Cartography: Map projejctions
We study the geometry of (and determine distances on) the surface of the earth. A map projection is a systematic representation of all or part of the surface of the earth on a plane. This comprises lines called Meridians (longitudes) and parallels (latitudes). Since the sphere cannot be flattened onto a plane without distortion, the strategy is to use an intermediate surface (developable surface) that can be flattened. The sphere is first projected onto this surface, which is then laid out as a plane. Commonly used surfaces: cylinder, the cone and the plane itself.
8 The basic idea for map projection: consider a sphere with coordinates (λ, φ) for longitude and latitude, and construct a coordinate system (x, y)sothat
x = f(λ, φ),y= g(λ, φ) where f and g are appropriate functions to be determined depending on the properties we want our map to possess.
9 Equal-area maps. Use to display areal-referenced data. An example is a sinusoidal projection. Sinusoidal projection Equal-area projection. Obtained by specifying δg/δφ = R (R radius of the earth) this imposes equally spaced straight lines for parallels, and results in the following analytical expressions for f and g
f(λ, φ)=Rλ cos(φ) g(λ, φ)=Rφ
Both the Equator and central meridian are standard lines, thus the
10 whole map is twice wide as tall.
11 Another popular equal-area projection (with equally spaced straight lines for the meridians) is the Lambert cylindrical projection given by f(λ, φ)=Rλ g(λ, φ)=R sin(φ)
This projection’s perspective is easily visualized by rolling a flexible sheet around the globe and projecting each point horizontally onto the tube so formed. In other words, light rays shoot from the cylinder’s axis towards its surface, which is afterwards cut along a meridian and unrolled. Like most cylindrical projections, it is quite acceptable for the tropics, but practically useless at polar regions, which are rather compressed, resulting in a map much broader than tall. Again like in other cylindrical projections, deformation is uniform along the same parallel.
12 Properties: Meridians are equally spaced • Parallels get closer near poles. • Parallels are sines. • True scale at equator. • History • – Invented in 1772 by Johann Heinrich Lambert with along with 6 other projections. – Prototype for Behrmann and other modified cylindrical equal-area projections.
13 Figure 1: Schematic development of Lambert’s equal-area cylindrical projection. With a tangent cylinder.
f(λ, φ)=Rλ g(λ, φ)=R ln tan(π/4+φ/2)
The great Flemish cartographer Gerhard Kremer became famous with the Latinized name Gerardus Mercator. A revolutionary invention, the cylindrical projection bearing his name has a remarkable property: any straight line between two points is a loxodrome, or line of constant course on the sphere. In other words, if one draws a straight line connecting a journey’s starting and ending points on a Mercator map, that line’s slope yields the journey direction, and keeping a constant bearing is enough to get to one’s destination.
15 Figure 2: Conventional (equatorial)16 Mercator map; graticule spacing 10?; map arbitrarily clipped at parallels 85 deg. N and 85 deg. S. Properties: Conformal: it is a projection for which local (infinitesimal) angles • on a sphere are mapped to the same angles in the projection. Parallels unequally spaced, distance increases away from equator • directly proportional to increasing scale. Loxodromes or rhumb lines are straight. (rhumbs are curves that • intersect the meridians at a constant angle) Used for navigation and regions near equator. • History • – Invented in 1569 by Gerardus Mercator (Flanders) graphically. – Standard for maritime mapping in the 17th and 18th centuries. – Used for mapping the world/oceans/equatorial regions in 19th
17 century. – Used for mapping the world/U.S. Coastal and Geodetic Survey/other planets in 20th century. – Much criticism recently.
18 Northings and Eastings Map projections lead to complex equations relating longitude and latitude to the positions of points on a given map. Thus, rectangular grids have been developed, in which each point is designated merely by its distance from two perpendicular axes on a flat map. The y-axis is the chosen central meridian, y increasing north, and the x-axis is perpendicular to the y-axis at a latitude of origing on the central meridian, with x increasing east. The x coordinates are called eastings and the y coordinates northings. The grid lines do not coincide with any meridians and parllels except for the central meridian and the equator.
19 Universal Transverse Mercator Projection (UTM) The world is divided into 60 north-south zones, each covering a strip six degress wide in longitude. These zones are numbered consecutively beginning with Zone 1, between 180 degrees and 174 degrees west longitude, going eastward to zone 60 between 174 and 180 degrees east longitude. The northing values are measured continuously from zero at the Equator, in a northerly direction. to avoid engative numbers for location south of the Equator, we assigne the Equator an arbitrary false northing value of 10,000,000 meters. A central meridian through the middle of each 6 degree zone is assigned an easting value of 500,000. The northing of a point is the value of the nearest UTM grid line south of it plus its distance north of that line; its easting is the value of the nearest UTM grid line west of it plus its distance east of that line.
20 The UTM system was introduced in the 1940’s by the U.S. Army. It is widely used in topographic and military mapping.
21 Spatial modeling of point-level data often requires computing distances between points on the earth’s surface. Thus, we can wonder about a planar map projection, which would preserve distances between points. The existence of a map is precluded by Gauss’ Theorem Eggregium in differential geometrial. Projections perserve area and shapes, distances are always distorted.
22 Calculating distance on the earth’s surface We must account for the curvature of the earth when computing distances. We find the shortest distance (geodesic) between two points.
P1 =(θ1,λ1)andP2 =(θ2,λ2). The solution is
D = Rφ where R is the radius of the earth and φ is the angle (measured in radians) satisfying
cosφ =sinθ sin θ +cosθ cos θ cos(λ λ ). 1 2 1 2 1 − 2 The geodesic is the arc of the great circle ( a circle with radius equal to the radius of the earth) joining the two points.
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