Spatial C2 closed loops of prescribed arc length defined by Pythagorean-hodograph curves

We investigate the problem of constructing spatial C 2 closed loops from a single polynomial curve segment r ( t ) , ? [ 0 1 ] with prescribed arc length S and continuity Frenet frame curvature at juncture point = . Adopting canonical coordinates to fix initial/final tangent, closed-form solution for two-parameter family interpolants given data can be constructed in terms degree 7 Pythagorean-hodograph (PH) space curves, torsion is also obtained when one parameters set zero. The geometrical properties these closed-loop PH curves are elucidated, certain symmetry degenerate cases identified. closed-loop? used construct swept surfaces tubular surfaces, selection computed examples included illustrate methodology. • defined by curve. Continuity such considered. used. Geometrical investigated. Swept constructed.