Enhancing flexibility and control in κ-curve using fractional Bézier curves

Abstract

The κ-curve is commonly applied as a curvature pen tool in Adobe Illustrator® and Photoshop®. The κ-curve has an excellent property where the local maxima curvature occurred at the control points. Having the local maxima curvature at the control points gives salient features to the curve and keeps away the unintended creation of cusps and loops. However, the existing κ-curve has low degree of freedom since it was constructed from the classical quadratic Bézier curve with lack of local control shape, which will limit the flexibility and adjustability of curve modelling. Furthermore, the existing κ-curve needs to be optimized globally, where additional computational time is required when modifying control points. The κ-curve also only exhibits G1 continuity at the inflection point. Hence, in this work, a new method is proposed to rectify the lower degree of freedom and to improve the flexibility and adjustability of the κ-curve. This study will also resolve the necessity of global optimization and adjust the smoothness of G1 continuity at inflection point in the current κ-curve. The improved κ-curve can be constructed using the fractional Bézier curve with the help of fractional continuity. The fractional Bézier curve is equipped with shape and fractional parameters that will enhance the flexibility and adjustability of the curves while still maintaining the local maxima curvature at the control points. The algorithms for the construction of the modified κ-curve will be shown by using the fractional Bézier curve and fractional continuity. The nature of fractional continuity in the proposed algorithms will prevent the global optimization and guaranteed the G2/F2 continuity everywhere. Therefore, the modified κ-curve is expected to rectify the aforementioned issues.