Reminder about vectors. Review the distance formula between two points in 3-space. Also, for the sphere of radius 1 and center (0,0,0), what is the equation for the sphere? The distance between two points A and B in the sphere is the angle t = angle AOB. The cosine of t = dot product of A and B (assuming the lengths of A and B are = 1, since they are on the sphere). Otherwise, divide A and B by their lengths before (or after) taking dot product. Also, recall that the plane ax + by + cz = 0 has normal (orthogonal vector) (a, b, c). You can divide by the length of this vector to get the vector on the unit sphere in the same direction.

1 Circles on the the sphere

Explain carefully by using plane geometry and plane sections of the sphere.

1. Given a sphere S of radius r with center O, if p is a plane whose distance from O is d, show that for 0 < d < r the intersection of the plane p and the sphere S is a circle c. What is the radius of c? If P is the center of c, where is P located? How is line OP related to p? What happens when d = 0?
2. Explain why for any 3 points on the sphere, there is a unique circle through the 3 points. Also, explain why, if two are the points are antipodal (opposite) points, that the circle is a great circle.
3. For the same S, p, and c, show that there is a point E so that all the tangent lines from E to S are tangent at points of c. Tell where point E is located. Tell what are the lengths of the tangents in terms of what you know about S and p. Explain that this means there is a right circular cone with vertex E such that S is tangent to all the generating lines of E. What happens when d = 0?
4. Define Power of a Point with respect to S using the same definition as for a circle. Also define inversion of a point in S. Show that the inversion of E is P.
5. Explain how to construct a new sphere U that is orthogonal to S so that the intersection of U and S is the circle c. (Two spheres are orthogonal if the dihedral angle between their tangent planes is a right angle.)

1. If the radius of the sphere S is r, and the center is O, and the distance |PQ| = h in 3-space, tell what is the spherical distance between P and Q. (This is a theorem about arcs and chords in a circle.)
2. If the radius of a circle on the sphere is tr (spherical distance, where t is angle measured in radians), what is the (Euclidean 3-space) distance of the plane of the sphere from the center of the sphere? Using your answer to the first problem, when what is the circumferece of the circle as a function of t? (The circumference is just an arc length on the plane, so you can find it from the Euclidean radius of the circle.)
3. In the plane: Given point A = (1,0), B = (-1,0) and Q = (0,k), tell the arc length of the arc from A to B on the circle with center Q through A (and thus also through B). Choose the minor (shorter) arc. You can assume that k is nonnegative.)

3 Exercises on Computing distances on the sphere

Let S be the "unit sphere" in (x,y,z) space, i.e., the sphere of radius 1 with center O = (0,0,0).

1. Explain why the 6 intersection points of the coordinate axes are the vertices of an octahedron made of 8 (Euclidean) equilateral triangles in 3-space.
2. What is the length of an edge of one of these triangles?
3. What is the spherical distance (in degrees or radians) between two of the vertices of one of these triangles?
4. What is the spherical area of one of these triangles, as a fraction of the area of S?
5. What is the sum of the interior angles of this triangle?

5 Tessellations of the sphere - part 2

Let S be the "unit sphere" in (x,y,z) space, i.e., the sphere of radius 1 with center O = (0,0,0).

The rays from O through the centers of these 8 triangles intersect the sphere in 8 points. These are the vertices of a cube each of whose edges is parallel to one of the coordinate axes.