Improperly parametrized rational curves
Computer Aided Geometric Design
Rational cubic circular arcs and their application in CAD
Computers in Industry
Curves and surfaces for computer aided geometric design (3rd ed.): a practical guide
Curves and surfaces for computer aided geometric design (3rd ed.): a practical guide
The rational cubic Be´zier representation of conics
Computer Aided Geometric Design
The uniqueness of Be´zier control points
Computer Aided Geometric Design
NURBS for Curve and Surface Design
NURBS for Curve and Surface Design
Computational Geometry for Design and Manufacture
Computational Geometry for Design and Manufacture
Accurate Parametrization of Conics by NURBS
IEEE Computer Graphics and Applications
A rational quartic Bézier representation for conics
Computer Aided Geometric Design
Rational B-Splines for Curve and Surface Representation
IEEE Computer Graphics and Applications
Representation of quadric primitives by rational polynomials
Computer Aided Geometric Design
An approximation of circular arcs by quartic Bézier curves
Computer-Aided Design
Approximation of conic sections by curvature continuous quartic Bézier curves
Computers & Mathematics with Applications
Approximating conic sections by constrained Bézier curves of arbitrary degree
Journal of Computational and Applied Mathematics
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Conic section is one of the geometric elements most commonly used for shape expression and mechanical accessory cartography. A rational quadratic Bezier curve is just a conic section. It cannot represent an elliptic segment whose center angle is not less than @p. However, conics represented in rational quartic format when compared to rational quadratic format, enjoy better properties such as being able to represent conics up to 2@p (but not including 2@p) without resorting to negative weights and possessing better parameterization. Therefore, it is actually worth studying the necessary and sufficient conditions for the rational quartic Bezier representation of conics. This paper attributes the rational quartic conic sections to two special kinds, that is, degree-reducible and improperly parameterized; on this basis, the necessary and sufficient conditions for the rational quartic Bezier representation of conics are derived. They are divided into two parts: Bezier control points and weights. These conditions can be used to judge whether a rational quartic Bezier curve is a conic section; or for a given conic section, present positions of the control points and values of the weights of the conic section in form of a rational quartic Bezier curve. Many examples are given to show the use of our results.