Maa-6.3284 Map Projections

Lecture IAutumn 2010

Period I

Mauri Väisänen

Short history

• the science of map projections was born more than two thousand years ago

• Greek scientists started to make maps of the Earth and celestial sky with meridians and parallels

– Anaximander– Hipparchus– Apollonius– Eratosthenes

History

• AD 150 Claudius Ptolemy: GEOGRAPHY – descriptions and methods for map design– determination of the Earth dimensions– map projections

History

• Renaissance: intensive development of cartography

• epoch of great discoveries

• development of trade and navigation

• military maps

History

• Abraham Ortelius and Gerardus Mercator late sixteenth century:

– ATLAS maps

– for navigation Mercator projection

History

• during eighteenth century: ellipsoidal shape of the Earth

• regular topographic surveys of large regions

• Bonne, Lagrange,Euler, De l'Isle

• logarithms and differential calculus

• Gauss: conformal transformations of one surface onto another

• Tissot: indicatrix

About Map Projections

The physical body of the Earth is very complicated That is why we need some simplifications to describe or make a two dimensional map from three dimensional world

The Earth can be described a sphere in such works, where the accuracy is not (so) important

About Map Projections

• Also when the scale is small, the Earth can be a sphere

• More accurate works, the Earth is described a rotation ellipsoid

• There is also a third way to describe an Earth:– Physical body is called a geoid– The surface is difficult to handle

About map projections

• But usually we want a two dimensional map from the world or from certain area

• A map projection is any method of representing the surface of a sphere or an ellipsoid

• Map projections are necessary for creating maps

How

• There is no perfect solution to this 3d->2d problem

• It is convenient to define a map projection as a systematic arrangement of intersecting lines on a plane that represent and have a one-to-one correspondence to the meridians and parallels on the datum surface

How

• who creates a map projection needs to find relation between a sphere (ellipsoid) and plane

• differential geometry is needed for creating formulaes

• boundaries

• Gaussian fundamental quantities give answer

to some basic features of creating a map

Distortions

• All map projections distort the surface in some way

• distortion is a false presentation of

–angles

–shapes

–distances

–areas

Distortion

• it depends on the purpose, which distortions are acceptaple and which are not

• the same data can be stretched, compressed, twisted and otherwise distorted in different ways

Properties of a projection A map of the Earth is a representation of a

curved surface on a plane Many properties can be measured on the Earth's

surface independently of its geography Some of these properties are:

• Area• Shape• Direction• Bearing• Distance• scale

Properties of a projection

• Map projection can be constructed to preserve one or more of these properties

• At the same time not all

• Each projection preserves or compromises or approximates basic metric properties in different ways

Scale

• In the theory of map projections, concepts and formulas for both area and linear scale are considered

• The nominal linear scale of a map must be distinguished from the local scale

Nominal linear scale

• The nominal linear scale shows the general reduction of either the whole ellipsoid (or sphere) or some part of it to represent the surface being mapped

• Scale is placed on the map but it is preserved only at some points or along some lines of the map

• Changing the nominal scale does not affect the projection that is used

Local linear scale

• The local linear scale of the reprentation of a given point and along a given direction is the ratio of the length of an infinitesimal segment on the projection to that of the corresponding infinitesimal segment on the surface of the ellipsoid (or sphere)

• The ratio of this local linear scale to the nominal linear scale is called the linear scale factor µ:

µ =ds'/ds

Equidistancy

• if the graticule of the equidistant projection is orthogonal, then the base directions coinside with meridians and parallels and these projections are equidistant along meridians or along parallels.

• the term equidistant projection usually applies to equidistance along meridians or verticals

Equidistancy

Describing a map projection that preserves scale

between certain points

There is no map that can show scale correctly throughout the entire map but some can show true scale between one or more lines

Most projections have one or more lines which

(at map scale) are the same on the earth → true distance, standard lines

A map is equidistant when there is a correct representation of a distance between two points on the datum surface and the corresponding points on the projection surface

scale is maintained along the lines connecting a pair of points

It is not possible to preserve all distances For example central meridian has true distance

Equidistant

Equivalency (equality of areas)

• Equivalence of areas means that areas of figures represented are retained but the expense of shapes and angles which are in such case deformed

Conformality

Strictly speaking only in small areas

When the scale of a map at any point on the map is the same in any direction, the projection is conformal. Meridians (lines of longitude) and parallels (lines of latitude) intersect at right angles.

Shape is preserved locally on conformal maps

Developable surface

• a surface that can be unfolded or unrolled into a plane or sheet without

stretching, tearing or shrinking is called a developable surface

→ plane, cone and cylinder

Aspects of the shape

•The aspect describes how the developable surface is placed relative to the globe

•it can be normal (also regular, direct,conventional are used)

• the surfaces axis of symmetry coincides with the Earth's axis

Normal/Direct/Regular aspect

Aspects

• It can be transverse

-it means at right angles of the Earth's axis

Aspects

• Or it can be oblique

Aspects

• Projection can be tangential, when it touches plane in one point or

• in regular (normal) cylindrical projection it can touch the whole equator or transverse case the whole meridian

• it can touch along two lines parallel to central line (normal)

• in oblique case it can touch along great circle

Regular / Direct / Normalcylinder projection

Regular conic projection

Regular conic projection

• Conic can has one standard parallel or two standard parallels

• When there are two standard parallels, distortion is smaller

• Parallels are arcs of concentric cirles

• Meridians are straight lines

• Can be conformal, equal area or arbitrary in distortion

• Arbitrary: equidistant along meridians and/or along parallels

Transverse cylinder

Transverse plane

Transverse conic

Oblique projection

Points of tangency are neither on poles nor on equator

Cylinder or conic has tangency points along a great circle

Examples of projections

Lambert's cylindrical equal-area projection

Azimuthal Equidistant Projection

Transverse Mercator projection

UTM projectionhttp://en.wikipedia.org/wiki/File:Utm-zones.jpg

The choice of a projection It cannot be said which projection is the best It depends on purpose It depends on scale It depends on geographical situation on the

globe– Think about USA, Russia, Chile,

Estonia, Greenland, New Zealand, poles;

It depends on media which can be used

The choice of a projection

Large scale mapping conformal projections are neededIn Finland we have used Gauss-Krüger projection with one zone (YKJ) or in four zones (KKJ) (zones 0 and 5 not used) but now changing into new system

New system recommendation: it is also possible to take the best meridian between 19-31 degrees from Greenwich (Gauss-Krüger)

For example Lahti has chosen 26°

The choice of projection

• Also Lambert conformal conic is very often used projection in large scale mapping

– (usually mid latitude countries and countries which are situated in west-east direction)

Navigation

If you navigate, then directions should be mapped right

-Mercator-Normal conformal cylinder projection

-Meridians are straight lines, parallels at right angles toward them

Navigation

-Gnomonic projection

-Arcs of great circles are straight lines

-meridians are straight, in equatorial case parallels but not at equal distance

-parallels concentric circles

-it is useful in aviation

-perspective azimuthal projection

Mercator's projection

Choice of Projection

Geographical situation, shape and size must be taken into account

For example Finland is long in north-south direction → Gauss-Krüger is a natural choice Estonia is broad in west-east direction → Lambert conformal conic is then a natural choice in large scale mapping

Mapping poles -For mapping polar regions plane projection is valid, for example stereographic projection

-Cylindrical projections like Mercator or Gauss-Krüger are not possible

More about projections http://www.colorado.edu/geography/gcraft/notes/mapproj/mapproj.html http://www.progonos.com/furuti/MapProj/Normal/TOC/cartTOC.html http://mathworld.wolfram.com/MapProjection.html http://www.geog.ubc.ca/courses/klink/gis.notes/ncgia/u27.html#SEC27.4.5 http://en.wikipedia.org/wiki/Map_projections http://www.geography.wisc.edu/maplib/rob_proj.html

Classification of map projectionssummary

• By the properties of the transformation

• By the shape of the normal graticule of meridians and parallels

• By the orientation of aspect of the map craticule depending on the location of the pole of the coordinate system adopted

• By the form of differential equations defining the map projection

• By the methods of obtaining the projection

Classification

• Most commonly the projection classification is based on three principal features :

– DISTORTION CHARACTERISTICS– THE SHAPE OF THE NORMAL GRATICULE– GRATICULE ORIENTATION

Classification http://www.kartografie.nl/geometrics/Map%20projections/body.htm

• Map projections can be described in terms of their:

• 1. class (cylindrical, conical or azimuthal),

• 2. point of secancy (tangent or secant),

• 3. aspect (normal, transverse or oblique), and

• 4. distortion property (equivalent, equidistant or conformal).

Classification

• Carlos Furuti:

By geometry: azimuthal, conic, cylindrical, pseudoconic, pseudocylindrical

By property: equal-area, equidistant, conformal, aphylactic

Classification

• In Snyder's book there is more detailed classification of projections which is made by Department of Map Design and Compilation of the Moscow Engineering Institute of Geodesy, Aerophotography and Cartography

• Maurer has made in Germany

• Carolus Linnaeus in Sweden

Distortion characteristics

• Based on distortion characteristics projections are divided into conformal, equal-area or equivalent and arbitrary categories

• On CONFORMAL projections the similarity infinitesimal part of the representation is preserved (m=n=a=b=µ)

• Local linear scale does not depend on direction

• Angular distortion is absent and the area scale is equal to the square of the linear scale

Equal-area projection• On equal area projections the relationship of

areas on the territory to be mapped and on the plane remains constant

• Extreme linear scale factors are inversely proportional to each other

a=1/b and b=1/a

• The maximum angular deformation on these projections is preferably calculated from formulas with tangents

tan(ω/2) = (a-b)/2

where a and b are extreme linear scale factors at a given point

Equal area projections

• Preserve the area of displayed features

• Shape, scale, and directions are distorted

• Meridians and paralles may not intercect at right angles

Arbitrary projections

• Distortion characteristics are neither conformal nor equal area

• Both areas and angles are distorted

• We should distinguish equidistant projections where the extreme linear scale along one of the main directions remains constant

• If the graticule of the equidistant projection is orthogonal, then the base directions coinside with meridians and parallels and these projections are equidistant along parallels or equidistant along meridians

Map projections with straight parallels, cylindrical projections

• Conformal (Mercator)

• Equal area (Gall, Behrmann,Edwards, Lambert)

• Equidistant

• Equidistant along meridians (plate carrée, normal aspect)

• Arbitrary

• Oblique

• Transverse

• Perspective

Equal area projections examples

• http://gmt.soest.hawaii.edu/gmt/doc/gmt/html/GMT_Docs/node110.html

Pseudocylindrical projections

• There are lot of pseudocylindrical projections, usually for small scale or atlas mapping

• Equal-area sinusoidal pseudocylindrical projections with the poles in the form of points

– Sanson-Flamsteed – General world maps– Maps of South-America and Africa

Pseudocylindrical

• Equal area sinusoidal pseudocylindrical projection with poles as lines

– Eckert VI– Was used in Soviet Geographical atlases before

Second World War

• Wagner's and Kavrayskiy's equal area sinusoidal pseudocylindrical projection

• Equal-area elliptical pseudocylindrical projection with poles as points

– Mollweide

Pseudocylindrical

• Urmayev equal-area pseudocylindrical projections

• Hojovec equal area pseudocylindrical projections

• Eckert IV equal area pseudocylindrical projection

• Wagner IV equal area elliptical pseudocylindrical projection

Pseudocylindrical projections with arbitrary distortion

• Kavrayski's elliptical pseudocylindrical projection

• Robinson

• Trapezoidal

• Pseudocylindrical projection of Oxford Atlas

Pseudoconic

• Parallels are arcs of concentric circles and meridians are curves symmetrical about the straight central meridian

• Bonne projection– Created by Bonne about in the mid 1700's– pseudoconic– Heart-shaped– Along all parallels scale is true – Also along central meridian scale is true– Used for maps of the northern continents, Asia,

Europe and North America– http://www.hammondmap.com/sites/hammond/geography/proani.html

Pseudoazimuthal

• Developed by Ginzburg 1952, but firstly presented by Wiechel 1879 in Germany

• Usually used in an oblique aspect

• On polar aspect parallels are represented by concentric circular arcs and meridians are shown as curves or straight lines

Retroazimuthal

• Azimuth from every point on the map to the center point is shown correctly

• Useful for showing the direction in which to point antennas to receive radio signals

links

• http://en.wikipedia.org/wiki/Map_projection#Projections_by_surface

• http://data.geocomm.com/helpdesk/table1.html

• http://www.colorado.edu/geography/gcraft/notes/mapproj/mapproj_f.html

• http://www.nationalatlas.gov/articles/mapping/a_projections.html#two

• http://egsc.usgs.gov/isb/pubs/MapProjections/projections.html

• http://en.wikipedia.org/wiki/Bonne_projection

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

• http://brightcove.newscientist.com/services/player/bcpid96978243001?bctid=55885952001