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

Abstract We investigate the problem of constructing spatial C 2 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​ C 2 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.

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