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Opis: Two methods for the ▫$L^2$▫ approximation of smooth space curves by spatial Pythagorean-hodograph (PH) curves are considered. The first method employs direct approximation in ▫${\mathbb R}^3$▫, which results in a non-linear system of equations that may be solved by a numerical iteration or optimization scheme. The second method performs the optimization in the quaternion preimage space of PH curves, which results in a linear system of equations. The preimage of a curve in ▫${\mathbb R}^3$▫ is a surface in the quaternion space, and an appropriate locus on this surface must be identified for the given space curve. This is accomplished by observing that PH curves equipped with a rational rotation-minimizing frame (RMF) have preimages that are geodesic loci on the surface, and computing a discretized approximation to the RMF on the given curve. The methods are also extended to the approximation of spatial B-spline curves. Detailed algorithm descriptions are provided for both methods, and several computed examples illustrate their performance.
Ključne besede: ▫$L^2$▫ approximation, quaternions, Pythagorean-hodograph curves, B-splines, preimage
Objavljeno v RUP: 17.08.2026; Ogledov: 217; Prenosov: 8
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