Projections

Objectives of lecture:

• Change in schedule (next lecture Review and Questions)
• Demonstrate more Great Circles
• Distortions from Sphere to Plane (what can be preserved)
• Geometric froms of projections, and some examples

Basic Definition: Cartographic Projection

a transformation that establishes a correspondence between a position on a sphere and a position on a plane.

It has been demonstrated mathematically (by Gauss no less) that this procedure must distort some of the geometric properties.

Geometric Properties

• Area
• Shape (meaning local angles)
• Distance
• Direction (meaning long-distance angles)

Some extra issues

• Connectedness (continuity)
• Great circles
• Rhumb lines (loxodromes, lines of constant compass bearing)

A projection can choose to preserve area or angles, but not both. Some can preserve direction, but only with respect to a chosen center. To achieve the other objectives, distance (constant scale) is sacrificed to some extent. Scale errors can vary from the control of a UTM zone (.9996 - 1.004) to a percent or two on the whole USA to order of magnitude scale changes on a global projection (like Mercator).

Classes of projections

based on geometry of developable surface (flattened into a plane)

• Planar (also called Azimuthal)
  • Stereographic
  • Orthographic
  • Gnomonic (preserves great circles as straight lines)
• Cylindrical
  • Mercator (nautical charts, preserves rhumb lines)
  • Equal Area (Gall and Peters)
• Conic
  • Lambert conformal conic (preserves angles, used in State Plane)
  • Alber's Equal Area (used for USA)
• "Other" (non-geometric)
  • Sinusoidal
  • Molleweide
  • Goode's Homolosine

Aspect (Orientation)

Version of 30 January 2000