Math 402, Assignment 3, Spring 2008

Due Wed. February 6(at the beginning of class)

    For the questions on this homework, consider a 160 degree cone. Make one out of paper or light cardboard by taping the edges so that they overlap about 20 degrees -- it does not need to be perfect.

  1. For this question, consider only curves which avoid the point of the cone. What curves on the cone are straight lines? What do they look like? How many different types are there? What properties of geodesics do they have? What do they lack?

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  2. Now, on the same cone, consider a curve which goes straight up the cone, through the point, and then back down the opposite side so that it divides the cone into two equal pieces. What properties of a geodesic does this curve have? What properties does it lack? In comparison, fix a point A and look at all the lines through A as they get closer and closer to going through the vertex. What curve do you get in the limit? Does it depend on which lines you consider? Do you get the line described above? If not, which properties of a geodesic does this curve have, and which does it lack. Finally, which curve or curves (if any) going through the cone point do YOU think should be called lines? Explain.

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  3. Despite the many reservations described above, for the sake of uniformity we will now declare that the first curve described in the last question is a straight line, and that these are the only straight lines that go through the cone points. For each part of the five axioms, decide whether it is true or false for this space. Explain how you know that it is true and/or exactly when and how it fails. Draw pictures where appropriate.