Map Projections: From 3D to 2d

Map Projection = Mathematical Operation

  • Map projection is a mathematical operation that transforms the surface of a 3D ellipsoid to a 2D surface
  • The mathematical operation used for a particular projection is chosen based on the parameters

Process (Georeferencing, then Projecting)

  • A two-step process transforms the real world to a map projection:
    • A chosen coordinate system is used to locate real-world features onto the ellipsoid
    • The surface of the ellipsoid is then mathematically transformed to a 2D surface

Four Characteristics of the Globe

  • Shapes

    • Shape is the outline of a polygon

  • Area

    • Area is the size of a polygon and its value relative to other polygon

  • Distance

    • The shortest distance between any two points on the globe sits on a circumference of the globe, called a great circle path
      • The equator, all meridians, and any other circumference is a great circle path
      • On a map, the shortest distance often looks like a straight line, but depending on the projection and the points, it may be drawn as a curved line

  • Direction

    • Directions (azimuths) are measured as angles from meridians
    • A rhumb line is a line crossing all meridians of longitude at the same angle (a line of constant bearing) (eg. the β\beta in the below image)
      • They are useful when relying on a compass
      • Mercator projection is notable since a straight line between two points is a rhumb line

Rhumb lines vs. Great Circles

  • A rhumb line / line of equal bearing differs from a great circle
  • Rhumb line: lines of equal angles to meridians
  • Great circles: lines of shortest distance on circumference, having changing angles as you cross meridians

Distortions of Four Globe Characteristics

  • Shape and Areas

    • Shape: the shape of the polygons odes not hold true; i.e. a square gets stretched to a rectangle
    • Area: the ratio of sizes of polygons on the globe and on the map does not hold true

  • Distance and Direction

    • Distance: the ratio of lengths of lines between points on the globe and no the map does not hold true
    • Direction: the angles between points on the globe does not hold true

Projection Parameters

  • Four Projection Parameters:
    • Developable Surace
    • Point of Projection
    • Point/Line(s) of Contact
    • Aspect

Developable Surface

  • The developable surface (also called the projection surface) is the 2D surface that is the result of the mathematical operation of projection
  • Three types of developable surface:
    • cylinder developable surface
    • conic developable surface
    • Planar developable surface

Point of Projection

  • The point of projection is the point from which the projection is initiated / the perspective point

Point of projection is center of ellipsoid

Point of projection is antipode(opposite point on ellipsoid) to point of contact

Point of projection is infinity

Aspect

  • Orientation of the developable surface to the globe

Point/Line(s) of Contact

  • two type of contact
    • Tangent
    • Secant

Standard Line: Line of True Scale

  • lines of contact are lines of true scale - the ratio of distance between the ellipsoid and line of contact hold true

General Principle for choosing a Map Projection

  • Consider what type of accuracy is most important for your purpose
  • Remember a map projection is always most accurate - in all respects - near its center

Projected Coordinate System

Benefits of PCS

  • PCS make communication and calculation easier