Curvature

Summary

In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius. Smaller circles bend more sharply, and hence have higher curvature. The curvature at a point of a differentiable curve is the curvature of its osculating circle, that is the circle that best approximates the curve near this point. The curvature of a straight line is zero. In contrast to the tangent, which is a vector quantity, the curvature at a point is typically a scalar quantity, that is, it is expressed by a single real number. For surfaces (and, more generally for higher-dimensional manifolds), that are embedded in a Euclidean space, the concept of curvature is more complex, as it depends on the choice of a direction on the surface or manifold. Th

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Global radii of curvature, and the biarc approximation of space curves

Jana Smutny

The distance from self-intersection of a (smooth and either closed or infinite) curve q in three dimensions can be characterised via the global radius of curvature at q(s), which is defined as the smallest possible radius amongst all circles passing through the given point and any two other points on the curve. The minimum value of the global radius of curvature along the curve gives a convenient measure of curve thickness or normal injectivity radius. Given the utility of the construction inherent to global curvature, it is natural to consider variants defined in related ways. The first part of the thesis considers all possible circular and spherical distance functions and the associated, single argument, global radius of curvature functions that are constructed by minimisation over all but one argument. It is shown that among all possible global radius of curvature functions there are only five independent ones. And amongst these five there are two particularly useful ones for characterising thickness of a curve. We investigate the geometry of how these two functions, ρpt and ρtp, can be achieved. Properties and interrelations of the divers global radius of curvature functions are illustrated with the simple examples of ellipses and helices. It is known that any Lipschitz continuous curve with positive thickness actually has C1,1-regularity. Accordingly, C1,1 is the natural space in which to carry out computations involving self-avoiding curves. The second part of the thesis develops the mathematical theory of biarcs, which are a geometrically elegant way of discretizing C1,1 space curves. A biarc is a pair of circular arcs joined in a C1 fashion according to certain matching rules. We establish a self-contained theory of the geometry of biarc interpolation of point-tangent data sampled from an underlying base curve, and demonstrate that such biarc curves have attractive convergence properties in both a pointwise and function-space sense, e.g. the two arcs of the biarc interpolating a coalescent point-tangent data pair on a C2-curve approach the osculating circle of the curve at the limit of the data points, and for a C1,1-base curve and a sequence of (possibly non-uniform) meshes, the interpolating biarc curves approach the base curve in the C1-norm. For smoother base curves, stronger convergence can be obtained, e.g. interpolating biarc curves approach a C2 base curve in the C1,1-norm. The third part of the thesis concerns the practical utility of biarcs in computation. It is shown that both the global radius of curvature function ρpt and thickness can be evaluated efficiently (and to an arbitrarily small, prescribed precision) on biarc curves. Moreover, both the notion of a contact set, i.e. the set of points realising thickness, and an approximate contact set can be defined rigorously. The theory is then illustrated with an application to the computation of ideal shapes of knots. Informally ideal knot shapes can be described as the configuration allowing a given knot to be tied with the shortest possible piece of rope of prescribed thickness. The biarc discretization is combined with a simulated annealing code to obtain approximate ideal shapes. These shapes provide rigorous upper bounds for rope length of ideal knots. The approximate contact set and the function ρpt evaluated on the computed shapes allow us to assess closeness of the computations to ideality. The high accuracy of the computations reveal various, previously unrecognized, features of ideal knot shapes.

EPFL2004

Déformations conformes des variétés de Finsler-Ehresmann

Ratiba Djelid

An intrinsic approach to Finsler geometry is proposed. A concept of Finsler- Ehresmann manifold, denoted by (M,F,H), is introduced and a generalized Chern connection is built for this manifold. Conformal deformations on this manifold are considered. First, we have an analogous of Chern's theorem: we prove the existence and uniqueness of a generalized Chern connection for the manifold (M,F,H). Similarly, within an essentially koszulian formalism, we present two curvatures associated to this generalized connection, namely a R curvature and a P one. The second result is the deduction of conformal transformations laws for the generalized Chern connection and associated curvatures. The transformation of R seems to have very similar properties as that of the Riemannian curvature while that of P reveals other objects of pure Finslerian nature. Third, we construct the finsler Weyl and Schouten tensors W and S respectively and we study their conformal transformations. Furthermore, we show that for the dimension 3, the horizontal component of W for generalized Berwald manifolds is identically zero. The next result is a theorem of Weyl-Schouten type giving necessary and sufficient conditions for a Finsler-Ehresmann manifold to be conformaly R-flat. We complete this result by exploring the case of dimension 3 for Berwald spaces which gives a result very similar to the Riemannian case. In addition, we announce some necessary conditions to characterize conformal flatness of Finsler-Ehresmann manifolds.

EPFL2011

In recent years, optical gas detectors based on a tunable diode laser absorption spectrometer has become more and more important for monitoring trace gases in atmospheric, medical, and industrial processes. In the medical field, blood gases, essentially pO2 and pCO2, are vital parameters for patient respiratory status monitoring. Typically, hypoxaemia or hypercapnia can be very harmful, especially for neonates, and requires continuous monitoring for rapid clinical intervention. Transcutaneous blood gases, permeating through the skin, can be continuously measured in patients by electrochemical sensors applied to the surface of the skin. These sensors suffer specific instability and must be re-calibrated. For this reason, the present thesis explores a new concept of gas sensors, applying light absorption at the surface of an integrated optics waveguide and using the advantages of a modulated spectroscopy technique for sensitivity and selectivity enhancement. In other words, the present thesis sets a mathematical model and demonstrates experimentally a new concept of evanescent wave gas sensing based on modulation spectroscopy. The field of application considered in this book is restricted to gas detection, however the concept can also be applied to the detection of liquids. In particular, we demonstrate the potential applicability of the concept to transcutaneous blood gas monitoring, and especially carbon dioxide monitoring. The system transfer function modelization of the concept includes carbon dioxide absorption spectrum in the near-infrared region (1.6µm), laser diode modulation spectroscopy assuming optical frequency modulation (FM) as well as intensity modulation (IM) induced by current modulation, and integrated optics sensor-head transmission. The modelization evidences that the silicon nitride technology has the advantage of: high evanescent wave sensitivity (8% for the selected TE mode), low propagation loss, possibility of strong curvatures for miniaturization and availability of fiberto- waveguide mode matching technique (taper). In the modelization, we evidence that the optimal waveguide length equals the inverse of the propagation loss which is mainly limited by the waveguide scattering. Also, we demonstrate that the implementation of a fiber-to-waveguide mode matching taper at both waveguide ends, drastically decreases the resonance limitation of the system resolution. Experimentally, the transfer function of the concept is validated and characterized based on a free-space modulation spectroscopy setup whose performance, in terms of second harmonic resolution, is first measured at 3.55 · 10-6 1/√Hz corresponding to 0.275 mmHg/√Hz CO2 partial pressure. In the chosen region of 200-400MHz of frequency modulation, we confirm that the 1/f-noise is negligible and the dominating amplifier noise can be approximated by the thermal noise. In the case of high insertion loss and short interaction length the interferometric noise (or etalon-effect) becomes the system resolution limitation. By the insertion of a prism coupled planar waveguide instead of the free-space cell, we demonstrate the concept of integrated optics evanescent wave sensing based on modulation spectroscopy. The system characterization is completed by the validation of the evanescent wave sensitivity in gas of about 8% for a laterally confined waveguide. The outlook of a transcutaneous blood gas monitoring instrument is also established. We first extrapolate the expected waveguide characteristics to show that a resolution of 1mmHg CO2 partial pressure can be reached in 4.5 to 12sec response-time. Then, the requirement of a maximal sensing area of 10mm diameter is at the origin of two geometries of sensor-head: a crossing spiral and a non-crossing double spiral. We verify experimentally a propagation loss as low as 0.5dB/cm and observe a stringent tolerance for light injection. The latter requires an integrated device for mode matching and pigtailing. After implementation of a taper at both waveguide ends, we demonstrate the drastic reduction of the total coupling loss. The carbon dioxide measurements reveal a relatively deep pattern on the second harmonic signal that tends to prove that polarization and resonances are coupled together most likely at the waveguide level. Applying nitrogen (N2) as reference gas, the pattern compensation results in the carbon dioxide signature measurement in direct detection only. The next generation, now in process, is a patented pigtailed non-crossing double-spiral sensor-head building the first step towards an industrial prototype. The waveguide miniaturization allows an interaction length longer than 21cm on a 3×3mm area. The improvement towards a high resolution system concerns the propagation loss and the coupling loss on the sensor-head, and concerns the polarization control or scrambling on the modulation spectroscopy setup. We finally conclude, following the demonstration on the planar waveguide and the extrapolation of the measured characteristics, that the concept of evanescent wave gas sensing based on modulation spectroscopy, can compensate the lack of sensitivity of an integrated optics sensor-head and can be applied in gas and liquid sensing. Moreover, based on the sensor-head developments and realistic improvements, we also conclude that the perspective of high resolution sensing system, and in particular for carbon dioxide determination, evidences the potential applicability of the concept to transcutaneous blood gas monitoring. This opens a new optical outlook of continuous and non-invasive blood gas monitoring, applied in medicine for patient respiratory status monitoring.

EPFL2003

Global radii of curvature, and the biarc approximation of space curves

Jana Smutny

EPFL2004

EPFL2003