Why projections distort

When the irrigation district's GIS went into service, Ruth Albrecht noticed that the same question kept arriving at Coulee Geomatics in different forms. The district's engineer wanted to know why the length of Lateral 7 in the GIS was 0.3 m shorter per kilometre than the length his crew had chained. A contractor asked whether the coordinates on the 1958 as-built drawings, the coordinates in the GIS and the coordinates on Grant's control sheets could be mixed in one design file. A landowner's lawyer asked why the area of a quarter section on the provincial parcel map differed from the area on the title. Each question had the same root: the Earth is curved and the maps, drawings and screens on which its features are measured are flat. Ruth asked Leah Tran to prepare a short course for the district's staff and the firm's new hires, and this course follows that work. It begins with the fact that makes the subject necessary: a curved surface cannot be laid flat without stretching or tearing it.

The orange peel problem

Peel an orange and try to press the peel flat on a table. However carefully it is done, the peel either splits or stretches: the middle of each piece lies flat while the edges gape or wrinkle. A sphere and a plane have different intrinsic geometry. On a plane the angles of a triangle add to 180°, the circumference of a circle is \(2\pi\) times its radius and parallel lines never meet. On a sphere the angles of a triangle add to more than 180°, by an amount proportional to its area; a circle drawn on the surface has a circumference less than \(2\pi\) times its radius measured along the surface; and the 'straight lines' of the surface, the great circles, always meet. These differences are properties of the surface itself, measurable by someone who never leaves it.

Carl Friedrich Gauss proved in 1827 that this intrinsic curvature cannot be changed by bending a surface without stretching it. A sheet of paper can be rolled into a cylinder or a cone because both have zero intrinsic curvature, the same as the plane; it cannot be wrapped smoothly around a ball. The Earth's surface, approximated by a sphere or an ellipsoid, has positive curvature everywhere. It follows that every flat map of any part of it must change some distances. The change can be kept small over a small area, it can be arranged so that some property is kept exactly, and it can be placed where it does least harm, but it cannot be removed. Map projection is the craft of deciding where the distortion goes.

What a map projection is

A map projection is a mathematical rule that assigns plane coordinates \((x, y)\), usually called easting and northing, to every point of latitude \(\varphi\) and longitude \(\lambda\) on a reference surface:

together with the inverse rule that recovers latitude and longitude from the plane coordinates. The word 'projection' suggests light shining through a globe onto a screen, and a few projections can be built that way, but most are defined by formulas chosen to give the map a desired property. The functions work on geodetic latitude and longitude on an ellipsoid, the surface introduced in Geodesy I. A complete projected coordinate reference system therefore has several parts: a geodetic datum and its ellipsoid, which fix where latitude and longitude are; a projection method with its parameters, such as a central meridian and a scale factor; a false easting and false northing that keep coordinates positive; a unit of length; and the order and direction of the axes. Leaving out any of these is the source of most coordinate mistakes, and later lessons treat each in turn.

It helps to separate three ideas that are often blurred. A datum, such as NAD83(CSRS) or NAD83(2011), says which ellipsoid is used and how it is placed relative to the Earth. A projection, such as transverse Mercator, is a formula that flattens that ellipsoid. A coordinate system for a particular region, such as UTM zone 12 or a state plane zone, is a projection with chosen parameters, applied to a stated datum. Changing the datum moves points by centimetres to tens of metres; changing the projection changes the numbers completely while the points stay where they are.

What can be preserved

Because a flat map cannot keep all distances, projection designers choose which properties to keep. The four classical properties are these. A conformal projection keeps angles, and therefore the shapes of small features, correct at every point: the scale may differ from place to place, but at any one point it is the same in every direction. An equal-area (equivalent) projection keeps areas in correct proportion everywhere, at the cost of angles and shapes. An equidistant projection keeps true scale along a particular set of lines, such as all meridians, or all straight lines through one centre point. An azimuthal projection keeps the directions, or azimuths, from one central point true.

No projection can be both conformal and equal-area; that would make it an exact copy of the curved surface, which Gauss showed is impossible. Some projections are compromises that are neither, chosen because they look balanced on world maps. For surveying and engineering the choice is nearly always conformal, because survey measurements are angles and distances made over short lines, and a conformal projection lets an angle measured on the ground be used on the map unchanged, with a single scale factor at each point correcting all the distances. Equal-area projections are used for statistical and thematic maps where area comparisons matter, such as maps of irrigated area or land cover by county or municipal district.

Developable surfaces, aspects and secant forms

Many projections are easiest to picture by imagining the globe wrapped in a surface that can be unrolled flat, a developable surface. A cylinder gives cylindrical projections, in which meridians are equally spaced parallel lines and parallels are straight lines at right angles to them. A cone gives conic projections, in which meridians are straight lines converging to a point and parallels are arcs of circles. A plane gives azimuthal projections, in which meridians radiate from a centre and directions from that centre are true. In each case the distortion is zero where the surface touches the globe and grows away from it.

The surface can touch the globe along one line, the tangent case, or cut through it along two lines, the secant case. A secant surface lies partly inside and partly outside the globe; features between the two lines of contact are drawn slightly too small and features outside them slightly too large. Spreading the error both ways halves the largest error for a given width of map, which is why most coordinate systems used in surveying are secant, or are tangent with the scale on the central line reduced slightly below one, which has the same effect. The orientation of the surface is called the aspect. In the normal aspect a cylinder touches the equator; in the transverse aspect it is turned through 90° to touch a meridian, giving the transverse Mercator projection of lesson 6; in an oblique aspect it touches some other great circle. Aspect changes the pattern of distortion but not the property kept.

Distortion grows with the size of the area

The code below puts numbers on these ideas. Its first part uses the simplest of all flat maps, the plate carrée, which plots longitude and latitude directly as \(x\) and \(y\). At the equator a degree of longitude and a degree of latitude are both about 111 km, so the map is nearly true there. At the district's latitude a degree of longitude is much shorter on the ground, but the map draws it the same length as a degree of latitude, so every east-west distance is stretched. The second part shows how fast distortion grows with the size of the mapped area for the best possible plane: the error grows with the square of the distance from the centre. The third computes the spherical excess of a large triangle, a direct measurement of curvature.

import math
R = 6371000.0          # mean radius of a spherical Earth, m

# 1. The simplest flat map: plot longitude and latitude as x and y
#    (plate carree, one map unit = one degree on the equator).
print("Plate carree: map length of 1 km east-west, compared with the truth")
deg = R * math.pi / 180        # ground length of one degree on the equator
for lat in (0, 30, 45, 49.7, 60, 80):
    true_deg_lon = deg * math.cos(math.radians(lat))   # ground length of 1 degree of longitude
    k = deg / true_deg_lon                              # east-west scale on the map
    print(f"  latitude {lat:5.1f}: one degree of longitude = {true_deg_lon/1000:7.2f} km, "
          f"east-west scale = {k:5.3f}, a 1 km road is drawn {k:5.3f} km long")

# 2. A flat plane touching the sphere: how much does any plane map
#    get wrong at distance d from its point of contact?
print("\nBest plane touching the sphere: scale error grows with the square of size")
for name, d in (("one field", 0.8e3), ("the district", 15e3), ("a county", 60e3),
                ("southern Alberta", 300e3), ("North America", 3000e3)):
    err = (d / R) ** 2 / 4          # about half of the full spread d^2 / (2 R^2)
    print(f"  {name:17s} half-width {d/1000:7.1f} km -> scale error about {err*1e6:10.2f} ppm "
          f"({err*1000:8.3f} m per km)")

# 3. Gauss: the angles of a triangle drawn on the sphere add to more than 180 degrees
a = 300e3                           # side of an equilateral triangle on the ground, m
area_flat = math.sqrt(3) / 4 * a**2
excess = area_flat / R**2           # spherical excess in radians, approximately area / R^2
print(f"\nEquilateral triangle with {a/1000:.0f} km sides: angle sum = 180 deg + "
      f"{math.degrees(excess)*3600:.1f} arc seconds")

Plate carree: map length of 1 km east-west, compared with the truth
  latitude   0.0: one degree of longitude =  111.19 km, east-west scale = 1.000, a 1 km road is drawn 1.000 km long
  latitude  30.0: one degree of longitude =   96.30 km, east-west scale = 1.155, a 1 km road is drawn 1.155 km long
  latitude  45.0: one degree of longitude =   78.63 km, east-west scale = 1.414, a 1 km road is drawn 1.414 km long
  latitude  49.7: one degree of longitude =   71.92 km, east-west scale = 1.546, a 1 km road is drawn 1.546 km long
  latitude  60.0: one degree of longitude =   55.60 km, east-west scale = 2.000, a 1 km road is drawn 2.000 km long
  latitude  80.0: one degree of longitude =   19.31 km, east-west scale = 5.759, a 1 km road is drawn 5.759 km long

Best plane touching the sphere: scale error grows with the square of size
  one field         half-width     0.8 km -> scale error about       0.00 ppm (   0.000 m per km)
  the district      half-width    15.0 km -> scale error about       1.39 ppm (   0.001 m per km)
  a county          half-width    60.0 km -> scale error about      22.17 ppm (   0.022 m per km)
  southern Alberta  half-width   300.0 km -> scale error about     554.33 ppm (   0.554 m per km)
  North America     half-width  3000.0 km -> scale error about   55432.86 ppm (  55.433 m per km)

Equilateral triangle with 300 km sides: angle sum = 180 deg + 198.0 arc seconds

At latitude 49.7°, the plate carrée draws a 1 km road running east-west 1.546 km long, a 55 % error, while a road running north-south is drawn correctly. This is why a GIS that displays data in latitude and longitude makes Alberta look wide and squat, and why any distance or area measured in such a display is wrong. The second table is more encouraging for engineers: a plane touching the Earth at the centre of the district has a scale error of about 1.4 parts per million at its edge, 15 km away, which is 1.4 mm per kilometre and smaller than the error of most measurements. Over a county the error reaches tens of parts per million, over southern Alberta hundreds, and over a continent tens of thousands. The 300 km triangle's angles add to 180° plus 198 seconds of arc, a quantity that a good total station can detect over long sights and that early geodetic surveys corrected routinely.

The square law has a practical meaning. Doubling the width of a projection zone quadruples its largest distortion. A coordinate system for a whole province or state must therefore accept distortions of a few hundred parts per million, while a system designed for one county or one project can hold distortion to a few parts per million. Much of this course is about that trade-off: wide systems are convenient because everyone shares them, narrow systems are accurate because they fit the ground, and the surveyor's task is to know which is in use and to correct for it.

Distortion that is not from the projection

Two other effects add to the projection's distortion and are often confused with it. The first is height. Projections flatten the ellipsoid, not the ground, and the ground at the district stands roughly 900 to 1,000 m above the ellipsoid. A distance measured on the ground is longer than the same distance reduced to the ellipsoid by about one part in 7,000, or 140 parts per million, at these heights. Lesson 8 combines this with the projection's scale into a single combined factor. The second is the datum: coordinates of the same point in two datums differ by a shift that has nothing to do with flattening. Lessons 9 and 10 deal with datum transformations and with coordinates that change over time.

There is also the distortion of measurement and display. A paper map printed at 1:50,000 shrinks and swells with humidity by more than a projection distorts over the same sheet; a scanned drawing carries the distortions of the scanner; and a screen shows whatever the GIS has been told the coordinates mean. These are reasons to keep coordinates as numbers in a well-defined system and to treat every picture of them as approximate.

Choosing a projection

The choice of projection follows from four questions. What property matters for the use: shapes and angles for surveying and engineering, areas for statistics, distances or directions from a point for coverage maps? How large is the area, and what is its shape: a region long in the east-west direction suits a conic or a normal cylindrical projection, one long north-south suits a transverse cylinder, and a compact region suits an azimuthal projection or any of them? Where is it: near the equator, at middle latitudes or near a pole? And who else will use the data: a project coordinate system that matches the ground is no help if every other agency publishes in a different one, and a widely shared system is no help if its distortion is too large for the work.

For the district, Leah's answers were straightforward. The GIS master data stayed in UTM zone 12 on NAD83(CSRS), the system already used for its imagery and most provincial data, because the GIS is used for planning and display where a few hundred parts per million do not matter. Engineering and construction work, where staking must agree with chained or measured distances, would use a projection that fits the district's ground, as lessons 7 and 8 explain. And thematic maps of irrigated area for the annual report would use an equal-area projection so that the areas compared on the page are correct. One set of data, three projections, each chosen for its purpose, and transformations among them handled by the software of lesson 11.