Gauss-Bonnet's Formula and Closed Frenet Frames

    Project Details


    Closed space curves with
    non-vanishing curvature
    defines via the Frenet
    formulas some closed curves
    on the unit 2-sphere, called
    the spherical indicatrices.
    By using Gauss-Bonnet's
    Formula after cutting a
    spherical curve into simple
    closed sub-curves an index on
    the unit 2-sphere is found.
    This spherical index may be
    seen as a spherical analogy
    to the winding number of
    closed plane curves. The
    spherical index has the
    property that the integral
    over the unit 2-sphere
    of this index, equals the
    integrated geodesic
    curvature of the spherical
    curve. Using this result on
    the spherical indicatrices of
    a space curve, we obtain
    almost similar proofs of some
    (generalizations of)
    classical theorems. The
    spherical index gives both
    upper and lower bounds on the
    total curvature and
    total torsion of space curves.
    Effective start/end date01/01/199631/01/1999