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 x = -b ±  b2 – 4ac is also often used.
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 b C
A + B + C = 180°
The solution of the quadratic equation ax2 + bx + c = 0
x = -b ±  b2 – 4ac is also often used. 2a

3. THE STRAIGHT LINE Y ∆XAB = XB-XA AND ∆YAB = YB-YA
∆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 P(XP,YP) R R2 = ∆XOP2 + ∆YOP2 O(XO,YO)
P(XP,YP) R
R2 = ∆XOP2 + ∆YOP2
XP2+YP2 – 2XOXP – 2YOYP + f = 0
R =  XO2 + YO2 - f
O(XO,YO) f

5. THE PERPENDICULAR OFFSETY X
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
Given A(XA,YA), B(XB,YB), AzAP and AzBP compute P(XP,YP)
AzAB = atan(∆XAB / ∆YAB ) + C
LAB =  ∆XAB2 + ∆YAB2
a = AzAB – AzAP
β = AzBA – AzBP
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
AzAP β
Outside Orientation
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
AzAB a γ
A(XA,YA)
P(XP,YP)
WARNING: USE A THIRD KNOWN POINT TO CHECK ORIENTATIONS

7. INTERSECTION OF A LINE WITH A CIRCLEY X
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 CIRCLESY X
Given A,B and C and radius R, compute P1 and P2
Note LAP1 = LAP2 = RA and
LBP1 = LBP2 = RB
Compute AzAB and LAB from given
coordinates of A and B
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 a
XP1 = XA + RAsin (AzAP1)
YP1 = YA + RAcos(AzAP1)
and
XP2 = XA + RAsin (AzAP2)
YP2 = YA + RAcos(AzAP2) 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. Global Cartesian Coordinates (x,y,z)
Greenwich
Meridian
Equator •

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 Z b O a X  F1  F2 P 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 Spheroid Rotate an ellipse around an axisZ b O a a Y 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, fm S p 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. Cutting Plane of a Meridian
Equator
plane
Prime 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 °)
-60° P l
-60°
-30°
30°
0°E, W

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

24. Length on Meridians and Parallels
(Lat, Long) = (f, l)
Length on a Meridian:
AB = Re Df
(same for all latitudes) R Dl
30 N R D C Re Df B
0 N Re
Length on a Parallel:
CD = R Dl = Re Dl 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 = /180 = radians
For the meridian, DL = Re Df = 6370 * = 111 km
For the parallel, DL = Re Dl Cos f
= 6370 * * Cos 30
= 96.5 km
Parallels converge as poles are approached

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

27. Geoid and Ellipsoid Gravity Anomaly
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 •
z = 0
Land Surface
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)
Point file attributed with the elevation difference between NGVD 29 and NAVD 88
NGVD 29 terrain + adjustment
= NAVD 88 terrain elevation

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

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
-120°
-90 °
-60 °

51. State Plane Coordinate System
Defined for each State in the United States
East-West States (e.g. Texas) use Lambert Conformal Conic, North-South States (e.g. California) use Transverse Mercator
Texas has five zones (North, North Central, Central, South Central, South) to give accurate representation
Greatest accuracy for local measurements

52. California Mapping System
Designed to give State-wide coverage of Texas without gaps
Lambert Conformal Conic projection with standard parallels 1/6 from the top and 1/6 from bottom of the State
Adapted to Albers equal area projection for working in hydrology

53. Standard Hydrologic Grid (SHG)
Developed by Hydrologic Engineering Center, US Army Corps of Engineers
Uses USGS National Albers Projection Parameters
Used for defining a grid over the US with cells of equal area and correct earth surface area everywhere in the country

54. Coordinate Systems Geographic coordinates (decimal degrees)
Projected coordinates (length units, ft or meters)

55. Summary Concepts
Two basic locational systems: geometric or Cartesian (x, y, z) and geographic or gravitational (f, l, z)
Mean sea level surface or geoid is approximated by an ellipsoid to define an earth datum which gives (f, l) and distance above geoid gives (z)

56. Summary Concepts (Cont.)
To prepare a map, the earth is first reduced to a globe and then projected onto a flat surface
Three basic types of map projections: conic, cylindrical and azimuthal
A particular projection is defined by a datum, a projection type and a set of projection parameters

57. Summary Concepts (Cont.)
Standard coordinate systems use particular projections over zones of the earth’s surface
Types of standard coordinate systems: UTM, State Plane, California State Mapping System, Standard Hydrologic Grid
Reference Frame in ArcInfo 8,& Geomedia requires projection and map extent

58. Cartesian Coordinate System
Satellite Locations
Cartesian Coordinate System
Three dimensional right coordinate system with an origin at the center of the earth and the X axis oriented at at the Prime Meridian and the Z at the North Pole
X Axis Coordinate Distance in meters from the the prime meridian at the origin; positive from 90º E Long to 90º W Long
Y Axis Coordinate Distance in meters from 90º E longitude at the origin; positive in the eastern hemisphere and negative in the western
Z Axis Coordinate Distance in meters from the plane of the equator; positive in the northern Hemisphere negative in the southern Z
(X,Y,Z)
Like the moon, which has reliably spun around the earth for millions of years without any significant change, the GPS satellites are orbiting the earth very predictably as well. The orbits are known in advance and in fact, some GPS receivers have an “almanac” programmed in them to tell where in the sky each satellite will be at any given moment. Each satellite is known where it is in space via Cartesian Coordinates. These known coordiantes are used to triangulate the position of the receiver.
GPS satellites are consistently monitored by the Department of Defense to ensure that each satellite is travelling along its mathematically modeled orbit.
This is the reason that the satellites are put in a non geo-synchronous orbit. This allows each satellite to pass over the monitoring station twice a day. This gives the DoD the chance to precisely measure their altitude, position and speed. Any variations are called ephemeris errors. These are typically minor and are caused from anomolies such as gravitational pulls from the moon and sun, and from pressure of solar radiation on the satellite.Once any ephemeris errors are detected, the correction is relayed to the satellite and the corrected information or ephemeris is then incorporated onto the timing information sent from the satellite.
Y 90°E
X 0º Long
Prime Meridian