Practical Approach to Understanding and Using Grid

Coordinates !!!!

New$Jersey$Society$of$!Professional+Land+Surveyors!

SurvCon(2014!!!!

Presented by !!

Joseph Paiva – CEO

February 2014

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New Jersey Society of Professional Land Surveyors

SurvCon 2014

Atlantic City

Practical Approach to Understanding and

Using Grid Coordinates

Joseph V.R. Paiva – CEO

Introduction

  Overview of concepts   Projection, scale differences   How the mathematics operates rather than the mathematics itself   Practical applications in the field, in the office, in communications   Stress understanding of theory and mathematics used in the system

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The grid headache

!  Why do we have it anyway? !  Is it those darn software

manufacturers? !  Why can’t we have the good old

“ground” days?

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Topics

!  Surveys: limited scope vs. large extent; plane vs. geodetic surveys

!  Projections—the concept and the math for surveyors

!  SPCS Calculations !  Strategies for dealing with the grid/

ground “thing”

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Introduction

!  Most small-area surveys assume the earth is flat (plane surveys)

!  For large areas, Earth’s curvature has to be considered

!  This usually involves determining geodetic positions (latitude and longitude) of survey stations

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State Plane Coordinate System

!  SPCS was designed in the early 1930s by the Coast and Geodetic survey to solve the problem of surveys of large extents for the “local” surveyor

!  In addition to allowing plane survey concepts to be used, it delivers several additional benefits

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SPCS benefits

!  Simplifies calculations for surveys over large distances

!  Provides common datum of reference for all surveys (if tied in)

!  Well suited for engineering projects of large extent, i.e. highways, but also photogrammetry, large scale cadastral surveys, etc.

!  Supplies vital grid for GIS (other options exist)

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SPCs

!  When surveys are tied into the SPCS, their locations become (potentially) indestructible

!  With GPS, the problem of what coordinates to use once geocentric coordinates of GPS have been transformed in geodetic coordinates makes SPCs a natural choice

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Projections

!  The basic problem with plane surveying is that it assumes the earth is flat

!  Some problems…

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Problems

!  meridians converge

Parallel is curved

except at equator

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Problems

!  On the Earth, “straight lines” are not straight except for meridians (or the equator) and the difference gets larger as you extend them

N

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Problems

!  Changes in elevation cannot be ignored, that is why all geodetic distances are at “sea level”

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Projections

!  To have a plane coordinate system, it is necessary to distort the curved surface of the earth to a fit on a plane

!  Orange peel analogy !  This process of flattening must be

systematic in order to have accuracy !  In surveying this process is called a

projection

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Projections / 2

!  Distortions inevitable (but systematic)

!  Different projections are used because each minimizes distortion in some properties at the expense of others

!  Different mathematical treatments are given to projections depending on the result desired

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Developable surface

!  A shape that can be made into a plane • Cone • Cylinder • Plane (of course)

General Classes

!  Cylindrical

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Tangent

Secant

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Developing the surface

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Transverse Mercator

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Developed transverse cylinder - distortions

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Distorted east-west distances

(Rel

ativ

ely)

un

dist

orte

d no

rth-

sout

h di

stan

ces

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Transverse Mercator (cutting through center)

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Cylinder

Ellipsoid (Earth)

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General Classes

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Secant

• Conic

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Developed Cone - distortions

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(Relatively) undistorted east-west distances D

isto

rted

nor

th-

sout

h di

stan

ces

Developed transverse cylinder - distortions

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Distorted east-west distances

(Rel

ativ

ely)

un

dist

orte

d no

rth-

sout

h di

stan

ces Re

-visit

ing…

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Lambert Conformal

!  Varying central apex angle of cone changes section of ellipsoid that is intersected

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Secant cone

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Topography to ellipsoid to grid

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Cylinder radius

Radius of

ellipsoid “slice”

Cylinder (grid)

Surface

Cylinder

Ellipsoid (Earth)

Topography to ellipsoid to grid

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Ellipsoid (Earth)

Cone (grid)

Surface

Cone

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“Lagrange” projection

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Notice that this projection here shows the whole world in a circle! Interested in the concepts of projections? See more at http://www.progonos.com/furuti/MapProj/Normal/ProjConf/projConf.html

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Next step

!  Once developable surface parameters are picked, plane is created

!  Because a developable surface is used, while there are distortions in converting coordinates on the earth to the developable surface, there is no further distortion of shape or size when it is unrolled or “developed”

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Most common surfaces in SPCS

!  Lambert conformal (conic) !  Transverse Mercator (cylinder) !  Also…skewed (or oblique) Mercator

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State Plane Coordinate Systems (83)

!  System for specifying geodetic stations using plane rectangular coordinates

!  Over 120 zones for U.S. !  Long N-S states use Transverse

Mercator !  Long E-W states use Lambert !  If square, use either

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SPCS (83)

!  Alaska, Florida and New York use both types of projections

!  In addition Alaska has an oblique projection for the southeastern part of the state

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http://www.mapsfordesign.com/images/P/bj_NewYork-ppt.gif

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http://i.infoplease.com/images/mflorida.gif

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http://alaskafisheries.noaa.gov/maps/images/nmfs_reporting_areas.gif

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SPC83 vs. SPC27

!  Coordinate values changed (N and E) !  Meters !  Types of projections changed for

some states !  Zones different in some !  Numbers of zones per state changed

in some !  Check implementing statutes in your

state for details!

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Feet!

!  U.S. Survey foot = [m] x 3937/1200 !  International foot = [m] / 0.3048 !  2 PPM! !  [0.01 ft in a mile] !  [but with a coord value of 500,000 m,

difference is 1 m!]

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NOAA/NGS document

!  NOAA Manual NOS NGS 5 State Plane Coordinate System of 1983

!  http://www.ngs.noaa.gov/ [www.ngs.noaa.gov/PUBS_LIB/ManualNOSNGS5.pdf]

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Distortions

!  Scale is exact where cone or cylinder intersects ellipsoid surface

!  Scale is less than one between lines of true scale (i.e. length on ellipsoid is greater than length on plane)

!  Scale is more than one outside lines of true scale (i.e. length on ellipsoid is smaller than length on plane)

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Scale less or greater than one…where?

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Zone size

!  Where the zone intersects the Earth, and whether it is tangent or secant controls the distortions

!  By strategic placement, distortions are minimized, scale differences can be kept to 1:10,000 or less

!  Done by keeping zone size to <158 mi and keeping zone width such that two-thirds of the zone is between lines of true scale (secant lines)

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More on zone size

!  Zones are designed to overlap each other considerably

!  Thus a survey done near a zone boundary can be done in either zone

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Transverse Mercator projection

!  Scale varies east to west but not north to south

!  Scale is true at the secant line !  All geodetic meridians are curved,

converging at the pole

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Transverse Mercator projection / 2

!  All parallels (of latitude) are curved !  CM is assigned to a meridian line !  All lines on the plane parallel to the

CM are grid north !  East-west lines on the plane are

perpendicular to the CM

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Transverse Mercator

A

B

C

D

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TM edge view

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Cylinder

Ellipsoid

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When developed… A

B

C

D

Scale greater

than true

Scale greater

than true

Scale less than true

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…or

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View perpendicular to axis of cylinder

Scale less than true

Scale greater than true Cylinder

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Mapping angle

!  Also called grid declination or variation

!  Greek letter - γ [gamma] for Mercator

!  θ Theta for Lambert

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Grid overlaid on developed surface - Mercator

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New York SPCS constants Item Value Zone New York E (3101) also NJ Type Transverse Mercator Central Meridian 74° 30’ W Grid origin latitude 38° 50’ N Grid origin longitude 74° 30’ W Grid origin X coordinate -easting 150,000 m Grid origin Y coordinate- northing 0 m Scale at central meridian 1:10,000

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New York SPCS constants Item Value Zone New York C (3102) Type Transverse Mercator Central Meridian 76° 35’ W Grid origin latitude 40° 00’ N Grid origin longitude 76° 35’ W Grid origin X coordinate -easting 250,000 m Grid origin Y coordinate- northing 0 m Scale at central meridian 1:16,000

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New York SPCS constants (be careful!)

Item Value Zone NY (LI) - Lambert (3104) Type Lambert Conformal Central Meridian 74° 00’ W Standard parallel N 41° 02’ N Standard parallel S 40° 40’ N Grid origin latitude 40° 10’ N * Grid origin longitude 74° 00’W Grid origin X coordinate -easting 300,000 m Grid origin Y coordinate- northing 0 m

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Pennsylvania SPCS constants Item Value Zone PA North - Lambert (3701) Type Lambert Conformal Central Meridian 77° 45’ W Standard parallel N 41° 57’ N Standard parallel S 40° 53’ N Grid origin latitude 40° 10’ N Grid origin longitude 77° 45’W Grid origin X coordinate -easting 600,000 m Grid origin Y coordinate- northing 0 m

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Pennsylvania SPCS constants Item Value Zone PA South - Lambert (3702) Type Lambert Conformal Central Meridian 77° 45’ W Standard parallel N 40° 58’ N Standard parallel S 39° 56’ W Grid origin latitude 39° 20’ N Grid origin longitude 77° 45’ W Grid origin X coordinate -easting 600,000 m Grid origin Y coordinate- northing 0 m

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Delaware SPCS constants Item Value Zone DE - (0700) Type Transverse Mercator Central Meridian 75° 25’ W Grid origin latitude 38° 00’ N Grid origin longitude 75° 25’ W Grid origin X coordinate -easting 200,000 m Grid origin Y coordinate- northing 0 m Scale at central meridian 1:200,000

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Ellipsoid, Geoid, Topography

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Mass Excess

Mass Deficiency

Geoid

Ellipsoid

Local topography

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Reducing surface distance to geodetic distance

Rm = 20,906,000 ft or 6,372,000 m !  Approximate SLF can be

calculated for project where relief is small

!  In high relief areas need to calculate individually using average elevation of the line

....

...

..

DistSurfSLFDistGeod

DistGrndElevRR

DistGeodm

m

×=

×+

=

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Reducing geodetic distance to grid distance

!  k is sometimes called SF (scale factor)

!  k is calculated from equations or interpolated from tables in state or NOAA documents

kDistGeodDistGrid ×= .

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Scale factor (Mercator)

!  “k” based on longitude (sometimes, k is calculated using EP’, which is the distance away from the C.M.)

!  A single Scale Factor (SF), can be picked for projects that are not large (under ~8 km)

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Direct conversion from surf. dist. to grid dist.

Grid Dist = Surf Dist x SLF x SF !  If average elevation and E for the

project are being used, multiply SLF and SF and use it as the Grid Factor (GF)

!  Grid factor also sometimes called “Combined Scale Factor” (CSF)

!  SF converts from geodetic to grid !  GF converts from ground to grid

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Grid Azimuth

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Grid Az = Geod Az - γ + Second Term !  For most surveys Second Term can

be ignored (lines under 8 km)

Why second term?

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How to apply mapping angle?

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Mapping angle gamma

).sin()..( StaLatStaLongCMLong ×−=γ

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!  Varies with longitude but can use same γ for many surveys

LaPlace correction may need to be added if using astro-azimuths

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General Pattern

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!  adjust traverse !  determine SLF using elevation (either for

project or dist. by dist.) !  determine SF using dist. from CM (either

for project or dist. by dist.) !  calc. GF if desired !  convert all distances to grid distances

using GF !  convert all azimuths to grid azimuths

General pattern / 2

!  Assuming one of the traverse points has a known SPC, calculation of the coordinates (SPC) of the other points is straightforward

!  Always multiply distances !  NEVER multiply coordinates by scale

or grid factor!

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Lambert conformal conic projection

!  Scale varies north to south but not east to west

!  Secant lines, where scale is true, are called standard parallels

!  All geodetic meridians are straight, converging at the pole

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Lambert / 2

!  All parallels (of latitude) are arcs of concentric circles have their center at the cone’s axis

!  CM is assigned to a meridian line !  All lines on the plane parallel to the

CM are grid north !  East-west lines on the plane are

perpendicular to the CM 69

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Lambert conformal

!  Varying central apex angle of cone changes section of ellipsoid that is intersected

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Standard parallels Pr

ojec

tion

limits

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Grid overlaid on developed surface - Lambert

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Mapping angle

!  Also called grid declination or variation

!  Greek letter - θ [theta]

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Calculations (Lambert)

!  Same as for Transverse Mercator except…

!  Tables for the zone have the value

of the long. of the CM and l !  General pattern for calcs is the

same

lStaLongCMLongAzGeodAzGrid

×−=

−=

.)..(....

θθ

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Typical calculations

!  Elevation 0 m; ground dist = 1000 m

!  Elevation 1,000 m; ground dist = 1000 m

....

...

..

DistSurfSLFDistGeod

DistGrndElevRR

DistGeodm

m

×=

×+

=

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Scale factor

!  Assume distance from CM is 30,000 m (doesn’t matter whether east or west)

!  Enter table and pick off value for 30,000 m: 0.9999444

!  If distance from CM is not a round number, will have to interpolate!

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Grid factor

!  GF = SLF x SF !  Also called Combined Scale Factor

(CSF)

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Mapping angle calcs (Mercator)

!  Sta. Long. = 93°00’00” !  CM = 92°30’00” !  Sta. Lat = 38°00’00”

).sin()..( StaLatStaLongCMLong ×−=γ

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Which way to apply mapping angle?

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Practical use

!  Tie in to monuments with SPCs, therefore don’t need to calculate mapping angle

!  Project coordinates sometime used—be careful!

!  On plats show SPCs. If you must show ground distances, show grid distances also!

!  Meta data!

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Whew !  How to use? !  My suggestion: use all grid (coordinates)

or all ground (distances) !  If all ground distances, publish a table of

grid coordinates of all the points !  If all grid coordinates, publish a table of all

ground distances and, if desired, azimuths/bearings on non-grid basis

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Grid vs. ground

!  Many publish “ground coordinates.” If you do, be sure that the values don’t look anything like SPCs!

!  DO NOT publish “ground coordinates” unless X and Y values are readily differentiable

!  On the plat if you show ground values and grid values use a suffix or prefix (GRID & ground)

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Grid vs. ground

!  If you have to, use different fonts or different styles (regular vs. italics)

!  But make sure they can be easily differentiated

!  Do NOT use different colors to differentiate; remember that whatever you prepare may become monochrome

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Keep in mind !  A point is a point !  It doesn’t matter whether it is on the

plane (grid), ellipsoid or surface !  Do some work on a survey nearby…

hand calculate grid or ground values…then see if your data collector and PC software handle correctly

!  Need to have fairly long distances to see differences between grid and ground (figure PPM to know how long)

!  Using a data collector do the math is OK, as long as it does it correctly

!  Remember: GIGO

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Closing note about transformations

E

X

X’

N

Y Y’

A

B

Given: A and B in N/E reference frame and X/Y reference frame. Determine the transformation equation to convert any point from the X/Y to N/E system

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Transformation / 2

E

X

X’

NY Y’

A B

Three parts to the transformation:

1.  Rotation

2.  Scale

3.  Translation

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Transformation / 3

E

X

X’

NY Y’

A B

Rotation:

1.  Determine azimuth of AB in XY and NE systems

2.  Rotation = azimuth in XY minus azimuth in NE = θ

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Transformation / 4

E

X

X’

NY Y’

A B

Scale:

1.  Determine length of AB in XY and NE systems

2.  Scale = length in NE system divided by length in XY system = s

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Transformation / 5

E

X

X’

NY Y’

A B

Translation is done in two steps:

1.  Calculate coordinates of A and B in X’Y’ system

2.  Then determine translation by subtracting coordinates in X’Y’ system from coordinates in NE system

3.  Result is Tx and TY

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Determining coordinates in X’Y’ frame

θθ

θθ

cossin'sincos'

AAA

AAA

sXsXYsYsXX

+=

−=

Transforms from XY to X’Y’ coordinates

E

X

X’

NY Y’

A B

AAY

AAX

YNTXET''

−=

−=

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Final equations for transformation

Y

X

TsYsXNTsYsXE++=

+−=

θθθθ

cossinsincos

The “your survey doesn’t check” problem

!  When you have users of your data, provide a value they can use to multiply ground distances by for the project so they can get grid values

!  Provide a value they can use to divide grid distances by for the project so they can get ground values

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Helping your user base

!  If possible, try to have one CSF for the project

!  If you have more than one indicate boundaries for each factor

!  Provide worked out solutions for calculating in both directions

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Helping…

!  If you know what instrumentation they will be using, provide specific instructions on how to set up their field equipment

!  Make sure your own field and office personnel know “which end is up”

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About&the&seminar&presenter& Joseph V.R. Paiva, PhD, PS, PE Joseph V.R. Paiva is CEO of GeoLearn, LLC (www.geo-learn.com), which is launching an online professional education business for the geospatial industry in early 2014. Joe started this business with his partner Bob Morris, whose most recent global industry position was President of Leica Mapping. Previously, Dr. Paiva was CTO of SADAR 3D and COO of Gatewing NV, a Belgian unmanned airborne systems company. Prior engagements in consulting were in the field of geomatics and general business, particularly to international developers, manufacturers and distributors of instrumentation and other geomatics tools. Dr. Paiva and Mr. Morris continue to be involved in consultancy through a separate partnership called GeoSpatial Associates, LLC will continue this consultancy. Joe’s career includes: managing director of Spatial Data Research, Inc., a GIS data collection, compilation and software development company; various assignments at Trimble Navigation Ltd. including senior scientist and technical advisor for Land Survey research & development, VP of the Land Survey group, and director of business development for the Engineering and Construction Division; vice president and a founder of Sokkia Technology, Inc., guiding development of GPS- and software-based products for surveying, mapping, measurement and positioning. He has also held senior technical management positions in The Lietz Co. and Sokkia Co. Ltd. Prior to that was assistant professor of civil engineering at the University of Missouri-Columbia, and a partner in a surveying/civil engineering consulting firm. He has continued his interest in teaching by serving as an adjunct instructor for online course development and teaching at the Missouri University of Science and Technology. His key contributions in the development field are: design of software flow for the SDR2, SDR20 series and SDR33 Electronic Field Books and software interface for the Trimble TTS500 total station. He is a registered Professional Engineer and Professional Land Surveyor, has served as ACSM representative to the Accrediting Board for Engineering and Technology (ABET), serving as a program evaluator, team chair, and commissioner and has more than 30 years experience working in civil engineering, surveying and mapping. He writes for POB, GeoDataPoint and The Empire State Surveyor magazines and has been a past contributor of columns to Civil Engineering News. Joe has also been a consultant to the Geomatics Industry Association of America, later reorganized under the Association of Equipment Manufacturers (AEM) as the Geospatial Industry Group, Joe has organized and presented workshops and authored and edited articles for the technical press in this role. Joe can be contacted at joepaiva@geo-learn.com

January 2014