The conformal map z→z2 of the hodograph plane
Computer Aided Geometric Design
Offset-rational parametric plane curves
Computer Aided Geometric Design
Generalizing rational degree elevation
Computer Aided Geometric Design
Cycles upon cycles: an anecdotal history of higher curves in science and engineering
Proceedings of the international conference on Mathematical methods for curves and surfaces II Lillehammer, 1997
From Conics to NURBS: A Tutorial and Survey
IEEE Computer Graphics and Applications
Constructive P th Algebra -Tool for Design,P rametrisation nd Visualisation
EGUK '02 Proceedings of the 20th UK conference on Eurographics
Rational quadratic circles are parametrized by chord length
Computer Aided Geometric Design
Curves with rational chord-length parametrization
Computer Aided Geometric Design
Geometric Hermite interpolation with circular precision
Computer-Aided Design
Rotations, translations and symmetry detection for complexified curves
Computer Aided Geometric Design
Curves with chord length parameterization
Computer Aided Geometric Design
Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable
Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable
Computer Aided Geometric Design
Closed rational trigonometric curves and surfaces
Journal of Computational and Applied Mathematics
Surfaces with rational chord length parameterization
GMP'10 Proceedings of the 6th international conference on Advances in Geometric Modeling and Processing
Curves and surfaces with rational chord length parameterization
Computer Aided Geometric Design
The Bernstein polynomial basis: A centennial retrospective
Computer Aided Geometric Design
Geometric constraints on quadratic Bézier curves using minimal length and energy
Journal of Computational and Applied Mathematics
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We extend the rational Bezier model for planar curves, by allowing complex weights. The key idea to endow this representation with a geometric meaning is the use of weight points (Farin points), which are no longer constrained to lie on the line segment defined by two successive control points. Instead of a control polygon, we have now the complex quotient of two control polygons, that is, a control curve made up of circular arcs, whose shape is controlled by the weight points. In the complex version of the rational de Casteljau algorithm, ratios become complex magnitudes, and repeated interpolation on lines is substituted by interpolation on circles. Invariance with respect to perspective transformations is replaced with invariance with respect to Mobius transformations. Therefore, not only enjoy these complex curves the customary linear precision, but also circular precision, i.e., if all control points and weight points lie on a circle, then an arc on the circle is reproduced. This complex model furnishes an intrinsically simpler representation for several remarkable curves, whose degree is halved with respect to the customary Bezier form. Circles are linear instead of quadratic, whereas remarkable epitrochoids (including the cardioid), Joukowski profiles and the Lemniscate of Bernoulli, are quadratic instead of quartic.