NOTATION: I = (1, 0, 0), J = (0, 1, 0), K = (0, 0, 1). O = (0,0,0).
What is the definition of a circle on a sphere? Is a circle always the intersection of the sphere with a plane? Why does a circle on a sphere have two centers? If a sphere of radius R intersects a plane at distance d from the center of the sphere, what is the size and shape of the intersection?
Define a great circle e on a sphere. If a second great circle f passes through a center of e, what is the angle between the two circles?
Explain why a great circle is considered straight on a sphere and other circles are not. (You can give a couple of informal reasons.)
At what points does the line through (1, -2, -1) and (-1, 2, 1) intersect the sphere of radius 1 with center (0,0,0)?
Given I, J, K, the triangle IJK is one of the 8 triangles that are faces of an octahedron with vertices on the coordinate axes. What is the midpoint M of segment KI in space and what is the midpoint N of segment KJ (i.e., what are the space coordinates)? What is the distance in space from M to N?
Given I, J, K, the spherical triangle IJK on the sphere of radius 1 with center O is an equilateral 90-90-90 degree spherical triangle. What are the space coordinates of the point M' of the point on the sphere that is the midpoint of the side KI of the spherical triangle. What are the space coordinates of the point N' of the point on the sphere that is the midpoint of the side KJ of the spherical triangle. What is the spherical distance from M' to N'?
Given the circle with equation x + y + z = 1 on the sphere of radius 1 with center O, what are the space coordinates of the centers of this circle on the sphere?
Define stereographic projection. You can assume that your sphere is in coordinate space if you like.
Given a figure that shows a circle e. Imagine that this circle is the stereographic image of the equator of a sphere. So we think of e as the equator and think of the center of projection on the sphere as N, the North Pole.
(a) Explain why the stereographic image of any great circle intersects e in two points P1 and P2 so that P1, P2, and the center of e are collinear.
(b) Explain why the stereographic image of a circle through N is a line
(c) Construct with straightedge and compass the stereographic image of the "circle of latitude" of points that are spherical distance 60 degrees from N on the sphere.
(d) Suppose a point P is on circle e and Q is a point in the plane not on e. Construct a circle in the plane through P and Q which is the stereographic image of a great circle.
Define the spherical excess of a spherical triangle. State the relationship between the angle sum of a spherical triangle and its area.
Let ABC be an equilateral spherical triangle. Let A', B', C' be the midpoints of the sides.
(a) Is A'B'C' an equilateral triangle?
(b) How do the angles of A'B'C' compare with the angles of ABC? Can you say for certain whether they are always bigger, the same, smaller?
(c) Is AB'C' an equilateral triangle?
(d) How does angle AB'C' compare with angle ABC? Can you say for certain whether it is always bigger, the same, smaller?
Given an angle drawn in the plane, it can be cut out and the sides can be taped together to form a cone. If points A and B are drawn in the angle, draw in the angle a piece of a straight line AB on the cone. Say for example if A and B are about two inches apart, you might be asked to draw a 6 inch piece of the line so that it would be enough to wrap around the cone a couple of times.
In a figure with two parallel lines and points A and B between the lines, one can cut on the lines and tape them together to form a cylinder. Again, draw a long piece of a straight line from A to B. Is there more than one? Can you draw it?