Abstract
We show that the geometric and PH-preserving properties of the Enneper surface allow us to find PH interpolants for all regular C1 Hermite data-sets. Each such data-set is satisfied by two scaled Enneper surfaces, and we can obtain four interpolants on each surface. Examples of these interpolants were found to be better, in terms of bending energy and arc-length, than those obtained using a previous PH-preserving mapping.
Highlights
Pythagorean-hodograph (PH) curves were first introduced by Farouki and Sakkalis [1] as polynomial curves in R2 with polynomial speed functions, which have polynomial arclengths, rational curvature functions, and rational offsets, all of which derive from their polynomial speed functions
C1 Hermite interpolation problems have been solved by several techniques [13, 24,25,26,27] including PHpreserving mappings [24], which have recently been extended [13] to MPH-preserving mappings
We went on to show how to produce interpolants which lie on two sEnneper surfaces and satisfy a regular C1 Hermite data-set in R3
Summary
Pythagorean-hodograph (PH) curves were first introduced by Farouki and Sakkalis [1] as polynomial curves in R2 with polynomial speed functions, which have polynomial arclengths, rational curvature functions, and rational offsets, all of which derive from their polynomial speed functions These properties make PH curves good candidates for CAGD and CAD/CAM applications such as interpolation of discrete data and control of motion along curved paths [2,3,4]. PH curves, there have been many developments: in particular, it has been shown [24] that C1 Hermite interpolation problems with PH curves in R3 can be reduced to problems in R2 and generic interpolants can be obtained to satisfy a given C1 Hermite data-set This is achieved by a special cubic PH-preserving mapping which satisfies the data-set.
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