Lecture 10 - Map Projections

Question 1

Outline the processing flow of measurements.

Answer 1

1. The Earth’s surface: a complex plane
2. Independent handling of the horizontal and the vertical
3. Globally or regionally best fit ellipsoids
4. Geodetic datum: position, orientation, shape and size of ellipsoids
5. Map projections: a flat representation of the Earth’s surface

Question 2

Explain what a map projection is.

Answer 2

A mathematical model for conversion of locations from a 3D surface of the Earth to a 2D planar map. Project the ellipsoidal surface to a developable surface. This conversion necessarily distorts some aspects of the Earth’s surface such as area, shape, distance or direction.

Question 3

What are possible distortions with map projection?

Answer 3

Area, shape, distance, direction.

Question 4

Why do distortions occur?

Answer 4

Because a non-developable ellipsoid surface is represented by a developable flat surface. There are 3 fundamental geometrical elements considered in map projections (distance, angle, area, shape) and which of the elements will be kept undistorted is chosen by the user.

Question 5

What is a developable surface?

Answer 5

A geometric shape that can be laid out into a flat surface without stretching or tearing.

Question 6

What are the three most commonly used developable surfaces?

Answer 6

Planar, conic, cylindrical.

Question 7

Describe Tissot’s indicatrix.

Answer 7

An ellipse of distortion with a systematic approach to quantitatively calculate distortion.

Question 8

Describe Tissot’s ellipse.

Answer 8

A mathematical measurement of to characterize distortions due to map projection. It is a shape of an infinitesimally small circle centered on some point on the Earth’s surface after being transformed by a given map projection.

Question 9

Explain direction, angle, area and distance distortions using the distortion ellipse.

Answer 9

A circle ABCD defined in a model of Earth.
A’B’C’D’ is the Tissot’s indicatrix resulted from its projection on the plane.
OA –> OA’, and OB –> OB’ so linear scale is not conserved along these two directions resulting in a distortion of distance
Angle MOA –> M’OA’ in the distortion ellipse M’OA’ < MOA which shows an angular distortion.
The area of circle ABCD = 1 –> A’B’C’D’=< which shows a distortion of area.

Question 10

Summarize classification of map projections.

Answer 10

With a, b, being principle scale factors.

1. Conformal: preserving local shapes; a-b=0.
2. Equidistant: preserving distance from a single point to all other points; a=1 or b=1.
3. Equal area: preserving areas; ab=1.
4. Azimuthal: preserving directions from a single point to all other points.
5. Gnomic (great circles are straight lines) and Stereographic (a conformal projection).

Question 11

General requirements of map projections.

Answer 11

Preserve directions/angles after map projection for avoiding large scale projection calculations.

Maintain the similar shapes after projection for convenience with map reading and application.

Sustain small distortion with distances and areas for simple formulas in calculation.

Ensure the unified coordinates w.r.t. the same initial point for a region, country or the world.

Question 12

Gauss-Kruger projection

Answer 12

The transverse tangent cylindrical conformal projection.

Question 13

UTM projection

Answer 13

The transverse secant cylindrical conformal projection. Used to compile topographic maps.

Question 14

How to define scale factor?

Answer 14

Accounts for the difference between distances on a curved ellipsoids to distance projected on a map.

Question 15

What is isometric latitude?

Answer 15

Directly proportional to the spacing of latitude parallels from equator on an ellipsoid Mercator projection.