3D Euler Spirals for 3D Curve Completion Gur Harary Ayellet Tal Dept. of Electrical Engineering Dept. of Electrical Engineering Technion Technion [email protected] [email protected] ABSTRACT Shape completion is an intriguing problem in geometry pro- cessing with applications in CAD and graphics. This paper defines a new type of 3D curves, 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 these curves { properties that have been shown to be important for the appeal of curves. We illustrate their utility in two applications. The first is \fixing" curves de- tected 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 in- completeness of the input model. Categories and Subject Descriptors I.3 [Computer Graphics]: Computational Geometry and Object Modeling|Curve, surface, solid, and object repre- sentations (a) Our completion (b) Hermite completion General Terms Figure 1: 3D Euler spirals (red) complete the curves on a broken Hellenistic oil lamp { curves that would Algorithms, Design most likely be drawn if the model were complete. The scale of the Hermite splines is determined Keywords manually (magenta), since the automatically-scaled Euler spirals, 3D curves splines (green) are inferior due to the large ratio between the length of the curve and the size of the 1. INTRODUCTION model. Note the perfect circular arcs of our curves. Shape completion has been an important task in compu- tational geometry with applications to CAD and computer might not always produce the preferable results. This is also graphics [2, 3, 28]. While most of the work has focused on supported by psychological studies that indicate that splines completing or repairing polyhedra and CAD models, this may be unsatisfactory for curve completion [30]. paper focuses on completing curves in three dimensions. It This paper defines a new type of 3D curves that can be presents a practical solution to the problem, which is demon- used for this purpose (Figure 1(a)). We show that our curves strated by real-life data. are not only appealing, but also qualitatively outperform Given two point-tangent pairs, one way to complete them some splines. In a nutshell, our curves can be considered as is to use any of the variety of known splines [7, 10]. Such an extension to 3D of the planar Euler spirals. An impor- curves possess many attractive properties, however, Fig- tant consideration in aesthetic curve design is the curve's ure 1(b) illustrates that they (in this case Hermite splines) fairness [25], which has been shown to be closely related to how little and how smoothly a curve bends. An Euler spiral, also referred to as a clothoid or a Cornu spiral, is an exam- Permission to make digital or hard copies of all or part of this work for ple of such an aesthetic curve. Its curvature varies linearly personal or classroom use is granted without fee provided that copies are with arc-length [16, 20, 21]. Our proposed curve has both not made or distributed for profit or commercial advantage and that copies its curvature and torsion change linearly with length. bear this notice and the full citation on the first page. To copy otherwise, to The contribution of this paper is threefold. First, the pa- republish, to post on servers or to redistribute to lists, requires prior specific per defines the 3D Euler spiral (Section 4) and proves that permission and/or a fee. SCG’10, June 13–16, 2010, Snowbird, Utah, USA. it satisfies some desirable properties { properties that have Copyright 2010 ACM 978-1-4503-0016-2/10/06 ...$10.00. been claimed to produce eye-pleasing curves [17] (Section 5). LATEX. [5, 17, 27] propose to use 2D cubic splines, giving up extensibility and roundness. Another way to construct curves is Elastica [14, 19, 24, 26], which refers to the curves that minimize the total square R L 2 curvature of the curve: E [κ(s)] = 0 κ (s)ds; where s 2 [0;L] is the arc-length parameter and κ(s) is the curvature. Horn [14] argues that Elastica is the \smoothest" 2D curve to complete the gap between two point-tangent pairs. It is extensible, but neither scale-variant nor round. Already in 1906 [4] discussed Elastica for 3D curves. A Curve that ap- proximates the solution to Elastica in 3D is explicitly defined in [24]. In 2D this curve is the 2D Euler spiral (discussed next), which has a linear curvature. In 3D, the curvature of this curve is expressed as a hyperbola. 2D Euler spirals: Euler spirals are curves whose cur- vature evolves linearly along the curve. They were dis- covered independently by three researchers [20]. In 1694 Bernoulli wrote the equations for the Euler spiral for the Figure 2: Archaeological drawing of a broken lamp first time, but did not draw the spirals or compute them nu- found in the archaeological site of Dor. This is a merically. In 1744 Euler rediscovered the curve's equations, manual drawing from [31]. described their properties, and derived a series expansion to the curve's integrals. Later, in 1781, he also computed the spiral's end points. The curves were re-discovered in 1890 for In particular, we prove that our curves are invariant to simi- the third time by Talbot, who used them to design railway larity transformations and that they are symmetric, extensi- tracks. Euler spirals are also known as \Cornu spirals" (after ble (i.e., refinable), smooth, and round (i.e., if the boundary Cornu who plotted them) and \Clothoid" (after Clotho, the conditions lie on a circle, then the curve is a circle, as illus- youngest of the three Fates of Greek mythology). They are trated in Figure 1(a)). Second, we present a parameter-less defined as the curves that penalize the curvature variation, algorithm for computing these curves (Section 6). Last but hence minimizing the following (κs is the derivative of κ): not least, we demonstrate the use of these curves in two curve completion applications (Section 7). Z L 2 In the first application, our spirals complete curves miss- E[κ(s)] = κs(s)ds: ing due to \weak" surfaces patches. The second application 0 is curve completion of broken shapes, such as archaeological In [30] psychological experiments show that in 2D an inter- artifacts. Currently, this task is performed by drawing the polation between two point-tangent pairs using Euler spiral missing curves manually in 2D (Figure 2). This is an expen- outperforms parabolic curves and circular arcs. This is at- sive and time-consuming process, which is prone to biases. tributed to its monotonous change in curvature, which has Our curves can replace this manual task while performing a good fit to the way the human eye interpolates curves. it in 3D, directly on the scanned artifact. With the growing 2D Euler spirals were used in computer aided design. In [14, popularity of digital documentation in archaeology, the abil- 23] they are used as an approximation to the solution of Elas- ity to draw curves in 3D is becoming ever more important. tica. In [22] the conditions under which the spirals can form a transition curve are investigated. Two spirals are used 2. RELATED WORK in [33] to form a parabola-like segment between consecutive points of a control polygon. In [21] a formulation for fitting Curve completion: Given two point-tangent pairs, the spiral primitives to a dense polyline data is developed. most common way to perform completion is to use splines, In [16] an algorithm is described for 2D curve completion such as a cubic Hermite spline [7, 10]. Splines are fast and using an Euler spiral. The algorithm is an iterative gradient- easy to compute, but they are not always the curves pre- descent, initialized by a 2D Biarc. Since there are infinite ferred by the human visual system. possible Biarcs, the Biarc that minimizes the total curvature Ullman [32] suggests properties that 2D curves should sat- variation is chosen. Some properties that characterize eye- isfy: invariance to rigid transformations, smoothness, min- pleasing curves are also proved. In [34] a faster and more imization of the total curvature, and extensibility. These accurate algorithm is proposed. It is proved that given two properties led to a Biarc solution { a curve consisting of two point-tangent pairs, there always exists an Euler spiral that circular arcs. Biarcs are smooth and invariant to rigid trans- interpolates them. formations, but they do not guarantee that the total curva- ture is minimized and they are not extensible [5]. Biarcs are 3D Euler spirals: An attempt to generalize Euler spirals generalized to 3D in [6, 29]. to 3D, maintaining the linearity of the curvature, is pre- Knuth [17] proposes different properties of eye-pleasing 2D sented in [13]. A given polygon is refined, such that the curves: invariance to similarity transformations and cyclic polygon satisfies both arc-length parameterization and lin- permutations (for closed curves), extensibility, smoothness, ear distribution of the discrete curvature binormal vector. roundness, and being locally constructed. Knuth also shows The algorithm ignores the torsion, despite being an impor- that the latter four properties cannot take place together. tant characteristic of 3D curves. These properties are the base of the METAFONT system of We propose a novel algorithm, which produces continu- ous, rather than discrete, curves and takes both curvature and torsion into account. The algorithm, which is inspired by [16], is general and does not require an initial polygon.