Mathematical Principles behind Projections

To locate Geographic Position

A model (3D) of the earth surfaceSpheroid

How the 3D model is related to the shape of the earth?

Datum

A model to translate the 3D points on a 2D surface with minimal distortion

Projection

A coordinate system to measure the points

3D or 2D

What is a Projection?

Mathematical transformation of 3D objects in a 2D space with minimal distortionThis two-dimensional surface would be the basis for your map.

Engineering Drawing

Types of ProjectionsOrthographicIsometricPerspectiveOblique

For any map, the most important parameters of accuracy can be expressed as:

can distances be accurately measured? are directions preserved? are shapes preserved? are area ratios preserved? which regions suffer the most, and which kind of, distortion?

Although all properties cannot be preserved, a reasonably small spherical patch can be approximated by a flat sheet with acceptable distortion

In most projections, at least one specific region (usually the center of the map) suffers little or no distortion.

If the represented region is small enough , the projection choice may be of little importance.

Geometric Projectiongeometric interpretation as light rays projected from a source intercept the Earth and, according to laws of perspective, "draw" its features on a surface (Plane, cylinder, or cone)

Algorithmic Projectionare described purely by mathematical formulae • E.g. equal-distant, equal area etc.,

Graticule

Graticules as a guide for distortion

along any meridian, the distance on the map between parallels should be constant along any single parallel, the distance on the map between meridians should be constant; for different parallels, Distance between meridians should decrease to zero towards the poles therefore, any two grid "cells" bounded by the same two parallels should enclose the same area

Geometric Projection

Distance Property preserved

The Azimuthal Equidistant Projection North-polar aspect (Arctic at the centre)• ρ = (π / 2 − φ)R and θ = λ

South-polar aspect (Antarctic at the centre)• ρ = (π / 2 + φ)R and θ = -λ

Only the radial distance from the centre of the map to any object is preserved

Azimuthal Equidistant projection (North polar aspect)

Azimuthal Equidistant projection (South polar aspect)

Inverse Mapping

Azimuthal Equal-area ProjectionN

N

E

Azimuthal Equal area projection (North polar aspect)

Self-Study

http://www.progonos.com/furuti/index.htmlhttp://mathworld.wolfram.com/MapProjection.htmlhttp://mathworld.wolfram.com/Ellipse.htmlhttp://mathworld.wolfram.com/Ellipsoid.html

AssignmentMap Projection

• Derive equations for azimuthal equal-distance and equal area from the equatorial aspect

Reference:• http://www.progonos.com/furuti/index.html• Books:

– Engineering Surveying, Higher Surveying

One projection to another

Source: http://www.kartografie.nl/geometrics/Introduction/introduction.html

Source: http://www.kartografie.nl/geometrics/Introduction/introduction.html

Datum Transformation

Earth is not a sphereEarth mass is not distributed uniformly, so the gravitational pull is not uniformDue to rotation, equator is slightly bulged, and poles are slightly flattened (1/300)• Oblate Spheroid

Terrain is not uniform

Highest spot on earth?

What is the tallest peak on earth?Mount Everest, at 8,850 meters above MSL

What is the highest spot on earth where you are the closest to the outer space?

Mount Chimborazo, in the Andes, • 6,100 meters above MSL• But is sitting on a bulge which makes it 2,400

meters taller than Everest• Everest is sitting down on the lower side of the

same bulge

Source: http://www.npr.org/templates/story/story.php?storyId=9428163

Deviations (undulations) between the Geoid and the WGS84 ellipsoid

Source: http://www.kartografie.nl/geometrics/Introduction/introduction.html

Earth Surfaces

Geoid is a surface of constant gravity.

Topographic surface

EllipsoidSea surface

Geoid

Taking into account all these irregularities is difficult

Some irregularities can be ignoredFor e.g. terrain although important locally, terrain levels are minuscule in planetary scale• the tallest land peak stands less than

9km above sea level, or nearly 1/1440 of Earth diameter

• the depth of the most profound sea abyss is roughly 1/1150 diameter.

For maps covering very large areas, especially worldwide,

the Earth may be assumed perfectly spherical, since any shape imprecision is dwarfed by unavoidable errors in data and media resolution.

Conversely, for very small areas terrain features dominate and measurements can be based on a flat Earth

Datum

Horizontal Datum or Geodetic Datum

Reference frame for locating points on Earth’s surface.Defines origin & orientation of latitude/longitude lines.Defined by spheroid and spheroid’s position relative to Earth’s center.

2 types of datums:Earth-centeredLocal

Earth’s SurfaceEarth-centered datum (WGS84)Local datum (NAD27)

Local datumcoordinate system

Earth-centered datumcoordinate system

Datum Transformation

Cartesian Coordinate System

Y

Z

X

Ellipsoid sized a & f

WGS -84 is Earth Centered Earth Fixed; Origin to with +/- 10 cm

To translate one datum to another we must know the relationship between the chosen ellipsoids in terms of position and orientation. The relationship is defined by 7 constants.

A. 3 - Distance of the ellipsoid center from the center of the earth (X, Y, Z)

B. 3 - Rotations around the X, Y, and Z of the Cartesian Coordinate System Axes (, , )

C. 1 - Scale change (S) of the survey control network

2 - The size and shape of the ellipsoid (semi major axis a and flattening f approximately 1/298

Datums and Defining Parameters

Translations (3 Parameters)Movement of points along an Axis

X

Z

Y

Rotations (3 Parameters)Movement of points around an Axis

Scale (1 Parameter)Changing the distance between points

S

Datum Transformation

Simple Three-parameter transformation

Geocentric translation

Seven-parameter transformationHelmert 7-parameter transformation

Standard Molodensky formulas

Earth Centered X/Y/Z Geodetic Lat/Lon/Height

3 Parameters

XYZ

+XYZ

X’Y’Z’

=

3 Parameter Determination

Most Transformation Parameters can be found in the NIMA technical report “Department of Defense World Geodetic System 1984” (TR 8350.2) NSN: 7643-01-402-0347

7 Parameters

Projection: ADatum: B

(Local Datum e.g. Indian)

Projection: CDatum: D

(Local or global datum e.g.Indian or WGS84

GeographicCoordinates

Datum: B

Inverse mapping

DatumTransformation

GeographicCoordinates

Datum: WGS 84

DatumTransformation

GeographicCoordinatesDatum: D

Reproject to C

Self-Study

http://www.progonos.com/furuti/MapProj/

CartIndex/cartIndex.html http://mathworld.wolfram.com/Ellipse.htmlhttp://mathworld.wolfram.com/Ellipsoid.html

References:

http://www.colorado.edu/geography/gcraft/notes/datum/datum.htmlhttps://www.navigator.navy.mil/navigator/gis.htmhttp://earth-info.nga.mil/GandG/coordsys/datums/index.htmlhttp://www.kartografie.nl/geometrics/Introduction/introduction.html