Template for Answer to Problem 9-1

Since each answer will be based on a different city, the only way to be able to grade these in a reasonable amount of time is for all answer to fit into a common format from which we can extract the needed data and check your work. (We will either use a spreadsheet or some math software such as Maple. You can use such tools also, so long as you explain what you are doing as laid out below.)

Either print out this form and use it for your answers, or make your answers look like this form.

Part (a) Spherical and Cartesian Coordinates

Suppose P = (x, y, z) is a point on the sphere of radius R.

• Given angle phi = p, tell how to compute z from R and p:

  z = __________________

• Given the same phi, if r is the distance from P to the z-axis, tell how to compute r from R and p:

r = _______________________

• Draw a figure that shows why the two formulas above are true:

• Given angle t, tell how to compute x and y from r and t:

  x = ____________________________

  y = ____________________________

• Draw a figure that shows why the formulas above are true:

Substitute your formula for r to derive formulas for x, y, z in terms of p, t, R.

x = ____________________________

y = ____________________________

z = ____________________________

Part (b) Distance from spherical coordinates

Again, assume that the points are on a sphere of radius R.

• For point A, the spherical coordinates are p = a, t = u, what are the cartesian (x, y, z) coodinates of A?

   

  ________________________________________________________

• For point B, the spherical coordinates are p = b, t = v, what are the cartesian (x, y, z) coodinates of B?

   

  ________________________________________________________

• Using dot product, tell what is the formula for the spherical distance (measured as an angle) from A to B.

   

  __________________________________________________________________

Part (c) Distance between cities

Coordinates of YOUR City

Name of your city: _________________________________

Latitude of your city (degrees north or south, in hundredths of a degree) _____________________

Longitude of your city (degrees east or west, in hundredths of a degree) _____________________

Spherical coordinates of your city (in hundredths of a degree):

phi = _____________   theta = _______________

Assume (temporarily) that units are chosen so that the radius of the earth R = 1. With R = 1, what are the cartesian coordinates (x, y, z) of your city?

x = _____________   y = _______________   z = _______________

Coordinates of SEATTLE

Name of city: ______SEATTLE, WA__________________

Latitude of city (degrees north or south, in hundredths of a degree) _____________________

Longitude of city (degrees east or west, in hundredths of a degree) _____________________

Spherical coordinates of city (in hundredths of a degree):

phi = _____________   theta = _______________

Assume (temporarily) that units are chosen so that the radius of the earth R = 1. With R = 1, what are the cartesian coordinates (x, y, z) of Seattle?

x = _____________   y = _______________   z = _______________

Distance computation

Compute the distance between the cities using ANGLE measure (in degrees) and show work below:

Distance (as angle) = ____________________

Show How You Calculated this::

 

On a sphere with unit of length chosen so that R = 1, what is the spherical distance (measured as arc length) between the two cities (i.e, given the distance in angle measure, what is the distance as arc length)?

Distance (as length) ___________________________________

State the radius of the earth (this is approximate, since the earth is not a sphere, so state a good approximate radius that you will use. Be sure to indicate units -- miles, kilometers, nautical miles).

R = Radius of earth = ______________________________

Using your value for R, what is the spherical distance (as a length) between the two cities, measured in miles, kilometer, or nautical miles (same units as R)?

Distance between Seattle and _________________ is __________________________.