Abstract We present a user‐guided, semi‐automatic approach to completing large holes in a mesh. The reconstruction of the missing features in such holes is usually ambiguous. Thus, unsupervised methods may produce unsatisfactory results. To overcome this problem, we let the user indicate constraints by providing merely four points per important feature curve on the mesh. Our algorithm regards this input as an indication of an important broken feature curve. Our completion is formulated as a global energy minimization problem, with user‐defined spatial‐coherence constraints, allows for completion that adheres to the existing features. We demonstrate the method on example problems that are not handled satisfactorily by fully automatic methods.

We introduce an algorithm to synthesize missing geometry for a given triangle mesh that has “holes.” Similarly to previous work, the algorithm is context based in that it fills the hole by synthesizing geometry that is similar to the remainder of the input mesh. Our algorithm goes further to impose a coherence objective. A synthesis is coherent if every local neighborhood of the filled hole is similar to some local neighborhood of the input mesh. This requirement avoids undesired features such as can occur in context-based completion. We demonstrate the algorithm's ability to fill holes that were difficult or impossible to fill in a compelling manner by earlier approaches.

Abstract Logarithmic spirals are ubiquitous in nature. This paper presents a novel mathematical definition of a 3D logarithmic spiral, which provides a proper description of objects found in nature. To motivate our work, we scanned spiral‐shaped objects and studied their geometric properties. We consider the extent to which the existing 3D definitions capture these properties. We identify a property that is shared by the objects we investigated and is not satisfied by the existing 3D definitions. This leads us to present our definition in which both the radius of curvature and the radius of torsion change linearly along the curve. We prove that our spiral satisfies several desirable properties, including invariance to similarity transformations, smoothness, symmetry, extensibility, and roundness. Finally, we demonstrate the utility of our curves in the modeling of several animal structures.

This video describes a new type of 3D curves, which generalizes the family of 2D Euler spirals. They are defined as the curves having both their curvature and their torsion evolve linearly along the curve. The utility of these spirals for curve completion applications is demonstrated. This video accompanies the paper presented in [4].

Shape completion is an intriguing problem in geometry processing 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 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.

3D Euler spirals are visually pleasing, due to their property of having their curvature and their torsion change linearly with arc-length. This paper presents a novel algorithm for fitting piecewise 3D Euler spirals to 3D curves with G2 continuity and torsion continuity. The algorithm can also handle sharp corners. Our piecewise representation is invariant to similarity transformations and it is close to the input curves up to an error tolerance.