The relationships between certain families of special curves, including the
general helices, slant helices, rectifying curves, Salkowski curves,
spherical curves, and centrodes, are analyzed. First, characterizations of
proper slant helices and Salkowski curves are developed, and it is shown
that, for any given proper slant helix with principal normal n, one may
associate a unique general helix whose binormal b coincides with n. It is
also shown that centrodes of Salkowski curves are proper slant helices.
Moreover, with each unit-speed non-helical Frenet curve in the Euclidean
space E3, one may associate a unique circular helix, and characterizations
of the slant helices, rectifying curves, Salkowski curves, and spherical
curves are presented in terms of their associated circular helices. Finally,
these families of special curves are studied in the context of general
polynomial/rational parameterizations, and it is observed that several of
them are intimately related to the families of polynomial/rational
Pythagorean-hodograph curves.
On the variety of rational space curves
