Helices through 3 or 4 pointsΦ

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How many points in space are needed to define a circular helix? We show here that given 3 distinct points in space there exist continuous families of helices passing through these points. Given 4 generic distinct points there is no helix. However, a discrete family of helices through 3 points can be specified if an additional property of the helix is prescribed. In particular, the case where the helical radius is specified is studied in detail.

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... for t ∈ (0, 2π), where r is the helix radius and 2πc is the separation between helix loops. In this case, the number of points required for the random sampling is n = 3, but the model might also need to constraint the radius with a cylinder [29]. ...
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A tracking algorithm based on consensus-robust estimators was implemented for the analysis of experiments with time-projection chambers. In this work, few algorithms beyond RANSAC were successfully tested using experimental data taken with the AT-TPC, ACTAR and TexAT detectors. The present tracking algorithm has a better inlier-outlier detection than the simple sequential RANSAC routine. Modifications in the random sampling and clustering were included to improve the tracking efficiency. Very good results were obtained in all the test cases, in particular for fitting short tracks in the detection limit.
We present a class of non-polynomial parametric splines that interpolate the given control points and show that some curve types in this class have a set of highly desirable properties that were not previously demonstrated for interpolating curves before. In particular, the formulation of this class guarantees that the resulting curves have C ² continuity everywhere and local support, such that only four control points define each curve segment between consecutive control points. These properties are achieved directly due to the mathematical formulation used for defining this class, without the need for a global numerical optimization step. We also provide four example spline types within this class. These examples show how guaranteed self-intersection-free curve segments can be achieved, regardless of the placement of control points, which has been a limitation of prior interpolating curve formulations. In addition, they present how perfect circular arcs and linear segments can be formed by splines within this class, which also have been challenging for prior methods of interpolating curves.
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A detailed analysis of structural and position dependent characteristic features of helices will give a better understanding of the secondary structure formation in globular proteins. Here we describe an algorithm that quantifies the geometry of helices in proteins on the basis of their C alpha atoms alone. The Fortran program HELANAL can extract the helices from the PDB files and then characterises the overall geometry of each helix as being linear, curved or kinked, in terms of its local structural features, viz. local helical twist and rise, virtual torsion angle, local helix origins and bending angles between successive local helix axes. Even helices with large radius of curvature are unambiguously identified as being linear or curved. The program can also be used to differentiate a kinked helix and other motifs, such as helix-loop-helix or a helix-turn-helix (with a single residue linker) with the help of local bending angles. In addition to these, the program can also be used to characterise the helix start and end as well as other types of secondary structures.
This paper describes a method for enhancing the initial design process, as well as the transfer of data, for the geometry of compound contact springs. A force-length constrained model is developed around a cantilever beam section which has short (less than 10% of the beam's length) structural elements to facilitate proper positioning. The short elements are often considered insignificant in deflection analysis, but are shown to contribute an additional 26% to the structure's deflection. Once this is done, a mount position constrained model is developed which must be solved iteratively. The resulting geometry is smooth, precise, and can be readily transferred to CAD to complete a robust design process.
Helical conformations of infinite polymer chains may be described by the helical parameters, d and θ (the translation along the helix axis and the angle of rotation about the axis per repeat unit), pi (the distance of the ith atom from the axis), dij, and dij (the translation along the axis and the angle of rotation, respectively, on passing from the ith atom to the jth). A general method has been worked out for calculating all those helical parameters from the bond lengths, bond angles, and internal-rotation angles. The positions of the main chain and side chain atoms with respect to the axis may also be calculated. All the equations are applicable to any helical polymer chain and are readily programmed for electronic computers. A method is also presented for calculating the partial derivatives of helical parameters with respect to molecular parameters.
Circle and helix fitting is of paramount importance in the data analysis of LHC experiments. We review several approaches to exact but fast fitting, including a recent development based on the projection of the measured points onto a second-order surface in space (a sphere or a paraboloid). We present results of a comparison with global and recursive linearized least-squares estimators.
We present an extension of the Riemann circle fit to a helix fit in space. The method is studied both in barrel- and disk-type detectors. We show results from two simulation experiments, including a comparison to linear regression and to the Kalman filter. An implementation in C++ is described.
The paper selects and develops appropriate numerical solutionmethods for initial boundary value problems of the equations of motionof geometrically nonlinear extensible Euler–Bernoulli-beams. A finiteelement method that uses first-order Hermitian polynomials as interpolationfunctions for the rod axis position vector is used as discretizationtechnique. An averaging method for the calculation of net forces andmoments is developed that achieves a better approximation than thedirect calculation from the strains. Time integration is done usingan energy and momentum conserving algorithm that is proposed in thispaper and Newmark type methods. The derived algorithms are used tosolve problems from space and marine engineering. The obtained simulationresults are compared with results which have been already publishedin the literature or were calculated by different methods.
A polyhelix is continuous space curve with continuous Frenet frame that consists of a sequence of connected helical segments. The main result of this paper is that given n points in space, there exist infinitely many polyhelices passing through these points. These curves are by construction continuous with continuous derivatives and are completely specified by 3n numbers, i.e., the initial position, the signed curvature, torsion, and length of each helical segment. Polyhelices can be parametrised by the arc length and easily expressed in terms of product of matrices.
Several methods for finding the axis of a helix are presented and compared. The most accurate determines the helix axis as the axis of rotation necessary to map point i to point i + 1 of the helix. The fastest method calculates the helix axis as the best-fit line through the coordinates by a three-dimensional parametric linear least-squares algorithm, taking advantage of the sequential nature of the data.
Methods for generating compound spring element curves
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N. Chouaieb, A. Goriely, and J. H. Maddocks. Helices. Proc. Nat. Acad. Sci. USA, 103(25):9398-9403, 2006.