Curvature-controlled curve editing using piecewise clothoid curves

• We propose piecewise clothoid curves (PCCs) as an alternative to polynomial splines for high-quality curve design.
• We claim that PCCs are superior to splines from a design point of view since they are “class-A by default”.
• PCCs provide excellent curvature control since the curvature profile can be directly edited.
• We give a physical interpretation of PCCs and we devise practical placement rules for control points.
• We propose schemes for efficient evaluation, for iterative refinement, and for automatic curve conversion.

Two-dimensional curves are conventionally designed using splines or Bézier curves. Although formally they are C2 or higher, the variation of the curvature of (piecewise) polynomial curves is difficult to control; in some cases it is practically impossible to obtain the desired curvature. As an alternative we propose piecewise clothoid curves (PCCs). We show that from the design point of view they have many advantages: control points are interpolated, curvature extrema lie in the control points, and adding control points does not change the curve. We present a fast localized clothoid interpolation algorithm that can also be used for curvature smoothing, for curve fitting, for curvature blending, and even for directly editing the curvature. We give a physical interpretation of variational curvature minimization, from which we derive our scheme. Finally, we demonstrate the achievable quality with a range of examples.

Design curves of a car captured with PCCs.Figure optionsDownload high-quality image (294 K)Download as PowerPoint slide