Piecewise rational rotation-minimizing motions via data stream interpolation

When a moving frame defined along a space curve is required to keep an axis aligned with the tangent direction of motion, the use of rotation-minimizing frames (RMF) avoids unnecessary rotations in the normal plane. The construction of rigid body motions using a specific subset of quintic curves with rational RMFs (RRMFs) is here considered. In particular, a novel geometric characterization of such subset enables the design of a local algorithm to interpolate an assigned stream of positions, together with an initial frame orientation. To achieve this, the translational part of the motion is described by a parametric <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Superscript 1"> <mml:semantics> <mml:msup> <mml:mi>G</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">G^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> spline curve whose segments are quintic RRMFs, with a globally continuous piecewise RRMF. A selection of numerical experiments illustrates the performances of the proposed method on synthetic and arbitrary data streams.