Suppose S is the sphere of radius R with center O = (0,0,0) in 3-space.
If A and B are on S, then there is a distance |AB| from A to B in space, and a spherical distance which is proportional to angle AOB.
Suppose S is the sphere of radius R with center O = (0,0,0) in 3-space. Let ABC be a spherical triangle on S, with the angle at C a right angle.
If a, b, c are the spherical lengths of the sides of ABC, find a formula relating these three quantities.
Hint: Put c = (0,0,R) and let A and B lie in coordinate planes. You can find the spherical distance between points on the sphere by using dot product to find angle AOB, etc.
ASSUME THIS
For this assignment you can assume that given two points A and B in 3-space, the set of points equidistant from A and B is a plane M orthogonal to line AB through the midpoint of AB, called the perpendicular bisecting plane. Also, reflection of a point A in a plane q is defined to be the point A' such that q is the p perpendicular bisecting plane of AA'.
These questions are all about geometry on a sphere with center O and radius R; all distances, triangles, etc. are spherical unless specified to the contrary. In the proofs, try to do your best job of clear thinking, legible writing, and good illustrations.
(a) Given two points A and B on the sphere, prove that the set of points equidistant from A and B is a great circle m.
(b) If c is a great circle through A and B, at what points does the great circle m above intersect c? What does this say about the concept of a midpoint of a segment on the sphere?
(c) If c is a great circle through A and B, prove that c is orthogonal to the great circle m above.
Definition: The (spherical) perpendicular bisector of AB is the great circle m that is the set of points equidistant from A and B.
6.4 Concurrence of perpendicular bisectors
(a) Prove that the perpendicular bisectors of the sides of a spherical triangle ABC are concurrent at a point P. (Is P the only point?)
(b) Prove that for any spherical triangle, there is a circle d that passes through A, B, and C.
(c) Given a triangle ABC, the 3 great circles that form the sides of ABC cut the sphere into 8 triangles, each of which has perpendicular bisectors of the sides.. How many distinct perpendicular bisectors are there for the 8 triangles altogether. (Refer to your construction portfolio.)
6.4 Concurrence of angle bisectors
(a) Prove that the angle bisectors of the angles of a spherical triangle ABC are concurrent at a point P. (Is P the only point?)
(b) Prove that for any spherical triangle, there is a circle d that is tangent to the 3 sides..
(c) Given a triangle ABC, the 3 great circles that form the sides of ABC cut the sphere into 8 triangles, each of which has angle bisectors of the angles. How many distinct great circles are there that are angle bisectors for these 8 triangles. (Refer to your construction portfolio.)
6.5 Line Reflection
(a) If d is a great circle and A is a point on the sphere, explain why the reflection of A in the plane containing d is a point A' that is also on the sphere.
(b) Is is true that a great circle e through AA' is orthogonal to d? Why?
Suppose the sphere is in (x,y,z) space with center (0,0,0), and p1, p2, p3 are the great circles that are intersections of the sphere with the coordinate planes, x = 0, y = 0, z = 0.
(c) Describe clearly each of the transformations that are the composition of reflection in two of these great circles.
(d) Describe clearly each of the transformations that are the composition of reflection in three of these great circles