Curvature-controlled curve editing using piecewise clothoid curves

Highlights

• 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.

Abstract

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.

Introduction

The creation of good-looking curves is a fundamental task in 2D (and 3D) shape design. Through a recent collaboration with the surfacing department of a car manufacturer we realized how difficult it is in practice to convert a given design curve into a high-quality parametric curve that is suitable for further processing in a high-end CAD system (CATIA). Observing the work of the surface engineers we developed three hypotheses about their requirements:

Hypothesis 1

The control polygon alone must unambiguously, and in a predictable way, define the shape.

Hypothesis 2

The control polygon must be as sparse as possible (a) to offer control and (b) to avoid artifacts.

Hypothesis 3

Controlling curvature is essential; much time is spent on removing curvature artifacts.

Curvature control is so important in high-quality (“class-A”) shape design because curves are the basis for creating surfaces, and curvature artifacts lead to bad light reflections. State of the art in industrial design is the use of (piecewise) polynomial curves such as Bézier curves, B-splines, and their numerous variations. They are efficient to compute, their properties are well understood, and many algorithms exist for knot insertion, degree elevation, etc. The first hypothesis, however, precludes curve representations with invisible extra parameters such as NURBS, $\beta \mathrm{-}$, $\tau \mathrm{-}$, or $\nu \mathrm{-}\mathrm{splines}$; even varying the knot spacing is usually avoided. The use of any such non-graphical parameters has the crucial disadvantage that the curve shape cannot be judged only by looking at the control polygon; but this is exactly the expertise of surface engineers. This is why professional class-A surfacing software like IcemSurf still uses exclusively the oldest and most direct surface technique, tensor product Bézier surfaces of degree three up to nine.

The curvature of spline curves is difficult to control. In some cases it is even impossible to obtain the desired curvature practically, i.e., using a sparse control polygon (see Section 8). The reason is that, informally speaking, spline control points exert an ‘extra pull’ on the curve. A B-Spline with a regular n-gon as control polygon still deviates noticeably from a circle; without an extra weight parameter (rational splines) it is impossible to obtain a perfect circle. The real problem, however, is the lack of clear rules for the placement of control points. Surface engineers need years of experience to master the control point placement intuitively.

The work presented in this paper is the result of the quest for a curve representation that has no hidden parameters and offers exactly the degrees of freedom that designers need in order to control both the shape and the curvature of a 2D curve. We propose replacing splines by piecewise clothoid curves (PCCs). The heart of the framework is a fast algorithm to compute a PCC from a given (open or closed) sequence of input points. The problem can be stated formally as follows:

Given 2D points $p_1,\dots ,p_n$, compute clothoid segments $c_1(t),\dots ,c_{n-1}(t)$ with arc lengths $d_1,\dots ,d_{n-1}$ so that $c_1(0)=p_1$, $c_i(0)=c_{i-1}(d_{i-1})=p_i$ for $i>1$, and $c_{n-1}(d_{n-1})=p_n$.

There is no suitable closed-form representation of clothoid curves; they are defined via Fresnel integrals and computing them is impractical [1]. Therefore we use an iterative discrete scheme (see Section 3).

Our contribution is a unified framework for curvature-controlled curve design with PCCs:

• Fast adaptive clothoid interpolation algorithm.

• Curvature simplification by dynamic programming.

• Clothoid blending by nonlinear subdivision.

• Direct editing of the curvature plot.

• Physical interpretation of curvature smoothing.

• Different control modes for points: free, constrained tangents, and constrained curvature.

Clothoid curves are well known for their aesthetic quality. However, they have never been widely used in shape design, maybe for efficiency reasons, but certainly also because of the lack of suitable design tools. We argue that with PCCs, designers can reach their goal much faster and with an excellent level of control, for example to meet max/min curvature constraints, to limit the curvature variation, and to obtain aesthetically pleasing results. PCCs are much more ’relaxed’ than splines; the infamous 'spikes’ with unbounded curvature are avoided. Tension can still be added to the curve intentionally by explicitly inserting short segments as it is demonstrated, e.g., in Figs. 1 and 14.

In summary, we show that PCCs are superior to splines in practice: any spline curve can be approximated by a PCC with a sparse control polygon, but the converse is not true (see Section 8).