Course Notes: Conformal and Polyconic Map Projections
Map Projections
Map projections are often viewed as a mathematical operation in which coordinates X,Y are written as a pair of parametric functions of latitude f and longitude l ,
Classes of map projection include:
Conformal (orthomorphic) - projections result in a map where angles between short intersecting lines are preserved.
Equal-area - projections result in a map where areas are preserved, however angles can be distorted.
Equidistant - projections result in a map where distances are correctly represented.
Azimuthal - a projection where the direction of any point with respect to a central point is preserved.
Map projections to a plane include three basic types; gnomonic, stereographic and orthographic.
Gnomonic - used for navigation purposes
Stereographic - excellent projection for general maps showing a hemisphere.
Orthographic - appears as a photo taken from deep space
Conical - sufficiently accurate for maps of considerable areas, used by USGS and USNOS.
Gnomic Projection
Stereo Projection
Orthographic Projection
(a) Cone tangent to a sphere. (b) Cone tangent to a sphere with standard tangent closer to the equator. (c) Cylinder tangent to a sphere at the equator. (d) Cone with standard parallel at higher altitudes. (e) Plane tangent to a sphere at the pole. (Courtesy National Geographic Society.)
Polyconic projections - points on the surface of the earth are projected on to a series of frustrums of cones that are fitted together.
Polyconic Projection
where:
and,
a = equatorial radius
e = eccentricity of the ellipse
f = latitude
Conformal mapping - projection that preserves the angle between any pair of intersecting short line segments.
With conformal mapping there is not a direct relation between map coordinates X, Y and geodetic coordinates f ,l. An intermediate surface is introduced where,
where:
for
a = semimajor axis
b = semiminor axis
Map coordinates are then expressed as a function of l and q,
Lambert conformal conic projection - most widely used projection, used for state plane coordinates in states with greater East-West distances than North-South, it is a conic projection with two standard parallels.
Parametric equations for this projection are,
where:
and,
where:
Nl = normal radius of curvature
f l = standard parallel
Lambert Conformal Conic Projection
Graphical Representation of Coordinates on the Lambert Conformal Projection
Mercator projection - map projection on a cylinder tangent to the earth at the equator
where:
l = longitude
q = isometric latitude
Meridians are equally spaced vertical lines, and parallels are unequally spaced horizontal lines.
The comparison of scales at the equator as related to the scale at any latitude is,
Equator Þ
1:20,000
f = 60° Þ 1:10,000
This projection is valuable for navigation.
Transverse Mercator projection - an ordinary Mercator projection turned through a 90° angle so that it is related to a central meridian.
This projection is used for the state plane coordinate systems in states of greater North-South than East-West distances.
Universal transverse Mercator (UTM)
- projection is based entirely on the transverse Mercator projection.
UTM projection is in zones of 6°.
Reference ellipsoid is the Clarke 1866 for North America.
Origin of longitude is at the central meridian.
Origin of latitude is the equator.
Unit of measure is the meter.
False northing of 10,000,000 m is used in the southern hemisphere.
A false easting of 500,000 m is used for the central meridian of each zone.
The scale factor at the central meridian is 0.9996.
Zones are numbered with 1 beginning at 180°W and 174° meridians, and increasing to 60 for the zone between meridians of 174° and 180°E.
Latitude for this system varies between 80°N and 80°S.