Building blocks for designing arbitrarily smooth subdivision schemes with conic precision

Abstract

Since subdivision schemes featured by high smoothness and conic precision are strongly required in many application contexts, in this work we define the building blocks to obtain new families of non-stationary subdivision schemes enjoying such properties. To this purpose, we firstly derive a non-stationary extension of the Lane-Riesenfeld algorithm, and we exploit the resulting class of schemes to design a non-stationary family of alternating primal/dual subdivision schemes, all featured by reproduction of {1, x, e tx , e −tx }, t ∈ [0, π) ∪ iR + . Then, we focus our attention on interpolatory subdivision schemes with conic precision, that can be obtained as a byproduct of the above classes. In particular, we present a novel construction of a family of non-stationary interpolatory 2n-point schemes which generalizes the well-known Dubuc-Deslauriers family in such a way the n-th (n ≥ 2) family member reproduces Π 2n−3 ∪ {e tx , e −tx }, t ∈ [0, π) ∪ iR + , and keeps the original smoothness of its stationary counterpart unchanged.

Exponential pseudo-splines: Looking beyond exponential B-splines

“…Indeed, differently from stationary subdivision schemes, nonstationary subdivision schemes are capable of reproducing conic sections, spirals or, in general, of generating exponential polynomials x r e θx , x ∈ R, r ∈ N ∪ {0}, θ ∈ C. This generation property is important not only in geometric design (see, e.g., [30,32,37,42,43]), but also in many other applications, e.g., in biomedical imaging (see, e.g., [20,21]) and in Isogeometric Analysis (see, e.g., [19,31]). However, the use of nonstationary subdivision schemes in IgA is nowadays limited to the case of exponential B-splines since they are the only functions that have been shown to be able to overcome the NURBS limits while preserving their useful properties.…”

A family of smooth and interpolatory basis functions for parametric curve and surface representation

“…Therefore, from (17) we see that ϕ also reproduces the exponential polynomials given by (15) up to degree q and exponent α, where α ∈ α n 0 is of multiplicity q + 1.…”

“…This convenient property is particularly relevant for the exact rendering of conic sections such as circles, ellipses, or parabolas, as well as other trigonometric and hyperbolic curves and surfaces [13]. In its absence, one must resort to subdivision to tackle this aspect [14][15][16][17][18]. However, existing comparable subdivision schemes usually rely on basis functions that are defined as a limit process and do not have a closed-form expression [19].…”

Smooth shapes with spherical topology: Beyond traditional modeling, efficient deformation, and interaction

“…As opposed to polygon mesh models, subdivision methods do not necessarily have interpolating control points. Different methods based on nonstationary refinement rules have been proposed to approximate spheres using subdivision [27][28][29]. One drawback of polygon and subdivision methods is that they require a large number of parameters which can be a challenge when computational speed is important (e.g., in finite element models [30]).…”

“…The wedge operator is defined as dy ∧ dz = ∂y ∂u ∂z ∂v − ∂y ∂v ∂z ∂u (28) and is explicitly computed using ∂σ/∂u = (∂x/∂u, ∂y/∂u, ∂z/∂u) and ∂σ/∂v = (∂x/∂v, ∂y/∂v, ∂z/∂v)…”