Propagation of curved folding: the folded annulus with multiple creases exists

In this paper we consider developable surfaces which are isometric to planar domains and which are piecewise differentiable, exhibiting folds along curves. The paper revolves around the longstanding problem of existence of the so-called folded annulus with multiple creases, which we partially settle by building upon a deeper understanding of how a curved fold propagates to additional prescribed foldlines. After recalling some crucial properties of developables, we describe the local behaviour of curved folding employing normal curvature and relative torsion as parameters and then compute the very general relation between such geometric descriptors at consecutive folds, obtaining novel formulae enjoying a nice degree of symmetry. We make use of these formulae to prove that any proper fold can be propagated to an arbitrary finite number of rescaled copies of the first foldline and to give reasons why problems involving infinitely many foldlines are harder to solve.

Optimal Control of a Formula One Car on a Three-Dimensional Track—Part 2: Optimal Control

The optimal control of a Formula One car on a three-dimensional (3D) track is studied. The track is described by its geodesic and normal curvatures, and its relative torsion. These curvature parameters are obtained from noisy measurement data using the optimal estimation technique described in Part 1. The optimal control calculations presented are based on the aforementioned track model and a vehicle model that is responsive to the geometric features of a 3D track. For vehicle modeling purposes, the track is treated as a plane tangent to a nearby point on the track's spine. This tangent plane moves under the car and is orthogonal to the principal normal vector m at the nearby spine point. Results are presented that compare two-dimensional (2D) and 3D minimum-lap-time results, with the two compared. The Barcelona Formula One track studied in Part 1 is used again as an illustrative example.

The Absolute Torsion of Circular Shaft and Neutral Cross-Section

Mechanics of materials are based on the relative deformation to discuss the calculation of the circular shaft torsion deformation, But the relative torsion angle can not reflect the real deformation of a specific cross-section. According to the plane hypothesis, to discuss the absolute deformation of several common shaft withstand the simple torque, defined the neutral cross-section of torsion deformation and given the method to calculate absolute torsion angle, for the rational design of engineering components have a certain reference value.

RELATIVE TORSION

This paper achieves, among other things, the following: • It frees the main result of [9] from the hypothesis of determinant class and extends this result from unitary to arbitrary representations. • It extends (and at the same times provides a new proof of) the main result of Bismut and Zhang [3] from finite dimensional representations of Γ to representations on an [Formula: see text]-Hilbert module of finite type ([Formula: see text] a finite von Neumann algebra). The result of [3] corresponds to [Formula: see text]. • It provides interesting real valued functions on the space of representations of the fundamental group Γ of a closed manifold M. These functions might be a useful source of topological and geometric invariants of M. These objectives are achieved with the help of the relative torsion ℛ, first introduced by Carey, Mathai and Mishchenko [12] in special cases. The main result of this paper calculates explicitly this relative torsion (cf. Theorem 1.1).

Scoliosis in Friedreich’s Ataxia

SUMMARY:The preliminary results based on a three year study on the evolution and management of scoliosis in Friedreich’s ataxia are presented. Thirty-two patients were followed in the Neuromuscular Disease Clinic at Sainte-Justine Hospital where standardized spinal radiographs were taken periodically with the Scoliosis Chariot and the Throne.Pathomechanics — Between the age groups I (5 – 10 years) and II (10 – 15 years), a substantial increase in the Cobb values occurs. Associated with it, an increase was observed in the thoracic and thoraco-lumbar projected surface area indices. The relative torsion between the thoracic and lumbar segments was associated with a sudden increase in the Cobb measurements.A computer program which generates a tri-dimensional display of the spine from the coordinates obtained from the standardized radiographs has been adapted to study the importance and influence of the thoracic torsion on the progress of the scoliosis and on the evolution of the total spinal deformity.Management — Prevention of the progression of established curves was our main objective. Careful examination of the spine, depending on the age of the child, in our preliminary study, stimulated early orthopaedic treatment in any curve of 20° or more. There was always concern for curves of 30° or more. In the growing child, bracing which was recommended is now under a more thorough investigation. The use of spinal pacemakers is being clinically evaluated.In the older child, the curve was usually stable after sixteen years of age. Surgery was usually attempted in curves over 40° in the growing child. The same curve was usually stable after the growth period.For the non-ambulatory patients, the present study suggested the prescription of a molded seat with the following characteristics: i) a posterior lumbar support, ii) low thoracic lateral supports and Hi) a slight inclination of the seating system. This was presumed to be beneficial in maintaining stability of the spine.