Map Projections and Datums

Prepared By: Henry Morris

Parameters for Mapping

• A mathematical model of the earth must be selected. Spheroid

•The mathematical model must be related to real-world features. Datum

•Real-world features must be projected with minimum distortion from a round earth to a flat map; and given a grid system of coordinates. Projection

Modeling and Mapping the EarthModeling and Mapping the Earth

• The topographic surface of the earth comprises a highly irregular and non-mathematical 3-dimensional figure.

• Any attempt we make to define the earth’s surface in mathematical terms requires a piecemill (non-continuous) approach.

• At best, any physical model of the earth inevitably requires and results in a statistical approximation.

Spheroid

Simplistic - A round ball having a radius big enough to approximate the size of the earth.

A mathematical model of the earth must be selected.

Reality - Spinning planets bulge at the equator with reciprocal flattening at the poles. e.g.

Why use different spheroids?

• The earth's surface is not perfectly symmetrical, so the semi-major and semi-minor axes that fit one geographical region do not necessarily fit another.

• Satellite technology has revealed several elliptical deviations. For one thing, the most southerly point on the minor axis (the South Pole) is closer to the major axis (the equator) than is the most northerly point on the minor axis (the North Pole).

Datum

• A smooth mathematical surface that fits closely to the mean sea level surface throughout the area of interest. The surface to which the ground control measurements are referred.

• Provides a frame of reference for measuring locations on the surface of the earth.

A mathematical model must be related to real-world features.

How do I get a Datum?• To determine latitude and longitude, surveyors level

their measurements down to a surface called a geoid. The geoid is the shape that the earth would have if all its topography were removed.

• Or more accurately, the shape the earth would have if every point on the earth's surface had the value of mean sea level.

Gravity Recovery And Climate Experiment Gravity Recovery And Climate Experiment (GRACE)(GRACE)

Due to the irregular distribution of large-scale geologic features, the earth’s gravimetric field does not result in a uniform geometric figure.

Geoid vs Spheroid• Coordinate systems are applied to the simpler model of a

spheroid. The problem is that actual measurements of location conform to the geoid surface and have to be mathematically recalculated to positions on the spheroid. This process changes the measured positions of many point. Sometimes by a few feet, sometimes by hundreds of feet.

• Different datums use a different orientation of the spheroid to the geoid to determine which parts of the world keep accurate coordinates on the spheroid.

• For an area of interest, the surface of the spheroid can arbitrarily be made to coincide with the surface of the geoid; for this area, measurements can be accurately transferred from the geoid to the spheroid.

Earth Centered Datums

• Satellite technology has made earth-centered datums possible.

• In an earth-centered datum, the spheroid is no longer aligned with the geoid at a point on the earth's surface. Instead, the center of the spheroid is aligned with the center of mass of the earth—a location that satellite technology has made it possible to determine.

• In an earth-centered datum, the spheroid and geoid still don't match up perfectly, but the separations are more evenly distributed.

Projection

Real-world features must be projected with minimum distortion from a round earth to a flat map; and given a grid system of coordinates.

A map projection transforms latitude and longitudelocations to x,y coordinates.

What is a Projection?• If you could project light from a source through the

earth's surface onto a two-dimensional surface, you could then trace the shapes of the surface features onto the two-dimensional surface.

• This two-dimensional surface would be the basis for your map.

Why use a Projection?

• Can only see half the earth’s surface at a time.• Unless a globe is very large it will lack detail and

accuracy.• Harder to represent features on a flat computer

screen.• Doesn’t fold, roll or transport easily.

z

x

y

P(r, ,)

r

P’(x, y)

r’

sin'

cos'

cos'

ry

rx

rr

Simple ProjectionSimple Projection(3-D Polar to 2-D Rectangular)(3-D Polar to 2-D Rectangular)

Limitations Of Mapping ProjectionsLimitations Of Mapping Projections

• All mapping projections result in distortions between measurements made on the earth’s surface and their corresponding geometry on the projection plane.

• While these distortions cannot be completely removed, they can be systematically constrained and quantified based on the intended purpose of the map or reference frame.

• All geometric properties (area, angle, scale, shape, direction, distance, etc.) cannot be maintained in a true state but selected ones can at the expense of others.

Map Projection Category

Maintained Distorted

Equivalent (Equal-area) areal scale angle, shape, distance

Equidistant distance (over some portions)

direction, area

Azimuthal angular relationships from a central point

shape, distance, area

Conformal angles at any point, shapes for small areas

the size of large areas

Shape (conformal) - If a map preserves shape, then feature outlines (like county boundaries) look the same on the map as they do on the earth. Area (equal-area) - If a map preserves area, then the size of a feature on a map is the same relative to its size on the earth. On an equal-area map each county would take up the same percentage of map space that actual county takes up on the earth.Distance (equidistant) - An equidistant map is one that preserves true scale for all straight lines passing through a single, specified point. If a line from a to b on a map is the same distance that it is on the earth, then the map line has true scale. No map has true scale everywhere.

Imagine capturing the world in a net. The net divides the larger earth into sections, all contained in squares of the same size. Suddenly order is imposed on chaos. Finally we have the means to describe a location as so many squares to the left, so many to the right, so many up, or so many down, and at last we have its number.

– Watts, 1966