Sunhong Lee

C1 Hermite interpolation with simple planar PH curves by speed reparametrization

We introduce a new method of solving C^1 Hermite interpolation problems, which makes it possible to use a wider range of PH curves with potentially better shapes. By characterizing PH curves by roots of their hodographs in the complex representation, we introduce PH curves of type K(t-c)^2^n^+^1+d. Next, we introduce a speed reparametrization. Finally, we show that, for C^1 Hermite data, we can use PH curves of type K(t-c)^2^n^+^1+d or strongly regular PH quintics satisfying the G^1 reduction of C^1 data, and use these curves to solve the original C^1 Hermite interpolation problem.

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.

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.