Abstract: This study proposes new C rational cubic spline interpolant of the form cubic/quadratic with three shape parameters to preserves the geometric properties of the given data sets. Sufficient conditions for the positivity and data constrained modeling of the rational interpolant are derived on one parameter while the remaining two parameters can further be utilized to change and modify t...

Scaffold-like surfaces, ranging from large-scale trusses to engineered micro-structures, are often sketched via repeating patterns of nodes and edges. Offsetting these graphs turns them into meshes for which a smoothly-rounded scaffold surface ‘skin’ needs be locally generated on the fly production or analysis. We focus minimal single-valence (MSV) quad whose irregular vertices all have same va...

Recently, it was shown that a bi-cubic patch complex with n-sided holes can be completed into a curvature-continuous (G) surface by n-sided caps of degree bi-5 that offer good and flexible shape [KP13]. This paper further explores the space of n-sided caps of degree bi-5 but focusing on functionals to set degrees of freedom and to optimally propagate and average out curvature from the bi-cubic ...

We establish a relationship between the word complexity and the number of generalized diagonals for a polygonal billiard. We conclude that in the rational case the complexity function has cubic upper and lower bounds. In the tiling case the complexity has cubic asymptotic growth.

Recently, it was shown that a bi-cubic patch complex with n-sided holes can be completed into a curvature-continuous (G) surface by n-sided caps of degree bi-5 that offer good and flexible shape [KP13]. This paper further explores the space of n-sided caps of degree bi-5 but focusing on functionals to set degrees of freedom and to optimally propagate and average out curvature from the bi-cubic ...

The aim of these lectures is to study rational points and rational curves on varieties, mainly over finite fields Fq. We concentrate on hypersurfaces Xn of degree ≤ n+ 1 in Pn+1, especially on cubic hypersurfaces. The theorem of Chevalley–Warning (cf. Esnault’s lectures) guarantees rational points on low degree hypersurfaces over finite fields. That is, if X ⊂ Pn+1 is a hypersurface of degree ≤...

Abstract— A smooth curve interpolation is very significant in computer graphics or in data visualization. In the present paper -piecewise rational cubic spline function with tension parameter is considered which produces a monotonic interplant to a given monotonic data set. The parameters in the description of the spline curve can be used to modify the shape of the curve, locally and globally. ...

In 1958 E. Lehmer found an explicit description of those primes p for which a given prime q is a cubic residue. Her result states that if one writes 4p = L + 27M, then q is a cubic residue if and only if M/L ≡ (t − 1)/(t − 9t) mod q for some integer t. Recently, Z. Sun has stated a similar result for cubic nonresidues which follows from several corollaries appearing in an earlier paper of his. ...