Framed Curves Research Articles

Global curvature for rectifiable loops

We study in detail the notion of global curvature defined on rectifiable closed curves, a concept which has been successfully applied in existence and regularity investigations regarding elastic self-contact problems in nonlinear elasticity. A bound on this purely geometric quantity serves as an excluded volume constraint to prevent selfintersections of slender elastic bodies modeled as elastic rods. Moreover, a finite global curvature characterizes simple closed curv es, whose arc length parameterizations possess a Lipschitz continuous tangent field. The investigation of local and non-local properties of global curvature motivates, in particular, an extended definition of local curvature at any point of a rectifiable loop. Finally we show how a bound on global curvature can be used to define and control topological constraints such as a given knot type for closed loops or a prescribed linking number for closed framed curves, suitable to describe, e.g., supercoiling phen omena of biomolecules.

Global curvature and self-contact of nonlinearly elastic curves and rods

Many different physical systems, e.g. super-coiled DNA molecules, have been successfully modelled as elastic curves, ribbons or rods. We will describe all such systems as framed curves, and will consider problems in which a three dimensional framed curve has an associated energy that is to be minimized subject to the constraint of there being no self-intersection. For closed curves the knot type may therefore be specified a priori. Depending on the precise form of the energy and imposed boundary conditions, local minima of both open and closed framed curves often appear to involve regions of self-contact, that is, regions in which points that are distant along the curve are close in space. While this phenomenon of self-contact is familiar through every day experience with string, rope and wire, the idea is surprisingly difficult to define in a way that is simultaneously physically reasonable, mathematically precise, and analytically tractable. Here we use the notion of global radius of curvature of a space curve in a new formulation of the self-contact constraint, and exploit our formulation to derive existence results for minimizers, in the presence of self-contact, of a range of elastic energies that define various framed curve models. As a special case we establish the existence of ideal shapes of knots.

Texture frame curves and regions of attention using adaptive non-cartesian networks

This paper addresses the problem of locating and extracting frame curves in images with complex backgrounds. Frame curves are virtual image curves which lie along “the center” of texture boundaries. They can be used to carry out other visual tasks or to lead to subsequent processing such as recognition, indexing or image retrieval. We present an approach based on dynamic programming and adaptive non-cartesian networks ( ANCN ). ANCN are based on line placement using a distribution function adapted to the image texture. We present a computational scheme to extract frame curves directly from the image and we provide the results of several experiments on real images.

Measurements and analysis of acoustic properties of drainage asphalt.

Acoustic properties of drainage asphalt are investigated to simulate the noise propagation of road traffic noise. Normalized acoustic impedance, porosity and flow resistivity are measured by an ordinary method with a small sample of maximum size of aggregate 13 mm and thickness of 45 mm. By referring to Allard's impedance model for rigid frame porous material and curve fitting technique, tortuosity of the sample is deduced. It is shown that this model gives a good fitting to normalized acoustic impedance measured and applicable to simulate the noise propagation on the drainage asphalt of road surface by a boundary element method. Next, effective flow resistivity is deduced approximately by a short range sound propagation over a drainage asphalt surface of a highway in operation. It is shown an empirical one-parameter impedance model is applicable to the prediction of noise propagation on the drainage asphalt surface by using an effective flow resistivity. Normal and random incident absorption coefficients are measured and presented for data supplements.

Attentional frames, frame curves and figural boundaries: The inside/outside dilemma

Attended and unattended regions of the image array are viewed often as binary complements to one another, with a well-defined boundary between them. A simple experiment shows otherwise: if the contour of a simple convex shape is perturbed to create a distinctive texture, it is typically the outside of the contour that provides the basis for similarity judgement, not the inside. The introduction of the appropriate task, however, can make the inside part of the contour become more salient. A similar result occurs for concave shapes, such as a C, where notions of containment are not easily specified. These observations suggest that figure boundaries are difficult to define objectively and that the setting of an attentional reference frame plays a key role in object description. We propose that this frame is part of a virtual, transparent blackboard or “clearboard”, bound to the scene, not to the image.