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.

At the Exam

  • You may make yourself some reference notes on the very small card I hand out. The mini reference card must be handwritten. Think of the card as a way to include some important concepts, computations, or derivations that you aren't as comfortable with. You won't have room for much, so you should try to internalize as much as you can.
  • One scientific calculator or graphing calculator allowed (but no cell phone nor other calculators bundled in combination with additional technologies). I don't see that you would need this, but I know some people like to have it with them.
  • You may have out food, hydration, ear plugs, or similar if they will help you (however any ear plugs must be stand alone--no cell phone, internet or other technological connections)
  • Partial credit will be given, so (if you have time) showing your reasoning or thoughts on questions you are unsure of can help your grade.

    There will be three parts to the exam.
    Part 1: Fill in the blank/short answer
    Part 2: Calculations and Interpretations
    Part 3: Short Derivations/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
  • 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, it would be fill in the blank like
    -curvature T + torsion B = ____________________
    (with N' as a good answer), or I could ask
    N' = ________________________
    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.
  • other questions on material from class

    Part 2: Calculations and Interpretations There will be some by-hand computations and interpretations, like
  • Solving for the scalar curvature of a plane curve
  • Finding T(t), T(s) and curvature (vector and scalar) for a curve
  • 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 positive scalar curvature and planar is part of a circle, constant tau/scalar curvature is a circular helix...

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

  • For a regular curve, show that s(t) has an inverse (and showing how the mean value theorem applies).
  • the derivative of a unit vector is perpendicular to the original vector if neither are the 0 vector
  • prove that B is a unit 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 of the exam, but I won't ask you for any other complete proofs, other than those listed here.