Arc length preserving approximation of circular arcs by Pythagorean-hodograph curves of degree seven

Žagar, Emil

doi:10.48550/arxiv.2201.07445

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Quadratic Functionals1

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2023

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“…There are some recent attempts of interpolation with (polynomial) PH curves of prescribed length s > 0 [25][26][27]. Assuming that exact and feasible solutions exist in the interpolation space Q, they will be found by minimizing the quadratic functional…”

Section: Quadratic Functionalsmentioning

confidence: 99%

Optimal interpolation with spatial rational Pythagorean hodograph curves

Schröcker¹,

Šìr²

2023

Preprint

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Using a residuum approach, we provide a complete description of the space of the rational spatial curves of given tangent directions. The rational Pythagorean hodograph curves are obtained as a special case when the norm of the direction field is a perfect square. The basis for the curve space is given explicitly. Consequently a number of interpolation problems (G 1 , C 1 , C 2 , C 1 /G 2 ) in this space become linear, cusp avoidance can be encoded by linear inequalities, and optimization problems like minimal energy or optimal length are quadratic and can be solved efficiently via quadratic programming. We outline the interpolation/optimization strategy and demonstrate it on several examples.

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Section: Quadratic Functionalsmentioning

confidence: 99%