Map Projection Theory and Usage

What is a map projection?

A transformation of spherical or ellipsoidal

Latitude,longitude (coordinates to planar (x,y) coordinates on a flat

surface.

The Map Projection process in more depth

How can we make a Map projection?… By using coordinate transformation equations

Latitude (φ) , Longitude (λ)

(x,y)

x

y

x = Radius × λ

y = Radius × ln (tan (45° + φ /2.0))

Mercator Projection

Geometric Distortion is Unavoidable when Transforming from a Spherical to a Flat Surface

Different Projections have Different Types of Geometric Distortion

Understanding Scale Distortion by Studying

Scale Factors across the ProjectionScale Factor =

Denominator of Principal Scale RF_________________________

Denominator of Actual Scale RF

RF stands for Representative Fraction

Principal Scale is the RF of the Generating Globe

1:100,000,000

Actual Scale is the RF at a Point on the Projection in a Given Direction

1:50,000,000

Scale Factor

100,000,000___________

50,000,000

2.00 times as large

at the point=

Scale Distortion Patterns OnMajor Types of Projections

Cylindrical ProjectionsNormalAspect

TransverseAspect

ObliqueAspect

S.F.>1

S.F.>1S.F.=1

S.F.>1

S.F.>1

S.F.=1S

.F.>

1S

.F.>

1 S.F

.=1

Cylindrical Projection Cases

Normal Aspect, Tangent Case Example – Web Mercator

Transverse Aspect, Secant Case Example – UTM Zones

UniversalTransverseMercatorProjectionDetails

Conical Projections

Normal Aspect, Secant Case Example --Sectional Aeronautical Charts --

Azimuthal Projections

Tangent and Secant Case Azimuthal Map Projection

Polar Aspect, Secant Case Example --Universal Polar Stereographic Grid Zones --

Oblique Aspect, Tangent Case Example --Great Circle Sailing Chart on Gnomonic Projection--

Oblique Aspect, Tangent Case Example -- Earth Day and Night on Orthographic Projection--

Oblique and Equatorial Aspect, Tangent Case Examples -- Rotating Globes on Orthographic Projection--

Which one is spinning correctly?

Shape Distortion and Conformality

A Conformal Map Projection is one where

Shapes and Directions are preserved locally

A Conformal Map Projection is one where

Shapes and Directions are preserved locally

A Conformal Map Projection is one where

Shapes and Directions are preserved locally

Normal Aspect, Secant Case Conformal Projection --Sectional Aeronautical Charts --

Area Distortion and Equivalency

Mollweide Elliptical Equal Area Projection

Mollweide Elliptical Equal Area Projection

Albers Conic Equal Area Projection for U.S.

No Flat Map can be Conformal andEqual Area at the same time

…Only aGlobe

can be!