Simple and weighted cyclic proximity curves and surfaces

Abstract

Using the continuous extension ρu;σ=12σ1+cosuσ,u∈Rmod2π of the de la Vallée Poussin kernel (σ∈R≥0), we generalize the classic integral and rational cyclic curves of order n∈N≥1 published in Róth et al. [1] and Juhász and Róth [2], respectively. We refer to these new closed smooth curve-modeling tools as simple/weighted cyclic proximity curves of order n and of shape parameter σ. Continuously increasing the fullness-controlling parameter σ, these new types of curves provide pointwise convergent curve-flows consisting of smooth transitions between the starting classic integral/rational cyclic curves and their control polygons. Using tensor-products, we also define simple/weighted cyclic proximity surfaces of order n,m∈N≥12 and of shape vector σ,τ∈R≥02, which extend the notion of classic integral/rational cyclic surfaces and ensure smooth transitions between the initial classic integral/rational cyclic surfaces and their control nets. We also study the asymptotic behavior, the geometric and shape-preserving properties of these curve/surface-flows. Whenever we could find appropriate mathematical tools, we have proved these properties by assuming non-negative real shape parameters, but concerning variation- and length-diminishing properties, we have also formulated open questions for the research community when σ,τ∈R≥0∖S2, where S=N≥0∪(N≥0+12) denotes the set of those shape parameters for which we were able to theoretically justify the previously listed two properties at the moment. If σ,τ∈N≥02, the proposed simple/weighted proximity curves/surfaces also have an either integral or rational classic cyclic representation, but we show that the recommended modeling tools have advantages over the classic integral/rational cyclic curves/surfaces even in this special case. [Display omitted] [Display omitted]