Sved, Introduction and Chapter 1

Experiment with a physical model of a sphere and write a half-page answering the following questions: Given two points, is it true that there is always a great circle through the two points? Is this circle unique? How can you prove these assertions?

You will be assigned one of these problems to answer for Friday. The first draft should be turned in as a written or printed document Friday. The final draft (after your problem is presented) should be turned in as a Word document on the next class after your problem is presented.

Distances on a sphere of radius R can be measured in terms of degrees, with 360 degrees in a great circle; the distance can also be measured in radians. If A and B are points, they lie on the great circle, and the two points cut the circle into two arcs. The distance between points A and B is the minimum of the arc angles of these great circle arcs.

The chordal distance between A and B is the Euclidean 3-dimensional distance between the points.

1. On a sphere of radius R, if the spherical distance between A and B is t, what is the chordal distance? (The answer is a formula.)
2. In (x,y,z) space, given the sphere S with center (0,0,0) and radius 1, what is the chordal diameter of circle which is the intersection of S with the plane z=a? What is the diameter using spherical distance? What is the circumference of the circle in degrees?
3. In (x,y,z) space, given the sphere S with center (0,0,0) and radius 1, the plane ax + by + cz = d intersects S in a circle. What is the center of this circle? What is the center if you change d but leave a, b, c unchanged? For what value of d is the circle a great circle?
4. In (x,y,z) space, given the sphere S with center (0,0,0) and radius 1, let A = (a1, a2, a3) and B = (b1, b2, b3) be two points on S. What is the equation of the plane that intersects S in the great circle through A and B? First you may want to do an example or two, but you should be able to find the general answer if you review your lines and planes.
5. In (x,y,z) space, given the sphere S with center (0,0,0) and radius 1, let A = (a1, a2, a3) be a point on S. What is a formula for the antipodal point of A? On the Earth, what is the antipodal point of Seattle? If the latitude of a point is LA and the longitude is LO, what is the latitude and longitude of the antipodal point.
6. In (x,y,z) space, let A = (a1, a2, a3) and B = (b1, b2, b3) be two points. Find the equation satisfied by the set of points P = (x,y,z) for which the Euclidean distance |PA| = |PB|. Write the equation in such a way that it is clear that this set is a plane through the midpoint of AB.