Assignment 5B

Problem 5-2

Let E be a sphere of radius R with center O.  Suppose F is a plane at distance d from O.

• For what d is the intersection of F and E a circle c?  What is the radius of the circle?  For what d is F a tangent plane to E?
•  If Z is the center of the circle c, and P is a point on c, what is the angle ZOP (answer is an expression on R and d)?
• If the circle c is not a great circle, there is a right circular cone with vertex W on line OZ so that the cone is tangent to E along c.  What is the distance OW?  What is the distance from W to any point P on c?

Show your work for all parts. Make drawings as needed.

Problem 5-3

Review the vector concept of dot product and how the equation of a plane Ax+By+Cz = D can be written using dot product.  In particular review the fact that the vector [A, B, C] is orthogonal to the plane.

Let E be the sphere of radius 1 with center (0,0,0). 

• If W is the plane with equation x + y + 2z  = 0, demonstrate why the intersection c of W with E is a great circle. 
• Find the centers (poles) of this circle on the sphere.
• Find the equations of the two planes parallel to W that are tangent to the sphere.
• Find the equation of the circle that is the stereographic projection of c onto the z=0 plane.

Problem 5-4

Let E be the sphere of radius R with center (0,0,0). 

• If a cube is inscribed in the sphere with edges parallel to the coordinate axes, what are the coordinates of the vertices of the cube?
• Give the equation of the plane of one of the great circles that passes through the endpoints of the edge of the cube.
• Give the equation of the plane of a great circle that is the perpendicular bisector of the edge above.

Problem 5-5 Area on the sphere

Let E be a sphere of radius R.  Suppose that the surface area of E is A (there is a formula but we don't need it for what we will do here).

Example: What is the area of a hemisphere as a fraction of A? (Answer = (1/2)A)

• Let N and S be north and sound poles and let c and d be 180 arcs from N to S that meet at an angle t.  What is the area of the part of the sphere between the two arcs (the region between two arcs is called lune)?
• Let two 90-degree arcs u and v run from N to points U and V on the equator.  If the arcs meet at an angle t, what is the area of the spherical triangle UNV?  What are the angles of the triangle and what is  the angle sum?