CEE 8813 · GIS Lecture 2 · Coordinate systems

Projection Light Lab

Think of a map projection as a lamp shining through a see-through globe onto paper. The paper can lie flat, or wrap around the globe as a cylinder or a cone, and it is accurate only where it touches the globe. Move the lamp, change the paper, and watch what happens to Georgia.

Lamp · glass globe · paper

Gnomonic
Light passing through Georgia Edge of what can reach the paper

What shape is the paper?

Where is the lamp?

Aim at:

Drag to orbit · right-drag or Shift-drag to pan · scroll or pinch to zoom · click the globe to aim there

The paper, laid flat

Georgia on this paper

True to life
From touch point0°0 km
Area on map×1.00of true size
Shape stretch1.00 : 1shape kept

Close-up: filled shape is Georgia on this map, dashed outline is Georgia's true shape at the same scale. The arrow points to north.

Aim the lamp

Choose where the paper touches the globe. Each card previews Georgia under the current lamp and paper, next to its true outline (dashed). Click a card to fly there.

Try this in class

  1. Leave the paper on Georgia and slide the lamp from one end to the other. Georgia hardly changes, because every lamp is accurate near the touch point.
  2. Aim at Alabama: Georgia barely changes, because it is still close to the touch point. Now aim at California, about 30° away. With the lamp at the center Georgia grows (about ×1.5) and stretches; with the lamp infinitely far away it shrinks (about ×0.87) and flattens.
  3. Turn on Tissot circles. They are equal-sized circles on the globe. On the stereographic map they stay round and only grow, which is what “conformal” means.
  4. Aim at Tokyo, then at Georgia's antipode. Find out which lamp can still draw Georgia at all, and why the others can't.
  5. Switch the paper to a cylinder, a sideways cylinder and a cone, and aim each at Georgia, then at California. Which shapes care about east–west distance, and which only care about latitude?
  6. On the cylinder, flip between “Bulb at the center” and “Real formula”. Mercator is what you get when the formula moves each parallel so that small shapes stay true.
  7. Pick the cone and drag “Touch or cut through?” from tangent to secant. Once Georgia sits between the two standard parallels it comes out slightly smaller than true, and the far edges of the map improve. On the standing cylinder the two lines stay near the equator, so the whole map just shrinks a little. Then press “Georgia Statewide Lambert” to see the cone Georgia really uses.
  8. Pick the tilted cylinder, aim at Georgia and turn “Which way does the touch line run?”. At 0° it matches the sideways cylinder; aimed due east along the equator it matches the standing one. Then press “Atlanta–Savannah line” to lay the paper along a highway corridor.

The three classic lamps

Lamp positionWhat it keeps trueIts limit
Globe's centerGnomonic Every straight line on the paper is a great circle, the shortest route between two places. Navigators use it to plot long-distance routes. Shows less than half the Earth. Shapes stretch quickly away from the touch point.
Far side of the globeStereographic Angles and small shapes (it is conformal). Circles on the globe stay circles on the paper. Areas grow fast with distance. The point under the lamp can never be drawn.
Infinitely far awayOrthographic The look of the globe seen from deep space, like a photograph. Only one hemisphere fits. Shapes near the edge are squashed flat.

Back to Lecture 2: planes, cylinders and cones

The lecture's three projection families differ in the shape of the paper. Each one is most accurate where its paper touches the globe.

PlaneAzimuthal

Touches the globe at one point. Distortion grows in rings around that point, the way the distance rings on the paper show.

Used for polar maps and for maps centered on one city or airport.

CylinderMercator · Transverse · Oblique

Touches along a whole line. Mercator wraps the equator; Transverse Mercator turns the cylinder on its side to wrap a meridian; an oblique (Hotine) Mercator tilts it to follow any slanted line.

Georgia: UTM cuts the world into 60 strips 6° wide, and Georgia straddles zones 16N and 17N, split at 84°W. The State Plane East and West zones for Georgia are Transverse Mercator too.

ConeLambert Conformal Conic

Touches along one parallel, or cuts through the globe along two (a secant cone). Good for regions that run east to west.

Georgia: Georgia Statewide Lambert (ESRI:102604) is a Lambert Conformal Conic whose two standard parallels, 31°25′N and 34°17′N, both cross the state. Its central meridian is 83°30′W.

A bulb at the center shows the idea for every shape. Mercator, Transverse and Oblique Mercator, and Lambert then space the lines parallel to the touch line by formula so that small shapes stay true; compare them with the “Real formula” button. Either way the rule for choosing a coordinate system is the same: pick one whose paper touches the place you are mapping.