This cross-section shows stereographic projection from the sphere with center O of radius R, with center of projection at N to the equatorial plane perpendicular to the diameter ON.
The point B is the stereographic image of A, a point on the sphere.
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 (spherical) 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? What is the relationship between the spherical radius of such a circle (measured in degrees or radians) and the radius of the circle viewed as a Euclidean circle in a Euclidean plane?
Define a "great circle" on a sphere. Explain why a great circle is considered straight on a sphere and other circles are not. (You can give a couple of informal reasons.)
If e is a great circle, and a second great circle f passes through a center of e, how is the angle between the great circles defined? Tell two or three ways that this angle can be measured. (Think tangent lines, dihedral angles, distances between poles).
The (spherical) centers of a great circle are called the poles of the great circle. If a great circle f passes through the poles of a great circle e, how are the circles related? Where are the poles of f in this case?
Define stereographic projection.
Given a plane figure that includes 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 equi-angular triangle? 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 equi-angular triangle? 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?
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This cross-section shows stereographic projection from the sphere with center O of radius R, with center of projection at N to the equatorial plane perpendicular to the diameter ON. The point B is the stereographic image of A, a point on the sphere. |
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·If A is 45 degrees from N in spherical distance, calculate the distance from O to B.
Distance (in terms of R) _____________