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### A basis for the implicit representation of planar rational cubic Bézier curves

Oliver J.D. Barrowclough
2014 Computer Aided Geometric Design
We present an approach to finding the implicit equation of a planar rational parametric cubic curve, by defining a new basis for the representation.  ...  The basis, which contains only four cubic bivariate polynomials, is defined in terms of the B\'ezier control points of the curve.  ...  I would like to thank Tor Dokken for reading through the manuscript and for helpful suggestions. I would also like to thank the anonymous reviewers for their comments and corrections.  ...

### Geometric Modeling [chapter]

Shashi Shekhar, Hui Xiong
2008 Encyclopedia of GIS
For example, a 90 degrees arc as a rational Bezier curve: P1 w0 P0 B02 (t ) + w1P1B12 (t ) + w2 P2 B22 (t )  The B-spline functions are then  ...  by introducing a more powerful representation functions is non-zero of rational curves  ...

### A Review on Approaches for Handling Bezier Curves in CAD for Manufacturing

Hetal N. Fitter, Akash B. Pandey, Divyang D. Patel, Jitendra M. Mistry
2014 Procedia Engineering
A review is done on various approaches for handling Bezier curve in Computer Aided Design for the purpose of manufacturing for example tool path optimization, profile design, reverse engineering, etc.  ...  Various techniques and methodologies like curve fitting, curve manipulation, blending and merging of curves have been proposed over the years for better handling and enhancing Bezier curve use with every  ...  This made the entire Voronoi edge consisting of implicit bisectors to be converted into rational quadratic Bezier curve for unified representation.  ...

### Page 4321 of Mathematical Reviews Vol. , Issue 96g [page]

1996 Mathematical Reviews
Summary: “An explicit expression for the unique implicitization of any planar rational cubic Bézier curve is presented, the only exception being when the tangents at the ends of the curve are parallel.  ...  The representation is in the form of a cubic Bernstein- Bézier triangle and is invariant under affine transformations of the curve.  ...

### PHYSICS BASED GEOMETRIC DESIGN

HONG QIN
1996 International journal of shape modeling
We demonstrate that D-NURBS can not only serve as a basis for the future research o f physics-based geometric design but also become readily appropriate for a large variety of important applications in  ...  In this paper, we s u r v ey the diversity of shape representations ranging from the primitive polynomial to the sophisticated Non-Uniform Rational B-Spline (NURBS).  ...  Chou and Piegl used piecewise cubic rational Bezier splines to t a set of data points and their tangent directions in the least-square sense 19].  ...

### Resolution independent curve rendering using programmable graphics hardware

Charles Loop, Jim Blinn
2005 ACM Transactions on Graphics
A simple implicit equation for a parametric curve is found in a space that can be thought of as an analog to texture space.  ...  When the triangle(s) corresponding to the Bézier curve control hull are rendered, a pixel shader program evaluates the implicit equation for a pixel's interpolated texture coordinates to determine an inside  ...  Integral Cubics Solving for the roots of I(t, s) will, for arbitrary rational cubic curves, require solving a cubic equation.  ...

### Resolution independent curve rendering using programmable graphics hardware

Charles Loop, Jim Blinn
2005 ACM SIGGRAPH 2005 Papers on - SIGGRAPH '05
A simple implicit equation for a parametric curve is found in a space that can be thought of as an analog to texture space.  ...  When the triangle(s) corresponding to the Bézier curve control hull are rendered, a pixel shader program evaluates the implicit equation for a pixel's interpolated texture coordinates to determine an inside  ...  Integral Cubics Solving for the roots of I(t, s) will, for arbitrary rational cubic curves, require solving a cubic equation.  ...

### A geometric characterization of parametric cubic curves

Maureen C. Stone, Tony D. DeRose
1989 ACM Transactions on Graphics
In this paper, we analyze planar parametric cubic curves to determine conditions for loops, cusps, or inflection points.  ...  Such a characterization forms the basis for an easy and efficient solution to this problem.  ...  Su and Liu [ll] have presented a specific geometric solution for the Bezier representation, and Forrest [6] has studied rational cubic curves.  ...

### Page 4587 of Mathematical Reviews Vol. , Issue 2003f [page]

2003 Mathematical Reviews
In particular, a family of si- nusoidal spirals, including the cubic curves of Tschirnhausen, are identified as polar Bézier curves.  ...  The derivation formulae, basis function properties, envelope theorems of the L-surface and an equivalent representation of the L-surface as a Bezier surface are presented. In particular, a C?  ...

### Geometries for CAGD [chapter]

Helmut Pottmann, Stefan Leopoldseder
2002 Handbook of Computer Aided Geometric Design
For the use of algebraic geometry in geometric design, the reader is referred to chapter ?? on implicit surfaces.  ...  This holds especially for subdivision curves and surfaces (chapter ??) and multiresolution techniques (chapter ??), where discrete models of curves and surfaces play a fundamental role.  ...  Planar cubic Bézier curve via projection of a cubic normal curve frame point F i (see Figure 2 ).  ...

### Envelope computation in the plane by approximate implicitization

Tino Schulz, Bert Jüttler
2011 Applicable Algebra in Engineering, Communication and Computing
Given a rational family of planar rational curves in a certain region of interest, we are interested in computing an implicit representation of the envelope.  ...  This connection enables us to use approximate implicitization for computing the (exact or approximate) implicit representation of the envelope.  ...  This research has been supported by the Austrian Science Fund (FWF) in the frame of the FSP S092 "Industrial Geometry", subproject 2.  ...

### An Efficient Trim Structure for Rendering Large B-Rep Models [article]

Frédéric Claux, David Vanderhaeghe, Loïc Barthe, Mathias Paulin, Jean-Pierre Jessel, David Croenne
2012 International Symposium on Vision, Modeling, and Visualization
To get a good tradeoff between performance and visual accuracy, we propose to use a vectorial but approximated representation of the model that allows efficient, real-time GPU exploitation.  ...  We get interactive frame rates for models that consists of hundreds of thousands of B-Rep faces, regardless of the zoom level.  ...  The Cubic Bézier line counts the number of cubic Bézier surface patches that are created as a result of a 0.1mm approximation of the original basis surfaces.  ...

### Interrogation of spline surfaces with application to isogeometric design and analysis of lattice-skin structures

Xiao Xiao, Malcolm Sabin, Fehmi Cirak
2019 Computer Methods in Applied Mechanics and Engineering
Subsequently, the intersections of the identified B\'ezier patches with the curve segment are determined with a matrix-based implicit representation leading to the computation of a sequence of small singular  ...  For mechanical analysis, the skin is modelled as a Kirchhoff-Love thin-shell and the lattice as a pin-jointed truss.  ...  For displacements there is usually no need for rational splines as the exact representation of quadric sections is not relevant.  ...

### Approximate algebraic methods for curves and surfaces and their applications

Bert Jüttler, Pavel Chalmovianský, Mohamed Shalaby, Elmar Wurm
2005 Proceedings of the 21st spring conference on Computer graphics - SCCG '05
We report on approximate techniques for conversion between the implicit and the parametric representation of curves and surfaces, i.e., implicitization and parameterization.  ...  It is shown that these techniques are able to handle general free-form surfaces, and they can therefore be used to exploit the duality of implicit and parametric representations.  ...  and Emerging Technologies" arm of the Fifth Framework Programme.  ...

### Rational Bézier Formulas with Quaternion and Clifford Algebra Weights [chapter]

Rimvydas Krasauskas, Severinas Zubė
2014 Geometry and Computing
We consider Bézier-like formulas with weights in quaternion and geometric (Clifford) algebra for parametrizing rational curves and surfaces.  ...  Applications include Bézier curves and surfaces in the conformal model of Euclidean space, bilinear Clifford-Bézier patches on isotropic cyclides, and rational offset surface modeling.  ...  The majority of the numerical experiments and symbolic computations were made for this paper using the software package CLU-Calc/CLUViz described in [16] and the MAPLE package Clifford [1] .  ...
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