Math 445 Assignment #4

These problems are a return to a further look at spheres. The important fact to use is that for two vectors A and B, the cosine of the angle between them is the dot product divided by |A||B|. In case the sphere has radius 1, this is just the dot product, so the arc cosine of the dot product is the spherical distance (measured in radians or degrees). Look this up if you need to review it.

Chapter 3 will be explored in the lab on Wed 1/26. We will start Chapter 4 on Monday.

Assignment 4 (60 Points) Due Wed 1/26/00

  1. (10 points) Any cube can be inscribed in a sphere. If the cube is inscribed in the unit sphere, (a sphere of radius 1) what is the spherical distance between adjacent vertices? (Hint: The 8 points with (x,y,z) coordinates +1 or –1 are the vertices of a cube. Divide by their length to get a sphere of radius 1.
  2. (10 points) If one chooses a certain 4 vertices of a cube, these 4 points are the vertices of a regular tetrahedron. What is the spherical distance between these vertices?
  3. (10 points) The vectors of length 1 on the x, y, and z axes are the 6 vertices of an octahedron inscribed in a unit sphere. What is the distance between adjacent vertices? What kind of spherical triangles are formed by these vertices?
  4. (15 points) Consider the spherical triangle formed by A = (1, 0, 0), B = (0, 1, 0), C = (0, 0, 1). Let A’, B’ C’ be the midpoints of the great circle arcs BC, CA, AB. Find the lengths and vertex angles of the triangle A’, B’, C’. Find the lengths and vertex angles of AB’C’. Are both triangles equilateral triangles?
  5. (15 points) Let A and B be two points in the plane. The circles c and d are circles through A and B. The circles e and f are circles orthogonal to c and d. If m is a circle with center A, invert all the circles to c’, d’, e’, f’. Sketch the figure made up of these inversion images and explain carefully how they are related to each other and point B’.