We will put coordinates on the sphere with the mathematical/scientific convention for spherical coordinates. For a point P = (x, y, z), the angle t = theta = angle measure of (x, y, 0) using ordinary polar coordinates (positive angle in counterclockwise direction). Angle p = phi = angle that radius OP makes with ON, the radius to the north pole. Assume radius of sphere = R.
(a) Derive and write a formula for the (x, y, z) coordinates of a point on this sphere with angles t and p. Hint: Figure out the polar coordinates of (x, y, 0) and separately figure out z.
(b) Explain how to use this formula and the dot product to find the distance in degrees between two points given their spherical coordinates.
(c) You will pick or be assigned a city. Compute the distance from your city to Seattle, assuming the earth is a sphere. First compute the distance in degrees, then in kilometers or miles (or nautical miles).
Take a figure with two points A and B and construct the following objects – all objects are P-objects. Your figure disk should take up about half the page; no micro figures please. Also, write briefly the steps in each construction.
(a) Perpendicular bisector of AB
(b) Circle with diameter AB
(c) Midpoint of AB
Construct a typical P-line AB in the disk model. Then construct at least 4 points on this line on each side of A and B so that all the segments between successive points are P-congruent to AB. Hint: Use reflection or intersection with circles.
(a) Write a clear answer to Sved, Problem 4-5, and
(b) also construct an accurate figure in the disk model illustrating this proof.
(c) Finally, construct a figure that answer this question that Sved poses: Given an angle ABC that is acute, for any D on ray BC, construct the line d through D perpendicular to line AB. Does the intersection of d with ray AB always exist?
Construct a Saccheri quadrilateral in the P-model. Use this model to illustrate and answer the questions in Sved, Problem 4-6.