The Pythagorean hodograph (PH) curves are polynomial parametric curves {x(t),y(t)} whose hodograph (derivative) components satisfy the Pythagorean condition x' 2 (t) + y' 2 (t) = σ 2 (t) for some polynomial σ(t). Thus, unlike polynomial curves in general, PH curves have arc lengths and offset curves that admit exact rational representations. The lowest-order PH curves that are sufficiently flexible for general interpolation/approximation problems are the quintics. While the PH quintics are capable of matching arbitrary first-order Hermite data, the solution procedure is not straightforward and furthermore does not yield a unique result-there are always four distinct interpolants (of which only one, in general, has acceptable shape characteristics). We show that formulating PH quintics as complex-valued functions of a real parameter leads to a compact Hermite interpolation algorithm and facilitates an identification of the good interpolant (in terms of minimizing the absolute rotation number). This algorithm establishes the PH quintics as a viable medium for the design or approximation of free-form curves, and allows a one-for-one substitution of PH quintics in lieu of the widely-used ordinary cubics.

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Hermite interpolation by Pythagorean hodograph quintics

Curve and surface construction using variable degree polynomial splines

Arc splines, i.e. G1 curves made of circular arcs and straight-line segments, are important, as they are the paths that are used by automatically controlled cutting machinery. Many algorithms for approximately representing discrete data by a polygon have been published. In the paper, simple modifications of two of those algorithms are applied to the problem of the approximate representation of discrete data by an arc spline.

Approximation of discrete data by G1 arc splines

Preserving convexity using piecewise cubic interpolation

Fractal interpolation is a modern technique in approximation theory to fit and analyze scientific data. We develop a new class of $$\mathcal C ^1$$ - rational cubic fractal interpolation functions, where the associated iterated function system uses rational functions of the form $$\frac{p_i(x)}{q_i(x)},$$ where $$p_i(x)$$ and $$q_i(x)$$ are cubic polynomials involving two shape parameters. The rational cubic iterated function system scheme provides an additional freedom over the classical rational cubic interpolants due to the presence of the scaling factors and shape parameters. The classical rational cubic functions are obtained as a special case of the developed fractal interpolants. An upper bound of the uniform error of the rational cubic fractal interpolation function with an original function in $$\mathcal C ^2$$ is deduced for the convergence results. The rational fractal scheme is computationally economical, very much local, moderately local or global depending on the scaling factors and shape parameters. Appropriate restrictions on the scaling factors and shape parameters give sufficient conditions for a shape preserving rational cubic fractal interpolation function so that it is monotonic, positive, and convex if the data set is monotonic, positive, and convex, respectively. A visual illustration of the shape preserving fractal curves is provided to support our theoretical results.

K. Unsworth

Papers

Shape preserving interpolation by curvature continuous parametric curves