Test 1: Curves

It is time for our first test in order to be sure that everyone reviews some of the fundamental concepts before we move on to surfaces.

This test will be closed to notes/books, but a calculator will be allowed. There will be three parts to the test.
Part 1: Fill in the blank/short answer
Part 2: Calculations and Interpretations
Part 3: Short Proofs

I suggest that you review your class notes, the class highlights page, and go over ASULearn solutions to the projects, and the ASULearn Curve Glossary.

Part 1: Fill in the blank/short answer There will be some short answer questions, such as providing:

  • definitions related to any of the topics in the glossary on curves (Wiki-like entries are on ASULearn)
  • parametrizations, curvature or torsion of "basic" curves such as a circle, line, plane curves y=f(x), or a helix or strake
  • questions similar to previous clicker questions (see the class highlights page), where you fill in a blank rather than answer as a multiple choice. For instance, instead of asking a multiple choice question on -curvature T + torsion B, it could be fill in the blank (with N' as a good answer), or I could ask what N'=?.
  • questions on material from class

    Note: there is often more than one answer possible for fill in the blank questions: choose one response. Full credit responses demonstrate deep understanding of differential geometry. Informal responses are fine as long as they are correct.

    Part 2: Calculations and Interpretations There will be some by-hand computations and interpretations, like
  • Solving for the scalar curvature of a plane curve using the formula we derived for y=f(x) plane curves: (|f''|/|(1+f'^2)^(3/2))|
  • Finding T(t), T(s) and curvature (vector and scalar) for a curve that is not of the form y=f(x).
  • Finding B and tau, given T and N
  • Finding N, given T
  • Interpreting results, like recognizing that a line is the shortest distance between two points in Euclidean geometry, tau=0 is planar, k=0 is a line, constant postive scalar curvature and planar is part of a circle, constant tau/scalar curvature is a circular helix...

    Part 3: Short Proofs There will be some short proofs - the same as we've seen before. Review the following:

  • prove the derivative of a unit vector is perpendicular to the original vector if it is not the 0 vector
  • The proofs of the Frenet equations. You would be given one short part, such as
            prove T' has no component in the B direction
            prove that T' has a scalar curvature component in the N direction
            or a similar part of a proof for T', N' or B'
  • curvature of a curve is 0 iff the curve is a line
  • the Darboux derivations from the homework

    You should know the results of other statements we proved in class, which could be asked about in the first two sections, but I won't ask you for any other complete proofs, other than those listed here.