Reading: Read Chapter 7. Learn thoroughly the area formula for triangles and the definition of holonomy on the sphere. Do not worry about hyperbolic geometry yet.

Reminder. 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.

1. What is the equation of the sphere of radius 1 and center O = (0,0,0)?
2. If P = (1,2,3), what is the point that is the intersection of the ray OP and the sphere above?
3. If Q = (-1, 3, 4), what is angle POQ? (Use a calculator to get a numerical answer.)
4. If A = (1,0,0), B = (0,1,0), and C = (0,0,1), what is the point P at the center of triangle ABC? What is the opposite (antipodal) point of P?
5. If the equation of a great circle c is x – y + 2z = 0, what are the poles of this great circle?
6. If the equation of a great circle d is x + y + z = 0, what are points of intersection of c and d?

1. Let S be the "unit sphere" in (x,y,z) space, i.e., the sphere of radius 1 with center O = (0,0,0).
2. 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.
3. What is the length of an edge of one of these triangles?
4. What is the spherical distance (in degrees or radians) between two of the vertices of one of these triangles?
5. What is the spherical area of one of these triangles, as a fraction of the area of S?
6. What is the sum of the interior angles of this triangle. Show that this fits with the area formula.

1. 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.
2. What are the coordinates of these vertices?
3. What is the (Euclidean) distance between two endpoints of an edge of this cube?
4. What is the spherical distance between the same two points?
5. What is the equation of one great circle that passes through two vertices?

1. What is the spherical area (as a fraction of the area of S) of one of the spherical quadrilaterals formed by the vertices of the square face of a cube?
2. What is the sum of the interior angles of this quadrilateral. Show that this fits with the area formula.

1. Show that it is possible to pick out 4 of the 8 vertices of the cube so that these vertices form a regular tetrahedron. (Hint: Pick vertices whose distance is the length of a diagonal of a square face of the cube.)
2. Write down the coordinates of 4 vertices from the cube above, so that these are the vertices of a tetrahedron.
3. What is the (Euclidean) distance between two endpoints of an edge of this tetrahedron?
4. What is the spherical distance between the same two points?
5. What is the spherical area (as a fraction of the area of S) of one of the spherical triangles formed by the vertices of a face of a tetrahedron?
6. What is the sum of the interior angles of this triangle?. Show that this fits with the area formula.