Math Prompts for 12th Sept
Warm up
- A bear walks twenty miles south, twenty miles east and twenty miles north and ends up where it started. What colour is the bear? Have you found all of the theoretically possible starting points for the bear?
- Draw a triangle on a sphere whose angles add to \(270^{\circ}\). Do triangles on a sphere always have this property? Can you say anything about the sum of the angles in a triangle on a sphere in general?
Distances
lon lat
Marlboro -72.73475 42.83905
Grad School -72.55697 42.85073
Queen Mary Math Building -0.040645 51.52334
- How far is is from Marlboro to the grad school? How far to Queen Mary?
You'll need some trig.
This page has the formulas you'll need, and we'll talk more about them in class. Start by working out the "flat earth" distance (convert the lon and lat into miles from the equator/prime meridian and use Pythagoras; take the radius of the Earth to be 3956 miles) and then try it with both the cosine and haversine methods. Can you justify from plane trig why the latter two are correct? [hard!]
Projections (maybe for a future class)
Today we'll play with cylindrical and pseudo-cylindrical projections. Crucial property: the lines of latitude are horizontal. If the lines of longitude are vertical (and hence the map is rectangular) then it's cylindrical; otherwise it's pseudo-cylindrical. All projections (today) are tangent ones not secant ones (i.e. the cylinder doesn't cut the sphere, but touches it all around the equator).
- Consider a cylindrical map and take the equator to be to scale. Project out horizontally. Choose a latitude. By what factor is the scale at this latitude off? Repeat for several latitudes. Can you find a general rule?
- Consider the cylindrical map geometrically projected from a point at the center of the sphere. How are the lines of latitude spaced? What does this imply for areas on the map? For direction?
- Consider the cylindrical map geometrically projected from a point at infinity. How are the lines of latitude spaced? What does this imply for areas on the map? For direction?
We get non-geometrical projections by giving a rule for the spacing and lengths of the lines of latitude. Varying the lengths moves us into pseudo-cylindrical territory.
- Experiment with different options for spacing and lengths for the lines of latitude. What properties do your maps have? (Good place to start: evenly spaced lines of latitude of lengths that are to scale with the equator, see the earlier question on this. What shape is the map?) The wikipedia page for map projections gives a good range to explore, but try to do it for yourself as much as possible: you might well find you've recreated some famous ones.
