Tangent lines, inflections, and vertices of closed curves

We show that every smooth closed curve C immersed in Euclidean 3-space satisfies the sharp inequality 2(P+I)+V >5 which relates the numbers P of pairs of parallel tangent lines, I of inflections (or points of vanishing curvature), and V of vertices (or points of vanishing torsion) of C. We also show that 2(P'+I)+V >3, where P' is the number of pairs of concordant parallel tangent lines. The proofs, which employ curve shortening flow with surgery, are based on corresponding inequalities for the numbers of double points, singularities, and inflections of closed curves in the real projective plane and the sphere which intersect every closed geodesic. These findings extend some classical results in curve theory including works of Moebius, Fenchel, and Segre, which is also known as Arnold's "tennis ball theorem".Comment: Minor revisions; To appear in Duke Math.

Approximation by planar elastic curves

We give an algorithm for approximating a given plane curve segment by a planar elastic curve. The method depends on an analytic representation of the space of elastic curve segments, together with a geometric method for obtaining a good initial guess for the approximating curve. A gradient-driven optimization is then used to find the approximating elastic curve.Comment: 18 pages, 10 figures. Version2: new section 5 added (conclusions and discussions

Part-to-whole Registration of Histology and MRI using Shape Elements

Image registration between histology and magnetic resonance imaging (MRI) is a challenging task due to differences in structural content and contrast. Too thick and wide specimens cannot be processed all at once and must be cut into smaller pieces. This dramatically increases the complexity of the problem, since each piece should be individually and manually pre-aligned. To the best of our knowledge, no automatic method can reliably locate such piece of tissue within its respective whole in the MRI slice, and align it without any prior information. We propose here a novel automatic approach to the joint problem of multimodal registration between histology and MRI, when only a fraction of tissue is available from histology. The approach relies on the representation of images using their level lines so as to reach contrast invariance. Shape elements obtained via the extraction of bitangents are encoded in a projective-invariant manner, which permits the identification of common pieces of curves between two images. We evaluated the approach on human brain histology and compared resulting alignments against manually annotated ground truths. Considering the complexity of the brain folding patterns, preliminary results are promising and suggest the use of characteristic and meaningful shape elements for improved robustness and efficiency.Comment: Paper accepted at ICCV Workshop (Bio-Image Computing

A unified Pythagorean hodograph approach to the medial axis transform and offset approximation

AbstractAlgorithms based on Pythagorean hodographs (PH) in the Euclidean plane and in Minkowski space share common goals, the main one being rationality of offsets of planar domains. However, only separate interpolation techniques based on these curves can be found in the literature. It was recently revealed that rational PH curves in the Euclidean plane and in Minkowski space are very closely related. In this paper, we continue the discussion of the interplay between spatial MPH curves and their associated planar PH curves from the point of view of Hermite interpolation. On the basis of this approach we design a new, simple interpolation algorithm. The main advantage of the unifying method presented lies in the fact that it uses, after only some simple additional computations, an arbitrary algorithm for interpolation using planar PH curves also for interpolation using spatial MPH curves. We present the functionality of our method for G1 Hermite data; however, one could also obtain higher order algorithms

Forceless folding of thin annular strips

Thin strips or sheets with in-plane curvature have a natural tendency to adopt highly symmetric shapes when forced into closed structures and to spontaneously fold into compact multi-covered configurations under feed-in of more length or change of intrinsic curvature. This disposition is exploited in nature as well as in the design of everyday items such as foldable containers. We formulate boundary-value problems (for an ODE) for symmetric equilibrium solutions of unstretchable circular annular strips and present sequences of numerical solutions that mimic different folding modes. Because of the high-order symmetry, closed solutions cannot have an internal force, i.e., the strips are forceless. We consider both wide and narrow (strictly zero-width) strips. Narrow strips cannot have inflections, but wide strips can be either inflectional or non-inflectional. Inflectional solutions are found to feature stress localisations, with divergent strain energy density, on the edge of the strip at inflections of the surface. ‘Regular’ folding gives these singularities on the inside of the annulus, while ‘inverted’ folding gives them predominantly on the outside of the annulus. No new inflections are created in the folding process as more length is inserted. We end with a discussion of an intriguing apparent connection with a deep result on the topology of curves on surfaces

Forceless folding of thin annular strips

Thin strips or sheets with in-plane curvature have a natural tendency to adopt highly symmetric shapes when forced into closed structures and to spontaneously fold into compact multi-covered configurations under feed-in of more length or change of intrinsic curvature. This disposition is exploited in nature as well as in the design of everyday items such as foldable containers. We formulate boundary-value problems (for an ODE) for symmetric equilibrium solutions of unstretchable circular annular strips and present sequences of numerical solutions that mimic different folding modes. Because of the high-order symmetry, closed solutions cannot have an internal force, i.e., the strips are forceless. We consider both wide and narrow (strictly zero-width) strips. Narrow strips cannot have inflections, but wide strips can be either inflectional or non-inflectional. Inflectional solutions are found to feature stress localisations, with divergent strain energy density, on the edge of the strip at inflections of the surface. ‘Regular’ folding gives these singularities on the inside of the annulus, while ‘inverted’ folding gives them predominantly on the outside of the annulus. No new inflections are created in the folding process as more length is inserted. We end with a discussion of an intriguing apparent connection with a deep result on the topology of curves on surfaces