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

We investigate the problem of constructing spatial C2 closed loops from a single polynomial curve segment r(t), t∈[0,1] with a prescribed arc length S and continuity of the Frenet frame and curvature at the juncture point r(1)=r(0). Adopting canonical coordinates to fix the initial/final point and tangent, a closed-form solution for a two-parameter family of interpolants to the given data can be constructed in terms of degree 7 Pythagorean-hodograph (PH) space curves, and continuity of the torsion is also obtained when one of the parameters is set to zero. The geometrical properties of these closed-loop PH curves are elucidated, and certain symmetry properties and degenerate cases are identified. The two-parameter family of closed-loop​ C2 PH curves is also used to construct certain swept surfaces and tubular surfaces, and a selection of computed examples is included to illustrate the methodology.