z)

  12. Global Positioning System (GPS) • 24 satellites in orbit around the earth • Each satellite is continuously radiating a signal at speed of light, c • GPS receiver measures time lapse, Dt, since signal left the satellite, Dr = cDt • Position obtained by intersection of radial distances, Dr, from each satellite • Differential correction improves accuracy

  13. Global Positioning using Satellites Dr2 Dr3 Number of Satellites 1 2 3 4 Object Defined Sphere Circle Two Points Single Point Dr4 Dr1

  14. Geographic Coordinates (f, l, z) • Latitude (f) and Longitude (l) defined using an ellipsoid, an ellipse rotated about an axis • Elevation (z) defined using geoid, a surface of constant gravitational potential • Earth datums define standard values of the ellipsoid and geoid

  15. Shape of the Earth It is actually a spheroid, slightly larger in radius at the equator than at the poles We think of the earth as a sphere

  16. Ellipse An ellipse is defined by: Focal length =  Distance (F1, P, F2) is constant for all points on ellipse When  = 0, ellipse = circle Z b O a X   F1 F2 For the earth: Major axis, a = 6378 km Minor axis, b = 6357 km Flattening ratio, f = (a-b)/a ~ 1/300 P

  17. Ellipsoid or SpheroidRotate an ellipse around an axis Z b a O Y a X Rotational axis

  18. Standard Ellipsoids Ref: Snyder, Map Projections, A working manual, USGS Professional Paper 1395, p.12

  19. Horizontal Earth Datums • An earth datum is defined by an ellipse and an axis of rotation • NAD27 (North American Datum of 1927) uses the Clarke (1866) ellipsoid on a non geocentric axis of rotation • NAD83 (NAD,1983) uses the GRS80 ellipsoid on a geocentric axis of rotation • WGS84 (World Geodetic System of 1984) uses GRS80, almost the same as NAD83

  20. Definition of Latitude, f m p S n O f q r (1) Take a point S on the surface of the ellipsoid and define there the tangent plane, mn (2) Define the line pq through S and normal to the tangent plane (3) Angle pqr which this line makes with the equatorial plane is the latitude f, of point S

  21. P Prime Meridian Equator Meridian plane Cutting Plane of a Meridian

  22. Definition of Longitude, l l = the angle between a cutting plane on the prime meridian and the cutting plane on the meridian through the point, P 180°E, W -150° 150° -120° 120° 90°W (-90 °) 90°E (+90 °) P -60° l -60° -30° 30° 0°E, W

  23. =0-180°W =0-90°S Latitude and Longitude on a Sphere Meridian of longitude Z Greenwich meridian N Parallel of latitude =0° P • =0-90°N  - Geographic longitude  - Geographic latitude  E W O • Y R  R - Mean earth radius • =0° Equator  • O - Geocenter =0-180°E X

  24. Length on Meridians and Parallels (Lat, Long) = (f, l) Length on a Meridian: AB = ReDf (same for all latitudes) R Dl D R 30 N C B Re Df 0 N Re Length on a Parallel: CD = R Dl = ReDl Cos f (varies with latitude) A

  25. Example: What is the length of a 1º increment along on a meridian and on a parallel at 30N, 90W? Radius of the earth = 6370 km. Solution: • A 1º angle has first to be converted to radians p radians = 180 º, so 1º = p/180 = 3.1416/180 = 0.0175 radians • For the meridian, DL = ReDf = 6370 * 0.0175 = 111 km • For the parallel, DL = ReDl Cos f • = 6370 * 0.0175 * Cos 30 • = 96.5 km • Parallels converge as poles are approached

  26. Sea surface Ellipsoid Earth surface Geoid Representations of the Earth Mean Sea Level is a surface of constant gravitational potential called the Geoid

  27. Geoid and Ellipsoid Earth surface Ellipsoid Ocean Geoid Gravity Anomaly Gravity anomaly is the elevation difference between a standard shape of the earth (ellipsoid) and a surface of constant gravitational potential (geoid)

  28. Definition of Elevation Elevation Z P z = zp • Land Surface z = 0 Mean Sea level = Geoid Elevation is measured from the Geoid

  29. Vertical Earth Datums • A vertical datum defines elevation, z • NGVD29 (National Geodetic Vertical Datum of 1929) • NAVD88 (North American Vertical Datum of 1988) • takes into account a map of gravity anomalies between the ellipsoid and the geoid

  30. Converting Vertical Datums • Corps program Corpscon (not in ArcInfo) • http://crunch.tec.army.mil/software/corpscon/corpscon.html Point file attributed with the elevation difference between NGVD 29 and NAVD 88 NGVD 29 terrain + adjustment = NAVD 88 terrain elevation

  31. http://www.csr.utexas.edu/ocean/mss.html

  32. http://www.csr.utexas.edu/ocean/egs04.html

  33. Geodesy and Map Projections • Geodesy - the shape of the earth and definition of earth datums • Map Projection - the transformation of a curved earth to a flat map • Coordinate systems - (x,y) coordinate systems for map data

  34. Representative Fraction Globe distanceEarth distance = Earth to Globe to Map Map Projection: Map Scale: Scale Factor Map distanceGlobe distance = (e.g. 0.9996) (e.g. 1:24,000)

  35. Geographic and Projected Coordinates (f, l) (x, y) Map Projection

  36. Projection onto a Flat Surface

  37. Types of Projections • Conic (Albers Equal Area, Lambert Conformal Conic) - good for East-West land areas • Cylindrical (Transverse Mercator) - good for North-South land areas • Azimuthal (Lambert Azimuthal Equal Area) - good for global views

  38. Conic Projections(Albers, Lambert)

  39. Cylindrical Projections(Mercator) Transverse Oblique

  40. Azimuthal (Lambert)

  41. Albers Equal Area Conic Projection

  42. Lambert Conformal Conic Projection

  43. Universal Transverse Mercator Projection

  44. Lambert Azimuthal Equal Area Projection

  45. Projections Preserve Some Earth Properties • Area - correct earth surface area (Albers Equal Area) important for mass balances • Shape - local angles are shown correctly (Lambert Conformal Conic) • Direction - all directions are shown correctly relative to the center (Lambert Azimuthal Equal Area) • Distance - preserved along particular lines • Some projections preserve two properties

  46. Geodesy and Map Projections • Geodesy - the shape of the earth and definition of earth datums • Map Projection - the transformation of a curved earth to a flat map • Coordinate systems - (x,y) coordinate systems for map data

  47. Coordinate Systems • Universal Transverse Mercator (UTM) - a global system developed by the US Military Services • State Plane Coordinate System - civilian system for defining legal boundaries • California State Mapping System - a statewide coordinate system for California

  48. Coordinate System A planar coordinate system is defined by a pair of orthogonal (x,y) axes drawn through an origin Y X Origin (xo,yo) (fo,lo)

  49. Universal Transverse Mercator • Uses the Transverse Mercator projection • Each zone has a Central Meridian(lo), zones are 6° wide, and go from pole to pole • 60 zones cover the earth from East to West • Reference Latitude (fo), is the equator • (Xshift, Yshift) = (xo,yo) = (500000, 0) in the Northern Hemisphere, units are meters

  50. UTM Zone 21 &22 -123° -102° -96° 6° Origin Equator -90 ° -120° -60 °

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