Gwangil Kim

First order hermite interpolation with spherical Pythagorean-hodograph curves

The general stereographic projection which maps a point on a sphere with arbitrary radius to a point on a plane stereographically and its inverse projection have the Pythagorean-hodograph (PH) preserving property in the sense that they map a PH curve to another PH curve. Upon this fact, for given spatialC1 Hermite data, we construct a spatial PH curve on a sphere that is aC1 Hermite interpolant of the given data as follows: First, we solveC1 Hermite interpolation problem for the stereographically projected planar data of the given data in ℝ3 with planar PH curves expressed in the complex representation. Second, we construct spherical PH curves which are interpolants for the given data in ℝ3 using the inverse general stereographic projection.

Hermite Interpolation Using Möbius Transformations of Planar Pythagorean-Hodograph Cubics

We present an algorithm for Hermite interpolation using Mobius transformations of planar polynomial Pythagoreanhodograph (PH) cubics. In general, with PH cubics, we cannot solve Hermite interpolation problems, since their lack of parameters makes the problems overdetermined. In this paper, we show that, for each Mobius transformation, we can introduce an extra parameter determined by the transformation, with which we can reduce them to the problems determining PH cubics in the complex plane . Mobius transformations preserve the PH property of PH curves and are biholomorphic. Thus the interpolants obtained by this algorithm are also PH and preserve the topology of PH cubics. We present a condition to be met by a Hermite dataset, in order for the corresponding interpolant to be simple or to be a loop. We demonstrate the improved stability of these new interpolants compared with PH quintics.

Planar C 1 Hermite interpolation with PH cuts of degree ( 1 , 3 ) of Laurent series

We show how to find four generic interpolants to a C 1 Hermite data-set in the complex representation, using Pythagorean-hodograph curves generated as cuts of degree ( 1 , 3 ) of Laurent series. The developed numerical experiments have shown that two of these interpolants are simple curves and that these (at least) have stable shape, in the sense that their topologies persist when the direction of the velocity at each end-point changes. Our curves are fair, but have different shapes to those of other interpolants. Unlike existing methods, our technique allows regular PH interpolants to be found for special collinear C 1 Hermite data-sets. We introduce a new class of PH curves, PH cuts of degree ( 1 , 3 ) of Laurent series.We show how to find PH skew cut interpolants to a C 1 Hermite data-set.We show that two of these interpolants are short, simple curves with stable shape.Our curves are fair with different shapes to those of other interpolants.We can obtain regular PH interpolants for collinear C 1 Hermite data-sets.

Minkowski Pythagorean-hodograph preserving mappings

We state and prove the sufficient and necessary condition for a mapping to be a scaled MPH-preserving mapping which preserves the MPH property of a curve with rescaling the speed by a rational function in R 2 , 1 , and show how to produce polynomial scaled MPH-preserving mappings from given generating polynomials. We introduce s-cubic MPH-preserving mappings of the first kind, and their corresponding surfaces. We show that these mappings can be used to solve interpolation problems for C 1 Hermite data-sets with admissible velocity vectors on their corresponding surfaces.

Hermite Interpolation with PH Curves Using the Enneper Surface

We show that the geometric and PH-preserving properties of the Enneper surface allow us to find PH interpolants for all regular 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.