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Interpolating Rational Bézier Spline Curves with Local Shape Control

Alexander P. Pobegailo
2013 International Journal of Computer Graphics & Animation  
The paper presents a technique for construction of C n interpolating rational Bézier spline curves by means of blending rational quadric Bézier curves.  ...  The constructed spline curves have local shape control that make them useful in such geometric applications as real-time trajectory generation and fast curve sketching.  ...  CONSTRUCTION OF TWO SMOOTHLY JOINED CONIC ARCS The purpose of this section is to introduce analytical expressions for construction smoothly joined conic arcs.  ... 
doi:10.5121/ijcga.2013.3401 fatcat:i5eqxqp45baj7e3rpftf2bd5s4

Improving G1 surface joins by using a composite patch

Jorg Peters, Joe D. Warren
1992 Curves and Surfaces in Computer Vision and Graphics III  
A popular approach to joining surface patches smoothly across a given boundary curve is to choose the transversal derivative as an average of the tangents at the end points.  ...  It consists of tensor-product patches that separate the original boundary tangents, so that constant G 1 joins can be used.  ...  One can view the composite patch as a polynomial alternative to the rational corrected patch in that it can smoothly interpolate incompatible mixed derivatives at the vertices of a mesh of curves.  ... 
doi:10.1117/12.131760 fatcat:eruzc3yikjemfaok4aklrdxrbi

Algebraic surface design with Hermite interpolation

Chanderjit L. Bajaj, Insung Ihm
1992 ACM Transactions on Graphics  
for solid models as well as fleshing surfaces for curved wireframe models.  ...  Computational details of the Hermite interpolation algorithm are also presented along with several illustrative applications of the interpolation technique to the construction of joining or blending surfaces  ...  See Figure 1 . 0 Example 6.3 (JOINING 2) A Quartic Surface for Smoothly Joining Three Circular Cylinders Consider computing the lowest degree surface which can smoothly join three truncated orthog·  ... 
doi:10.1145/102377.120081 fatcat:5bgtbvygyffphlaiktn5nxhney

Skinning and blending with rational envelope surfaces

Michal Bizzarri, Miroslav Lávička, Jiří Kosinka
2017 Computer-Aided Design  
We continue the study of rational envelope (RE) surfaces.  ...  Our results are demonstrated on several computed examples of skins and blends with rational parametrizations.  ...  Blending between canal surfaces using rational envelope surfaces We now describe a method for computing blending surfaces smoothly joining n ≥ 3 given canal surfaces.  ... 
doi:10.1016/j.cad.2017.02.002 fatcat:svlbkt7jpzfctgtmijxp7gphea

Quartic Rational Said-Ball-Like Basis with Tension Shape Parameters and Its Application

Yuanpeng Zhu, Xuli Han, Shengjun Liu
2014 Journal of Applied Mathematics  
Moreover, theG1continuous conditions are deduced for the joining of two patches.  ...  The new basis is applied to generate a class ofC1continuous quartic rational Hermite interpolation splines with local tension shape parameters.  ...  From Theorem 9, we can know that the conditions for smoothly joining two quartic rational triangular Said-Balllike patches are similar to the ones necessary for joining two triangular Bernstein-Bézier  ... 
doi:10.1155/2014/857840 fatcat:4i635b46n5bmbkuuwzdy5gtiae

Modeling with rational biquadratic splines

Kȩstutis Karčiauskas, Jörg Peters
2011 Computer-Aided Design  
We develop a rational biquadratic G 1 analogue of the non-uniform C 1 B-spline paradigm.  ...  These G 1 splines can exactly reproduce parts of multiple basic shapes, such as cyclides and quadrics, and combine them into one smoothly-connected structure.  ...  For example, the twisted half-tori in Fig. 10 (b) join smoothly since they share the same tangent cylinder.  ... 
doi:10.1016/j.cad.2011.08.024 fatcat:gzutgpgidzejbjynceup2d2thy

Page 365 of Mathematical Reviews Vol. , Issue 93a [page]

1993 Mathematical Reviews  
T. (4-DUND) Joining rational curves smoothly. Comput. Aided Geom. Design 8 (1991), no. 6, 443-464.  ...  Summary: “A theory is developed for projective-invariant con- struction from a sequence of control points of a piecewise rational curve of arbitrary degree, joining with continuity of certain geo- metric  ... 

A Study of consumer perception and product form

Xiaojuan Chen, Alison McKay, Alan de Pennington, Hau Hing Chau
2019 Zenodo  
body lines Cap and body joined smoothly SP5 Asymmetric Symmetric Curved body lines Cap and body joined smoothly SP7 Symmetric Symmetric Curved body lines Have a shoulder between cap and body  ...  lines Cap and body joined smoothly SP16 Symmetric Asymmetric Curved body lines Cap and body joined smoothly This led to Observation 1 that bottles that were perceived similarly have same shape  ... 
doi:10.5281/zenodo.2584074 fatcat:oh7wdsnkfvdabppa6i5f5ka6w4

Page 4508 of Mathematical Reviews Vol. , Issue 93h [page]

1993 Mathematical Reviews  
Summary: “In shape construction, it is often required for a curve to interpolate smoothly among given points and to be compatible with physical phenomena.  ...  One involves projecting a piecewise polynomial curve whose polynomial segments are joined according to given totally positive connection matrices.  ... 

Spontaneous breaking of classical PT symmetry

Carl M. Bender, Daniel W. Darg
2007 Journal of Mathematical Physics  
These rare kinds of trajectories are not PT symmetric and occur only for special rational values of ϵ.  ...  It is known [1] that for ǫ ≥ 0 nearly all trajectories are closed curves.  ...  [Note that abrupt changes in the periods of the orbits cannot occur for the trajectories joining the K = 0 pairs of turning points because T (ǫ) in ( 4 ) is a smoothly decreasing function for all ǫ ≥  ... 
doi:10.1063/1.2720279 fatcat:xlwkjsnqbnbmdae5hzlohb7t6a

Page 5643 of Mathematical Reviews Vol. , Issue 93j [page]

1993 Mathematical Reviews  
The authors show the RBG (rational boundary Gregory) patch can be joined smoothly using a lower-degree patch than previously proposed by H. Chiyokura et al.  ...  The approximation of B-rational curves by Bézier or B-rational curves with fewer control vectors is treated in Chapter 6.  ... 

Curvature-sensitive splines and design with basic curves

Kȩstutis Karčiauskas, Jörg Peters
2013 Computer-Aided Design  
only in the third scenario can pieces of basic curves be exactly re-represented and included into the spline.  ...  This paper presents a non-uniform cubic C 2 spline framework that unifies three scenarios for incorporating data from basic curves, such as spirals and conics.  ...  Rational curves of degree 3 include for example conics and circles.  ... 
doi:10.1016/j.cad.2012.10.024 fatcat:bxqmbe7mfnantp7ykmr6f5fd3m

G1 interpolation using piecewise quadric and cubic surfaces

Chandrajit L. Bajaj, Leonard A. Ferrari, Rui J. P. de Figueiredo
1990 Curves and Surfaces in Computer Vision and Graphics  
constructing G t continuous meshes of degree two (quadric) and degree three (cubic) implicitly defined, piecewise algebraic surfaces, which exactly fit any given collection of points and algebraic space curves  ...  Geometrically then, the two hyperboloids of one sheet are smoothly joined by a sphere and a cylinder.  ...  Both implicit and rational parametric representations of the space curves and their derivatives are allowed.  ... 
doi:10.1117/12.19736 fatcat:64kdqd4m65gfjorjr7m7ltv3ye

Page 410 of Mathematical Reviews Vol. , Issue 97A [page]

1997 Mathematical Reviews  
A metric distance d is defined on G by taking the distance between two points to be the infimum of the g-lengths of horizontal curves joining the two points.  ...  A Carnot (or sub-Riemannian) structure on a smooth manifold G is given by a smoothly varying distribution 9 and a smoothly varying inner product g on Z.  ... 

Sculpture from Patchwise Modules

Stephen Luecking
2019 Mathematics  
First, the surface can curve back on itself such that the opposite edges join on what is termed a seam. The surface is then rendered continuous through this seam as with a cylindrical surface.  ...  The draftsman positions his "ducks" to pass the spline smoothly through a set of given points[80].  ... 
doi:10.3390/math7020197 fatcat:mfw45qfm6fctnbfecflrdvdow4
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