Simple and weighted cyclic proximity curves and surfaces. (August 2021) - Róth, Ágoston
Abstract: Using the continuous extension ρ u ; σ = 1 2 σ 1 + cos u σ, u ∈ R mod 2 π 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 ≥ 1 2 and of shape vector σ, τ ∈ R ≥ 0 2, 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 ∖ S 2, where S = N ≥ 0 ∪ ( N ≥ 0 + 1 2 ) 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 ≥ 0 2, 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. Graphical abstract: Highlights: Globally controlled simple/weighted cyclic proximity curves/surfaces are proposed. The proposed free-form modeling tools generalize the classic cyclic curves/surfaces. Based on continuously extended de la Vallée Poussin kernels/radial basis functions. Asymptotic behaviors, geometric and shape-preserving properties are studied. Continuous curve/surface-flows that converge pointwise to the control-polygon/net.