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Abstract. In this paper, the geometric interpolation of planar data points and boundary tangent directions by a cubic G 2 Pythagorean-hodograph (PH) spline curve is studied. It is shown that such an interpolant exists under some natural assumptions on the data. The construction of the spline is based upon the solution of a tridiagonal system of nonlinear equations. The asymptotic approximation order 4 is confirmed.

Abstract. In this paper the problem of geometric interpolation of planar data by parametric polynomial curves is revisited. The conjecture that a parametric polynomial curve of degree ≤ n can interpolate 2n given points in R 2 is confirmed for n ≤ 5 under certain natural restrictions. This conclusion also implies the optimal asymptotic approximation order. More generally, the optimal order 2n can be achieved as soon as the interpolating curve exists.

In this paper, the geometric Lagrange interpolation of four points by planar cubic Pythagorean-hodograph (PH) curves is studied. It is shown that such an interpolatory curve exists provided that the data polygon, formed by the interpolation points, is convex, and satisfies an additional restriction on its angles. The approximation order is 4. This gives rise to a conjecture that a PH curve of degree n can, under some natural restrictions on data points, interpolate up to n + 1 points.

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In this paper, geometric interpolation of certain circle-like curves by parametric polynomial curves is studied. It is shown that such an interpolating curve of degree n achieves the optimal approximation order 2 n, the fact already known for particular small values of n. Furthermore, numerical experiments suggest that the error decreases exponentially with growing n.

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In this paper, three-pencil lattices on triangulations are studied. The explicit representation of a lattice, based upon barycentric coordinates, enables us to construct lattice points in a simple and numerically stable way. Further, this representation carries over to triangulations in a natural way. The construction is based upon group action of S 3 on triangle vertices, and it is shown that the number of degrees of freedom is equal to the number of vertices of the triangulation.