PPW Curves: a C² Interpolating Spline with Hyperbolic Blending of Rational Bézier Curves

Abstract

It is important to consider curvature properties around the control points to produce natural-looking results in the vector illustration. C² interpolating splines satisfy point interpolation with local support. Unfortunately, they cannot control the sharpness of the segment because it utilizes trigonometric function as blending function that has no degree of freedom. In this paper, we alternate the definition of C² interpolating splines in both interpolation curve and blending function. For the interpolation curve, we adopt a rational Bézier curve that enables the user to tune the shape of curve around the control point. For the blending function, we generalize the weighting scheme of C² interpolating splines and replace the trigonometric weight to our novel hyperbolic blending function. By extending this basic definition, we can also handle exact non-C² features, such as cusps and fillets, without losing generality. In our experiment, we provide both quantitative and qualitative comparisons to existing parametric curve models and discuss the difference among them.