Math 445 Midterm Test

NAME ___________________________

Problem 1

Suppose a spherical triangle ABC has angle = 80 degrees at each vertex.  If the surface area of the sphere is S, what is the spherical area of this triangle?

Problem 2

(a)    Find the point C on line AB so that AC/CB = -3/7, where A = (1, 1, 1), B = (0, 2, 3).  It is OK to leave your work "unsimplified" or "factored" so long as it is numeric.

C =

(b)   At what point D does line AB intersect the (x,y, 0) plane?

D =

Problem 3

Suppose one makes a cylinder by cutting out the strip between the two parallel lines and taping the edges together.  Suppose that you draw two segments of straight paths (i.e., "lines" or geodesics) on the cylinder from A to B. Then you cut the tape and flatten the strip again. 

·        Draw below what you would see of the two segments. You should show the work that you did to make the drawing accurate, but the drawing does not have to be a straightedge and compass construction. (Of course there is more than one correct answer here.)

Problem 4: Barycentric Coordinates

(a)    This lattice of points in this figure is made of parallelograms, as it appears. Find the indicated barycentric coordinates and write the answers in the table below.  Most answers are by inspection, but a couple will require computation.

(b)   Suppose the plane of this triangle is projected to another plane, and point X is projected to X' = (2, 0, 0), Y is projected to point Y' = (1, 1, 1) and Z is projected to point Z' = (0, 0, -1).  To what point W' is point W projected?

Problem 5

Let I = (1, 0, 0), J = (0, 1, 0), K = (0, 0, 1).

(a)    In 3-space, let M be the midpoint of segment KI and let N be the midpoint of segment KJ.  What are the 3-space coordinates of point M and point N?

M =

N =

(b)   What is the (3-space) distance MN?

Let S be the sphere of radius 1 with center O = (0, 0, 0).  Points I, J and K are on this sphere.

(c)    Let M' be the spherical midpoint of the spherical (great circle) segment KI. Let N' be the spherical midpoint of the spherical segment KJ.  What are the 3-space coordinates of M' and N'?

M' =

N' =

(d)   What is the spherical distance M'N'? (measured in degrees or radians)