7 8.math projection (2)
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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