Interpolating Polynomial Splines for Constructing Smooth Curves

1996 ◽  
pp. 251-286
Author(s):  
Gisela Engeln-Müllges ◽  
Frank Uhlig
Keyword(s):  
Polynomial Splines ◽  
Smooth Curves

Interpolating Polynomial Splines for Constructing Smooth Curves

1996 ◽  
pp. 251-286
Author(s):  
Gisela Engeln-Müllges ◽  
Frank Uhlig
Keyword(s):  
Polynomial Splines ◽  
Smooth Curves

scholarly journals Curves on K3 surfaces in divisibility 2

2021 ◽  
Vol 9 ◽  
Author(s):  
Younghan Bae ◽  
Tim-Henrik Buelles
Keyword(s):  
Modular Forms ◽  
K3 Surfaces ◽  
Initial Condition ◽  
Smooth Curves ◽  
Holomorphic Anomaly ◽  
Holomorphic Anomaly Equation ◽  
Anomaly Equation ◽  
Degeneration Formula ◽  
The Relationship ◽  
Level 2

Abstract We prove a conjecture of Maulik, Pandharipande and Thomas expressing the Gromov–Witten invariants of K3 surfaces for divisibility 2 curve classes in all genera in terms of weakly holomorphic quasi-modular forms of level 2. Then we establish the holomorphic anomaly equation in divisibility 2 in all genera. Our approach involves a refined boundary induction, relying on the top tautological group of the moduli space of smooth curves, together with a degeneration formula for the reduced virtual fundamental class with imprimitive curve classes. We use double ramification relations with target variety as a new tool to prove the initial condition. The relationship between the holomorphic anomaly equation for higher divisibility and the conjectural multiple cover formula of Oberdieck and Pandharipande is discussed in detail and illustrated with several examples.

scholarly journals Construction and analysis of unified 4-point interpolating nonstationary subdivision surfaces

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mehwish Bari ◽  
Ghulam Mustafa ◽  
Abdul Ghaffar ◽  
Kottakkaran Sooppy Nisar ◽  
Dumitru Baleanu
Keyword(s):  
Computer Graphics ◽  
Geometric Modeling ◽  
Tension Control ◽  
Free Form ◽  
Subdivision Surfaces ◽  
Smooth Curves ◽  
Practical Applications ◽  
Tension Parameter ◽  
Exact Reproduction ◽  
Free Form Surfaces

AbstractSubdivision schemes (SSs) have been the heart of computer-aided geometric design almost from its origin, and several unifications of SSs have been established. SSs are commonly used in computer graphics, and several ways were discovered to connect smooth curves/surfaces generated by SSs to applied geometry. To construct the link between nonstationary SSs and applied geometry, in this paper, we unify the interpolating nonstationary subdivision scheme (INSS) with a tension control parameter, which is considered as a generalization of 4-point binary nonstationary SSs. The proposed scheme produces a limit surface having $C^{1}$ C 1 smoothness. It generates circular images, spirals, or parts of conics, which are important requirements for practical applications in computer graphics and geometric modeling. We also establish the rules for arbitrary topology for extraordinary vertices (valence ≥3). The well-known subdivision Kobbelt scheme (Kobbelt in Comput. Graph. Forum 15(3):409–420, 1996) is a particular case. We can visualize the performance of the unified scheme by taking different values of the tension parameter. It provides an exact reproduction of parametric surfaces and is used in the processing of free-form surfaces in engineering.

scholarly journals Solitons of the midpoint mapping and affine curvature

2021 ◽  
Vol 112 (1) ◽  
Author(s):  
Christine Rademacher ◽  
Hans-Bert Rademacher
Keyword(s):  
Differential Equation ◽  
Large Class ◽  
Affine Group ◽  
Arc Length ◽  
Smooth Curves ◽  
Affine Map ◽  
Affine Curvature

AbstractFor a polygon $$x=(x_j)_{j\in \mathbb {Z}}$$ x = ( x j ) j ∈ Z in $$\mathbb {R}^n$$ R n we consider the midpoints polygon $$(M(x))_j=\left( x_j+x_{j+1}\right) /2.$$ ( M ( x ) ) j = x j + x j + 1 / 2 . We call a polygon a soliton of the midpoints mapping M if its midpoints polygon is the image of the polygon under an invertible affine map. We show that a large class of these polygons lie on an orbit of a one-parameter subgroup of the affine group acting on $$\mathbb {R}^n.$$ R n . These smooth curves are also characterized as solutions of the differential equation $$\dot{c}(t)=Bc (t)+d$$ c ˙ ( t ) = B c ( t ) + d for a matrix B and a vector d. For $$n=2$$ n = 2 these curves are curves of constant generalized-affine curvature $$k_{ga}=k_{ga}(B)$$ k ga = k ga ( B ) depending on B parametrized by generalized-affine arc length unless they are parametrizations of a parabola, an ellipse, or a hyperbola.

scholarly journals The epidemiology of the common cold IV. The effect of weather

1965 ◽  
Vol 63 (3) ◽  
pp. 427-439 ◽  
Author(s):  
O. M. Lidwell ◽  
R. W. Morgan ◽  
R. E. O. Williams
Keyword(s):  
Common Cold ◽  
Partial Correlation ◽  
Water Vapour Pressure ◽  
Outdoor Temperature ◽  
Weather Variables ◽  
Smooth Curves ◽  
Correlation Studies ◽  
The Common ◽  
Expected Values ◽  
Time Of Year

An investigation has been made of the association between weather and the numbers of colds reported on a given day. The seasonal trends were eliminated by working with the differences between the observed values on any day and the expected values derived from smooth curves fitted to the averages for the time of year.Examination of nine weather variables for the day on which the colds were reported and for each of the 29 preceding days showed that only two, mean day temperature and water-vapour pressure at 9 a.m., were significantly correlated with the numbers of colds. Partial correlation studies showed that the strongest association was with lowered mean day temperature between 2 and 4 days before the reported onset of symptoms.Regression analysis demonstrated that the magnitudes of the associations were sufficient to account for the greater part of the seasonal variation in the incidence of the common cold in both London and Newcastle. A small effect of atmospheric pollution appeared in this analysis.These results suggest that some effect of low outdoor temperature promotes transmission of the virus or the development of disease.

Shortest Networks for Smooth Curves

1997 ◽  
Vol 7 (4) ◽  
pp. 1054-1068 ◽  
Author(s):  
J. F. Weng
Keyword(s):  
Smooth Curves
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