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.

Shape preservation of scientific data through rational fractal splines

Path planning with offset curves for layered fabrication processes

Traditionally methods for the design of free-form curves and surfaces focus on achieving a specific level of inter-element continuity. These methods use a combination of heuristics and constructions to achieve an ultimate shape. Though shapes constructed using these methods are technically continuous, they have been shown to lack fairness, possessing undesirable blemishes such as bulges and wrinkles. Fairness is closely related to the smooth and minimal variation of curvature.
In this work we present a new technique for curve and surface design that combines a geometrically based specification with constrained optimization (minimization) of a fairness functional. The difficult problem of achieving inter-element continuity is solved simply by incorporating it into the minimization via appropriate penalty functions. Where traditional fairness measures are based on strain energy, we have developed a better measure of fairness: the variation of curvature. In addition to producing objects of clearly superior quality, minimizing the variation of curvature makes it trivial to model regular shapes such as, circles and cyclides, a class of surface including: spheres, cylinders, cones, and tori.
In this thesis we introduce: curvature variation as a fairness metric, the minimum variation curve (MVC), the minimum variation network (MVN), and the minimum variation surface (MVS). MVC minimize the arc length integral of the square of the arc length derivative of curvature while interpolating a set of geometric constraints consisting of position, and optionally tangent direction and curvature. MVN minimize the same functional while interpolating a network of geometric constraints consisting of surface position, tangent plane, and surface curvatures. Finally, MVS are obtained by spanning the openings of the MVN while minimizing a surface functional that measures the variation of surface curvature.
We present the details of the techniques outlined above and describe the trade-offs between some alternative approaches. Solutions to difficult interpolation problems and comparisons with traditional methods are provided. Both demonstrate the superiority of curvature variation as a fairness metric and efficacy of optimization as a tool in shape design, albeit at significant computational cost.

Minimum curvature variation curves, networks, and surfaces for fair free-form shape design

A universal parametrization in B-spline curve and surface interpolation

Shape preserving interpolation by curvature continuous parametric curves

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Shape preserving interpolation by curvature continuous parametric curves