The C-Bézier curves represent the extension of the cubic Bézier curves. They are defined as a linear combination of the basis functions $\sin t$, $\cos t$ and 1. The definition also includes a parameter $\alpha$ that additionally affects the shape of the curve. In the limit case $\alpha \to 0$, the C-Bézier curves converge to Cubic Béziers curves. Both families of curves are used in computer aided geometric design because of their numerous beautiful geometric properties. In this work we briefly present the related Ferguson curves and their extension to the C-curves. The main advantage of the C-Bézier curves is that we can accurately plot the circular arc and arc ellipse. The geometric characterization of the curve, which represents a circular arc and a half of the arc of the ellipse, is also given in the paper. Finally, we present Bézier and C-Bézier curves as PH curves used in CNC machines.
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