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Differential Calculus of Vector Functions of one Variable
Published in C. Young Eutiquio, Vector and Tensor Analysis, 2017
The scalar factor τ(s) is called the torsion of the curve at the point corresponding to the parameter s. The torsion of a curve is a measure of the rate at which the binormal vector or the osculating plane rotates about the tangent vector as it moves along the curve. The negative sign in (2.35) is introduced so that the torsion will come out positive when the right-handed triple t, n, b rotates in a right. handed sense about the tangent vector as it moves in the positive direction on the curve.
Digitization of three-dimensional spine curvature profile in adolescent idiopathic scoliosis using anatomical palpation
Published in Computer Methods in Biomechanics and Biomedical Engineering: Imaging & Visualization, 2023
Patrick Salvia, Robert Elbaum, Emilie Bourgois, Shanon Péché, Véronique Feipel, Benoit Beyer
To the best of our knowledge, no true 3D approach has been employed in these non-invasive methods. In this study, the calibrated index finger pulp served as a probe, denoted as A-palp (Salvia et al. 2009b), to digitise the spinal curvature. Following the idea of constructing a 3D trajectory from biplanar X-ray views (Kadoury and Labelle 2012), the continuous trajectory was smoothed by B-spline in the sagittal and frontal planes to obtain a well-fitting 3D curvature. In our approach, Frenet frames were used as a tool to help detect the geometric torsion of the curve (Poncet et al. 2001). However, because of the numerical instability of this parameter, the direction change was confirmed manually using mouse clicks.
Characterization and quantification of railroad spiral-joint discontinuities
Published in Mechanics Based Design of Structures and Machines, 2022
The three Euler angles can be written in terms of the curvature and torsion of the curve using the differential relationships (Ling and Shabana 2020)