Skip to main content
Open Access Publications from the University of California

UC Davis

UC Davis Previously Published Works bannerUC Davis

Construction of periodic adapted orthonormal frames on closed space curves

  • Author(s): Farouki, RT
  • Kim, SH
  • Moon, HP
  • et al.

The construction of continuous adapted orthonormal frames along C closed–loop spatial curves is addressed. Such frames are important in the design of periodic spatial rigid–body motions along smooth closed paths. The construction is illustrated through the simplest non–trivial context — namely, C closed loops defined by a single Pythagorean–hodograph (PH) quintic space curve of a prescribed total arc length. It is shown that such curves comprise a two–parameter family, dependent on two angular variables, and they degenerate to planar curves when these parameters differ by an integer multiple of π. The desired frame is constructed through a rotation applied to the normal–plane vectors of the Euler–Rodrigues frame, so as to interpolate a given initial/final frame orientation. A general solution for periodic adapted frames of minimal twist on C closed–loop PH curves is possible, although this incurs transcendental terms. However, the C closed–loop PH quintics admit particularly simple rational periodic adapted frames. 1 1 1 1

Many UC-authored scholarly publications are freely available on this site because of the UC's open access policies. Let us know how this access is important for you.

Main Content
Current View