Complex rational Bézier curves

2009 - J. Sánchez-Reyes

Computer Aided Geometric Design 26, 865-876 (2009)

authors IMACI

Abtract

We extend the rational Bézier 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 Möbius 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 Bézier 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.