On $L^2$ approximation by spatial Pythagorean-hodograph curvesFarouki, Rida T. (Avtor) Knez, Marjetka (Avtor) Vitrih, Vito (Avtor) Žagar, Emil (Avtor) $L^2$ approximationquaternionsPythagorean-hodograph curvesB-splinespreimageTwo 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.20272026-07-31 08:35:15Neznano31390sl