Clothoid Research Articles

A note on quintic polynomial approximation of generalized Cornu spiral segments

In two recent papers Cripps et al. (2010) [3], and Cross and Cripps (2012) [2], quintic polynomial approximations of the generalized Cornu spirals have been studied by considering G2 continuity and G3 continuity at the end points respectively. The quintic curve is constructed so that the maximum curvature error is within the specified tolerance. In this paper, we provide corrections to the typing errors in Cross and Cripps (2012) [2], and propose a simpler and more efficient method for the G2-constrained quintic polynomial approximation where the G2 conditions are always satisfied by four free variables. Also, we introduce a new error measure of the maximum curvature error, which can greatly reduce the computational time when looking for the solution. Numerical experiments demonstrate the effectiveness of the new measure.

Formula omitted] quintic polynomial approximation for Generalised Cornu Spiral segments

Within Computer Aided Design (CAD) there is a need to construct fair curves. The Generalised Cornu Spirals (GCSs) are a set of curves with a monotonic curvature profile and are hence considered fair but implementation in current CAD systems is not straightforward, partly due to not being in the usual polynomial form.A method to approximate a GCS using a quintic polynomial curve is presented. The method seeks to interpolate the GCS to satisfy the G3 constraints at the end points with a quintic Bézier, leaving two degrees of freedom.An initial approximation is shown to be effective for the majority of GCS curves. Moreover, it is possible to determine when an initial approximation is likely to be poor. If this approximation does not meet the tolerance required, a search involving two parameters is performed. Characteristics of the search domain are used to establish a suitable starting value.

Analytic parametric equations of log-aesthetic curves in terms of incomplete gamma functions

Log-aesthetic curves (LACs) have recently been developed to meet the requirements of industrial design for visually pleasing shapes. LACs are defined in terms of definite integrals, and adaptive Gaussian quadrature can be used to obtain curve segments. To date, these integrals have only been evaluated analytically for restricted values ( 0 , 1 , 2 ) of the shape parameter α. We present parametric equations expressed in terms of incomplete gamma functions, which allow us to find an exact analytic representation of a curve segment for any real value of α. The computation time for generating an LAC segment using the incomplete gamma functions is up to 13 times faster than using direct numerical integration. Our equations are generalizations of the well-known Cornu, Nielsen, and logarithmic spirals, and involutes of a circle.

3D Euler spirals for 3D curve completion

Shape completion is an intriguing problem in geometry processing with applications in CAD and graphics. This paper defines a new type of 3D curve, which can be utilized for curve completion. It can be considered as the extension to three dimensions of the 2D Euler spiral. We prove several properties of this curve – properties that have been shown to be important for the appeal of curves. We illustrate its utility in two applications. The first is “fixing” curves detected by algorithms for edge detection on surfaces. The second is shape illustration in archaeology, where the user would like to draw curves that are missing due to the incompleteness of the input model.

Geometrically Guided Exemplar-Based Inpainting

Exemplar-based methods have proven their efficiency for the reconstruction of missing parts in a digital image. Texture as well as local geometry are often very well restored by such methods. Some applications, however, require the ability to reconstruct nonlocal geometric features, e.g., long edges. In order to do so, we propose to first compute a geometric sketch, which is then interpolated and used as a guide for the global reconstruction. In comparison with other related approaches, the originality of our work relies on the following points: (1) The geometric sketch computation is parameter-free and based on level lines, which provides a complete, reliable, and stable representation of the image. (2) The completion of the geometric sketch is fully automatic. It is done using a new—and interesting on its own—geometric inpainting approach that interpolates level lines with Euler spirals. Euler spirals are natural curves for shape completion and have been used already for edge completion and inpainting. It is the first time, however, that these curves are used for completing the whole level lines structure. (3) The general reconstruction is performed using a guided version of a classical exemplar-based method. However, we do not constrain the exemplar-based reconstruction to strictly follow the geometric guide. We actually use a new metric between blocks that consists of the sum of the classical ${{\mathrm L}^2}$ metric between any two blocks of the general image plus an ${{\mathrm L}^2}$ metric between the corresponding blocks in the completed geometric image. This is equivalent to a Lagrangian relaxation of a strictly guided reconstruction. We discuss in the paper the details of the method and some related mathematical issues, and we illustrate its efficiency on several examples.

The generalised Cornu spiral and its application to span generation

A Cornu spiral is a plane curve having a linear curvature profile. This paper considers plane curves having rational linear curvature profiles. These curves are defined to be generalised Cornu spirals (GCS) and are quality curves in the sense that they are continuous and smooth, can contain one inflection at most, and have a bounded and monotonic curvature profile. In addition, the GCS has an extra degree of freedom over the Cornu spiral that is available for shape control. Starting from the intrinsic equation of the GCS, the technique of curve synthesis is used to design a quality curve that can be applied to a wide range of span generation problems.

Biharmonic Hopf Cylinders

This paper concerns curves and surfaces, into indefinite space forms, whose mean curvature vector field is in the kernel of certain elliptic differential operators. It has been inspired by the paper of M. Barros and O.J. Garay [2], where the Riemannian version of this question is solved. We first consider the Laplacian to study indefinite submanifolds with harmonic mean curvature vector field in the normal bundle. This problem is closely related to a conjecture of B.-Y. Chen [5], on Riemannian submanifolds, stated as follows: harmonicity of the mean curvature vector field implies harmonicity of the immersion. Submanifolds with harmonic mean curvature vector field were called by Chen biharmonic submanifolds. In the realm of indefinite submanifolds, counterexamples to that conjecture have been given by the two first authors (see [1]). Biharmonic submanifolds are a special class of submanifolds for which its mean curvature vector is an eigenvector of ∆, that is, ∆H = λH for some real constant λ. First we describe the family of curves whose mean curvature vector field is proper for the Laplacian. This problem has been yet solved for Euclidean curves by M. Barros and O.J. Garay [2]. We have to think of a different Laplacian if we want to characterize curves others than those of both constant curvature and torsion. Since H is a normal vector field, it seems natural to consider the Laplacian associated to the connection in the normal bundle. Then we show that the indefinite Cornu spirals are the only non standard curves in a semi-Riemannian manifold that are biharmonic in the normal bundle. As for surfaces, we deal with the semi-Riemannian Hopf cylinders we introduced in [4]. Then we show that the biharmonicity of them strongly depends on the biharmonicity of the curves to which are associated. In fact, a non standard Hopf cylinder in H1(−1) is biharmonic in the normal bundle if and only if it is associated to a Cornu spiral in Hs(−4). Then we extend the results in [2]. The second operator considered is the Jacobi operator, which was introduced by J. Simons [9] and involves the Laplacian in the normal bundle. A normal vector field is called a Jacobi field if it belongs to the kernel of the Jacobi operator. This operator appears when one studies the second variation of the area functional for compact Riemannian minimal submanifolds. It has been recently used by Barros and Garay [3] to classify Hopf cylinders into S3 with Jacobi mean curvature vector field. In [4] we have made, following [7], a qualitative description of elastic curves into indefinite space forms to be used as a tool to find Lorentzian Willmore tori in H1(−1). Now the Jacobi operator allows us to get a characterization of elastic curves, as well as a characterization of semi-Riemannian Hopf cylinders in H1(−1), in terms of elasticae in Hs(−4), s = 0, 1. We show that a curve in an indefinite real space form has Jacobi mean curvature vector field if and only if it is curvature homothetic to a free elastica. As before, this characterization leads to find Hopf cylinders into H1(−1) with Jacobi mean curvature vector field.

A characterisation of Helices and Cornu spirals in real space forms

We classify unit speed curves contained in a real space form of arbitrary dimension Nm(c), whose mean curvature vector is proper for the Laplacian. Then we use these results to classify Hopf cylinders of S3 and semi-Riemannian Hopf cylinders of with proper mean curvature function.

The use of Cornu spirals in drawing planar curves of controlled curvature

Cornu spirals or clothoids have been used in highway design for many years. In the past the spirals have been found manually by draftsmen. The purpose of this paper is to show that it is practical to find these spirals with a micro-computer. The design curve will be made up of arcs of circles and segments of Cornu spirals joined in such a way that the curvature is continuous throughout, and takes its largest values on the arcs of circles. Thus, the radii of the circles used will limit, and control the curvature of the whole design curve.

Synthesis of 2D curves by blending piecewise linear curvature profiles

A new algorithm for fitting a smooth curve through two datapoints, to meet specified tangent direction and curvature requirements at these datapoints, is described. The algorithm produces a piecewise linear curvature profile, thus the resultant curve consists of parts of Cornu spirals. It does this by blending some basic forms defined in the paper. A FORTRAN program based on this algorithm can provide many substantially different solutions to any given problem. It can either be incorporated in a linear curvature spline package or used in an interactive system for designing spans or splines.

Two-dimensional curve synthesis using linear curvature elements

A new algorithm is presented for the generation of a smooth curve through two data points with specified curvature and tangent direction at those points. The curvature profile K(s) is designed to be piecewise linear, constructed from linear curvature elements (linces). This gives a curve form consisting of arcs of Cornu spirals (clothoids). A trilince is the normal solution as this can be optimized to meet a suitable criterion for fairness of the curve. A FORTRAN program based on this algorithm is incorporated in a linear curvature spline package.