PROJECTIONS AND COORDINATE SYSTEMS

Apostol Panayotov, PhD

February 2014

GEOSPATIAL WOLRD

COORDINATES AND

PROJECTIONS

COORDINATES

• Description of the object position in accordance with predefined starting origin, in specified units and based on reference plane (DATUM)

COORDINATE TYPES: • Polar

– Radius vector and vectorial angle

• Rectangular (Cartesian)

– 2D Cartesian (X,Y) – 3D Cartesian (X,Y,Z)

• Angular (Latitude, Longitude)

– Geodetic ( ,) – Geocentric (,) (’,) – Astronomic (,)

ECEF: X= 1,109,928m Y= 4,860,097m Z= 3,965,162m GEORGRAPHIC: φ = 39 º 52’ 12.90” λ= - 104º 37’ 15.62” GEODETIC: Lat = φ = 39 º 52’ 12.90” Lon=λ= - 104º 37’ 15.62” h= 1569.2m GEOREF: GJNJ5141 USNG: 13SEE3241113425 UTM: E = 532411.934m N= 4413425.399m El. = 1598.21 SPCS: E= 989605.103m N=531287.744m El. = 1598.21 MGRS: 18S UH 1425 8406 (New) ;18S UT 1421 8385 (Old) LOCAL: X = 85210.25, Y = 46852.12

WHY WE NEED COORDINATE SYSTEM?

• To properly and uniquely describe the location and position of the object in 2 or 3 dimensional space

– Time??

• To describe the location/position of the natural and man-made environment and as a base for measurements and analyses

• To perform positional, directional, angular and other necessary calculations

• Used in navigation, cartography, geodesy, surveying, construction, design….and every spatially related art and science

COORDINATE CHARECTERISTICS

• Based on strong mathematical principles

• Described by equations

• Must have measurement unit

• Must be based on reference surface

• Always right until it touches the real surface… then the fun begins in terms of problems……

PROBLEMS:

• What is real is actually not true - what is true actually is not real

– Linear, Angular and Area Distortions

– Positional inaccuracy

– Directional misleading

– Measurement problems

TYPES OF COORDINATE SYSTEMS • GLOBAL

– ECEF

– GEOGRPAHIC

– GEODETIC

– ASTRONOMIC

• LOCAL

– PROJECTED

– LDP

– LOCAL SURVEYING

• 3D

– ANGULAR (Spherical, Ellipsoidal)

– LINEAR (Cartesian)

• 2D

– ANGULAR (Polar)

– LINEAR (Cartesian)

LAT/LONG (THERE IS NO SUCH COORDINATE SYSTEM)

Represent a position of point on the Earth’s surface in degrees

Usually measured in Degrees Minutes Seconds (DMS) or Decimal Degrees (DD) but can be expressed as Grad/Gon or Radians

The lines of Longitude and Latitude always cross each other at the right angle

Latitude () is measured North or South from the reference Equatorial plane

Longitude ( ) is measured East or West from a reference meridian (Greenwich)

The distance of 1 Latitude and Longitude are almost the same at the Equator but they become different as proceed closer to the North or South pole

It is due to the convergence of the meridians as they intersect each other at the poles

TYPES OF LATITUDE AND LONGITUDE

• Geographic Coordinates

– Geocentric

– Geodetic

– Astronomic

• Geocentric and Geodetic Longitude are the same as they start from the same Prime Meridian (Greenwich)

• Geocentric and Geodetic Latitude differ because of the ellipsoidal nature of the Earth

• The latitudes are measured from the Equator but reference different planes: – Geodetic is referenced to the perpendicular to the Ellipsoid

– Geocentric is referenced to the line passing trough the Geocenter (Earth mass center)

– Astronomic is referenced to the direction of Gravity (true vertical at the point)

3D CARTESIAN COORDINATES

• Based on Geocentric Datum

• The center of the coordinate System is Earth mass center

• Earth Centered Earth Fixed (ECEF)

• X, Y, Z (m)

• X axis is defined by Zero meridian which is international reference meridian (IRM) Greenwich

• Y axis is defined by 90° East of the reference meridian

• Z axis is defined by the position of North pole at epoch 1984.0

• The problem with Z axis is that the North pole is moving as well as the entire Earth is rotating and wobbling

2D CARTESIAN COORDINATES

• Rene Descartes – French mathematician and philosopher

• Presenting a real world in 2D Cartesian coordinates always introduce distortions

• Distortions across small area are negligible

• Pairs of coordinates (X, Y)

• There is no need or theoretical requirements to have X,Y to be perpendicular to each other

• Axis's perpendicularity is accepted as the rules for Euclidean geometry can be applied and used: – Easy to use

– Easy to calculate

• 2D Cartesian coordinates are used in many cases • State Plane Coordinate System

• Local, Surveying or Project system for small areas

PROJECTION

• Map projection is a mathematical process which projects points from the earth’s surface to a plane surface

– Where the plane surface is called developable surface and it is an imaginary surface that can be unrolled and laid out flat with or without negligible distortion of shape or size

– Then a rectangular grid (X,Y) is superimposed on the developed plane surface

• Due to the irregular shape of the Earth, different map projections are used

• Projections can be used with different Datums

• Projection is a derivate of the DATUM

THE MAIN PROBLEM WITH PROJECTIONS:

• Complicated mathematics to preserve some quantities

• Distortions are inevitable

• The algorithmic implementation of some projections (especially the Transverse Mercator) introduces computational errors as one moves from the center or central meridian of the projection (N.Zinn)

PROJECTION PARAMETERS

• Developable surface

– Cones

– Cylinder

– Plane

• Aspect

– Oblique

– Normal

– Transverse

• Case

– Secant

– Tangent

REFERENCE SURFACES

• Geoid

• Ellipsoid – 2 axial

– 3 axial

• Sphere

DISTORTIONS

MAP PROJECTIONS ALWAYS INTRODUCE DISTORTIONS

• Angles (Conformal)

• Areas (Equal-area)

• Distances (Equidistant)

• Directions (Azimuthal)

• There is no projection that will preserve all the quantities…. So pick up your projection carefully

LINEAR DISTORTION

• In order to project the Earth surface distance on grid it is necessary to apply two scale factors • Elevation Factor (EF)

• Scale Factor (SF) (mapping scale)

• Combined Factor (CF) = EF*SF

ED = SD*EF

GD = ED*SF

GD = SD*EF*SF = SD*CF

Where:

ED – ellipsoidal distance

SD – surface (ground) distance

GD – grid distance

SF – scale factor

EF – elevation factor

CF = SE*EF = combined factor

ANGULAR DISTORTIONS

t

Azimuth Grid -t

correction Chord) (Arch to termSecond -

angle eConvergenc -

Azimuth Geodetic Projected - T

Azimuth Geodetic -

STEPS TO DETERMINE THE PROJECTION TYPE

1. Global or local map projection (county, state, continent, hemisphere, world)

2. Preserved quantities – angles, distance, azimuth, area

3. Select the accuracy of map projection

4. Select developable surface – plane, cone, cylinder

5. Select the case – tangent or secant

6. Select the acceptable amount of distortions

7. Select spherical or geodetic datum

8. Select vertical and horizontal datum

LOCAL COORDINATE SYSTEM (LCS) vs. LOW DISTORTION PROJECTION (LDP)

LCS LDP

USED FOR SMALL AREA PROJECTS (MAXIMUM 50KM/35M IN LENGTH)

COVERS ONLY THE AREA OF INTEREST FOR DESIGN, ENGINEERING, CONSTRUCTION, CADASTRAL, OR OTHER LOCAL ACTIVITY

ENGINEERING/CONSTRUCTION ACCURACY REQUIREMENTS

MEASUREMENTS AND CALCULATIONS MUST SATISFY SUB-CM ACCURACY

NO NEED OF HORIZONTAL DATUM BASED ON REFERENCE ELLIPSOID

ELEVATIONS ARE HANDLED SEPARATELY FROM COORDINATES BUT NOT NECESSARY TO BE BASED ON VERTICAL DATUM

ELEVATIONS ARE HANDLED SEPARATE FROM COORDINATES AND REQUIRES VERTICAL DATUM (GEOID)

CONFORMAL – PRESERVE ANGLES AND DISTANCES

REQUIRES EXISTING SURVEY CONTROL POINTS REQUIRES WELL KNOWN EXISTING GEODETIC CONTROL POINTS

REQUIRES FIELD CHECK

SEVEN STEPS TO SELECT PROPER LDP

1. Identify the dimensions of the area of interest (AOI)

2. Identify the shape of the AOI – rectangular or square in global sense

3. Is the AOI stretched E-W or N-S

4. Is the high mountain present in AOI

5. Is there large water basins (lake, sea, ocean) present in AOI

6. The purpose of the projection – mapping or engineering

7. The purpose determine which quantities should be preserved

EIGHT STEPS TO PROPER LDP DEVELOPMENT

1. Choose the maximum grid-scale factor distortion

2. Choose projection type and projection parameters

- Lambert Conformal Conic, Transverse Mercator or …..

3. List geodetic control points in AOI and download the data from NGS

4. Select the projection surface height

5. Select the reference ellipsoid

6. Compute projection constants

7. Compute coordinates for control points in LDP

8. Perform field observation and compare measured distance and azimuths with theoretically calculated

LDP PARAMETERS LAMBERT CONFORMAL CONIC

(1 standard parallel)

φb = latitude of the grid origin (SP)

λo = longitude of grid origin

Eo = false easting at longitude of grid origin

Nb = northing value of latitude of grid origin

ho = representative ellipsoid height

ko = scale factor at the central axis (parallel)

TRANSVERSE MERCATOR

φo = latitude of grid origin

λo = longitude of grid origin (CM)

Eo = false easting at longitude of grid origin

No = false northing at latitude of grid origin

ho = representative ellipsoid height

ko = scale factor at the central axis (meridian)

POINT PARAMETERS

φ = latitude of a point

λ = longitude of a point

h = ellipsoidal height of a point

k = scale factor at a point

δ = distortion at a point

TIPS AND TRICKS

• Select the Central Meridian and Standard Parallel λ and ϕ in such way so DMS and DD values have no more than 4 decimals

Example : DMS = 39⁰ 52’ 30” DD = 39.875

• To avoid similar N, E values select false Northing and Easting to differ at least by the AOI length/width

Example: AOI L (E) = 10km, W(N) = 7km then 50000 < E < 60000 and 65000 < N < 72000

• When computing ho use not only NGS points but other elevation information as well

Example: Data from 20 NGS points and DEM for the region

• Use only order A and B NGS points with proved stability and as many as possible

• Selected NGS points must cover evenly the AOI

• Perform all calculations in the same unit (ft/m) as the LDP

• NGS data sheets must be based on the most current datum

• Run extensive field check after computing local coordinates of control points

• Remember: Must use the same datum all the time

• Set all GPS units with the same settings Example : meter, DMS, Geoid12, NAD83(2011)

ECEF >> TOPOCENTRIC >> ORTHOGRAPHIC (2D TOPOCENTRIC)

WHY?

• Cartography is distorted, Geodesy is not

• To avoid distorted Latitude and Longitude values (too many Ellipsoids/Spheroids to maintain) and complicated computations

• Map projection calculations are complicated

• Elevation is always handled separately anyway

BENEFITS

• ECEF is scalable from small to big and vice-versa

• Straight forward computation between two 3D Cartesian coordinate systems

• The conversion is conformal

• ECEF to TOPOCENTRIC is valid for any celestial body

Thanks N.Zinn, Hydrometronics

Why Topocentric?

• ECEF (X,Y,Z)

• Topocentric (East, North, Up = UVW)

• Orhtographic (view from space) - no vertical (W) and preserve U=East, V = North)

• Orthographic projection is not conformal but distortions at the center are negligible – 180km from the center the minimum scale is 4 parts in 10 000 (N. Zinn, Hydrometronics)

AKNOWLEDGMENT

• “Map Projections”, L.Bugayevskiy, J.Snyder

• “The 3D global spatial data model”, E.Burkholder

• “Basic GIS Coordinates”, Jan Van Sickle

• Noel Zinn, www.hydrometronics.com

• www.ncgia.ucsb.edu

• http://www.ngs.noaa.gov/

QUESTIONS? IDEAS? COMPLAINS? CONCERNS?

HELP?

apostol.panayotov@ucdenver.edu