projections and coordinate reference systems

map projection

rendering of the curved earth on a flat map

grid north

direction of the Y-axis, often equaling the direction of a geographic meridian near the center of the projected area

points toward the edge of the map surface are

stretched apart

points that lie closer to the center appear

compressed together

distortion depends on

projection source, what shape we are projecting on, and where the projection surface is placed at or near the globe

equidistance

preserves distance relations

equivalence

preserves area relations

conformality

preserves local angles

line of true scale

a standard where there is no distance distortion

features outside line of true scale are

expanded

features inside line of true scale are

compressed

approximate distance distortion by

comparing grid coordinate distances to great circle distances

ellipses of distortion

conformal - ellipses are circles

equivalent - ellipses are the same size
equidistant - ellipses have same radii

developable surface

a geometric shape onto which the Earth's surface is projected

lambert conformal conic

cone intersecting surface of Earth, with points on Earth's surface projected onto the come

standard parallels

arcs where the cone intersects the ellipsoid, typically parallels of latitude

distortion is smallest near

standard parallels

LCC is common for

mapping areas that are larger in an east-west direction

transverse mercator

earth enveloped in a horizontal cylinder and the surface is projected onto it

lines of true scale for transverse mercator

two secant lines or a single north-south tangent

central meridian

line parallel to and midway between the secants, extending north and south

band of low distortion in mercator runs

north to south

mercator distortion increases as

distance from the two lines of intersection increases

orthodrome

line of a great circle, and the shortest distance between two points on a sphere

loxodrome

line of constant compass direction on a sphere, appearing a straight line on Mercator projection

literal projection

beam of light rays form the points from sphere’s surface onto projection surfacec

cylindrical projection

cylindrical shell is used as developable surface e.g. Mercator

normal cylindrical projection

cylinder is oriented across the equator

transverse cylindrical projection

cylinder is oriented from pole to pole

rectangular projection

projected onto rectangle, angles are distorted because the ellipses are distorted

when ellipses are the same along latitudes

there is symmetrical distortion of angles so you can measure distances and areas along them

angles can be measured with Mercator because

the ellipses are all circular

azimuthal/planar projections

plane surface tangent to Earth’s surface and area to be mapped is projected onto it

gnomonic azimuthal projection

centre of projection is in the centre of earth

stereographic azimuthal projection

centre of projection is at the counterpole

orthographic azimuthal projection

rays of light are parallel because centre of projection is at an infinite distance

normal azimuthal projection

projection plane touches the pole such that meridians become radial beams and latitudes become concentric circles

transverse azimuthal projection

projection plane touches the equator, longitudes become curves and latitudes become straight lines

oblique azimuthal projection

anywhere else but the equator

equidistant

preserving equal distances

equivalent

preserving equal areas

conformal

preserving equal angles

conic projections

conical shell used as the developable surface

no distortion at points of conic projection where

the secant lines cut through the earth’s surface a little bit

lambert’s conic projection used by pilots because

you can measure the angle and distance

mathematical projections

assign pixels arbitrarily through formulas e.g. sinusoidal

robinson projection

best compromise of factorsm often used in schoolbooks

waterman-butterfly

best approximation to achieve all three

state plane coordinate systems

specify positions in cartesian systems for each US state

state plane systems use either

lambert or mercator depending on length of state’s axis

lambert has more distortion as you move away from

horizontal standard parallels

mercator has more distortion as you move away from

vertical central meridian

universal transverse mercator system

global coordinate system based on the transverse mercator projection

review details in notes

map distortion may be reduced by making the surface

cut or interrupted

project between coordinate systems via the

inverse and forward projection equations; calculate geographic coordinates from projected ones, then new projected ones from geographic

datum transformations

account for differences in geographic coordinates due to changes in the shape or origin of the spheroid used

public land survey systems

division of land by 6 mile apart north-south lines, forming square townships, meant to identify specific parcels of land

all coordinate reference systems have an underlying

projection (not always specified)