112.215 Modern NavigationThomas Herring ([email protected]),http://geoweb.mit.edu/~tah/12.21510/19/2009 12.215 Lec 11 2Todayʼs class• Map Projections:– Why projections are needed– Types of map projections• Classification by type of projection• Classification by characteristics of projection– Mathematics of map projections210/19/2009 12.215 Lec 11 3Need for Map Projections• Basic need is because the Earthʼs surface is curvedand so it is not possible to represent on a flat surfacewith out some distortions• Flat surfaces were needed so that people could carrymaps with them (still a major use)• With GPS, maps are now often represented in acomputer in 3-D form or as ellipsoidal coordinatesthus minimizing the distortion• The amount of distortion depends on the area to berepresented (over small areas the Earth is nearly flat).10/19/2009 12.215 Lec 11 4Types of map projections• Map projections are classified either by way theprojection is made and the surface onto which it isprojected or by the characteristics of the resultantprojected maps.• Some projection surfaces are planes, cones andcylinders (each of these surfaces can be un-wrappedinto a flat surface)• Some map projections are purely mathematical sothat they can minimize distortions.• We will deal (mathematically) with only projection froma spherical body. Most accurate map projections areprojections from an ellipsoidal body.310/19/2009 12.215 Lec 11 5Projection by characteristics• The general characteristics of map projections are given by:• Conformality: When the scale of a map at any point on the mapis 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 conformalmaps.• Distance: A map is equidistant when it portrays distances fromthe center of the projection to any other place on the map.• Direction: A map preserves direction when azimuths (anglesfrom a point on a line to another point) are portrayed correctly inall directions.• Area: When a map portrays areas over the entire map so that allmapped areas have the same proportional relationship to theareas on the Earth that they represent, the map is an equal-areamap.10/19/2009 12.215 Lec 11 6Scale characteristics• Scale: Scale is the relationship between a distance portrayed ona map and the same distance on the Earth.• A large scale map shows a small area with a large amount ofdetail (eg. 1:25000)• A small scale map shows a large area with a small amount ofdetail (eg. 1:500000)• The interpretation of the scale is 1:25000 is 1 unit on the maprepresents 25000 units on the Earth• On many maps the scale changes across the map.• Usually the scale is shown graphically somewhere on the mapand if the scale varies across the map, the scale should indicatewhere it is applicable and the changes in scale across the map.410/19/2009 12.215 Lec 11 7Large/small scalemapLarge scaleSmall ScaleNote: Scale bar inlower left handcornerSource: http://www.mapblast.com/10/19/2009 12.215 Lec 11 8Projection type by surface• Projections are often referred to by the type of surfacethat the projection is made on to.• The three main surfaces are:– Plane (often referred to a Azimuthal Projections)– Cylindrical (Mercator is probably the most famous)– Conic projection• The characteristics of the map are set by how thesurface contacts the Earth (e.g., a Plane may betangential to the surface or it may cut through theEarth at some depth.510/19/2009 12.215 Lec 11 9General characteristics• All projections can be written in a form that allows planecoordinates x and y to be written as functions of φ and λ:x = f(φ, λ) and y = g(φ, λ).• The exact forms of the functions f and g depend on the projection.For the geometric projections from a sphere, these can be writtenas simple trigonometric functions as shown in the next few slides.• More complicated projections can involve more complicated andsometime approximate formulas especially when ellipsoidalcoordinates are projected (such as the Universal TransverseMercator (UTM) projection which is used for many US maps• On many maps UTM coordinates are given (also called gridcoordinates) and GPS receivers can normally be set to outputand interpret these types of coordinates.10/19/2009 12.215 Lec 11 10Plane projection mapsPPʼONNOPPʼθXYPPʼNrrRrRλSectionPlan view€ x = R tanθcosλy = Rtanθsinλ610/19/2009 12.215 Lec 11 11Conical Projection• The equations to solve the conical projection will beset as a homework exercise.• In a conical projection, points are projected radiallyonto the cone. The cone is then “cut” and unwrappedto form the projection.• In the case shown, the coneʼs dimensions are set byspecifying the co-latitude of the tangent point of thecone (θT). The distance around this part of the cone isset equal to the distance around the small circle onthe Earth. This allows the relationship betweenlongitude and the angle around the cut cone (β) to bedetermined.10/19/2009 12.215 Lec 11 12Conical ProjectionsNOPPʼNOPPʼTXYPθθTDistance around tangentline (green) is set equalbetween real world andthe unwrapped cone(tangent point)rrrβ710/19/2009 12.215 Lec 11 13CylindricalProjectionsPPʼφRHPPʼPʼHRλXY€ x = Rλy = RtanφSection viewProjected view10/19/2009 12.215 Lec 11 14UTM coordinates• The Universal Transverse Mercator (UTM) projectionis most commonly used in the US (and many othermid-latitude to equatorial countries)• This is an ellipsoidal projection that divides the worldinto numbered zones in longitude. For the US thesezones are:810/19/2009 12.215 Lec 11 15UTM coordinates• Within each of the zones, the latitude and longitudedifference from the central meridian is used tocompute the UTM coordinates.•These coordinatesare given asNorthing and Easting.(The eastcoordinates have500,000 added sothat they are notnegative west of thecentral meridian)10/19/2009 12.215 Lec 11 16Example ofusing UTMcoordinates910/19/2009 12.215 Lec 11 17Notes that go with previous figure• UTM coordinate maps usually have notes thatdescribe the projection in more detail• Details given on datum (NAD-27 in this case)• More details at http://www.maptools.com/UsingUTM/10/19/2009 12.215 Lec 11 18UTM coordinates• Software for converting latitude and longitude to UTMcoordinates is available at:• ftp://ftp.ngs.noaa.gov/pub/pcsoft/utms/• This software (available as PC