Shape Of Knots Research Articles

Research on Non-Destructive Testing of Log Knot Resistance Based on Improved Inverse-Distance-Weighted Interpolation Algorithm

The objective of this paper is to propose a non-destructive resistance detection imaging algorithm for log knots based on improved inverse-distance-weighted interpolation algorithm, i.e., the eccentric circle-based inverse-distance-weighted (ECIDW) method, to predict the size, shape, and position of internal knots of logs; evaluate its precision and accuracy; and both lay a theoretical foundation and provide a scientific basis for predicting and assessing knots in standing trees. Six sample logs with natural knots were selected for this study. Resistance measurements were performed on the log cross-sections using a digital bridge, and resistance tomography was conducted using the improved ECIDW algorithm, which combines the azimuth search method with the eccentric circle search method. The results indicated that both the conventional inverse-distance-weighted (IDW) algorithm and the ECIDW algorithm accurately predicted the positions of the knots. However, neither algorithm was able to predict the shape of the knots with high precision, leading to some discrepancies between the predicted and actual knot shapes. The relative error (Dt1) between the knot areas measured by the IDW algorithm and the actual knot areas ranged from 18.97% to 88.34%. The relative error (Dt2) for the knot areas predicted by the ECIDW algorithm ranged from 1.82% to 74.16%. The average prediction accuracy for the knot areas using the IDW algorithm was 51.58%, compared to 72.90% using the ECIDW algorithm. This indicates that the ECIDW algorithm has higher accuracy in predicting knot areas compared to the conventional IDW algorithm. The ECIDW algorithm proposed in this paper provides a more reasonable and accurate prediction and evaluation of knots inside logs. Compared to the conventional IDW algorithm, the ECIDW algorithm demonstrates greater precision and accuracy in predicting the shape and size of knots. While the resistance method shows significant potential for predicting internal knots in logs and standing trees, further improvements to the algorithm were needed to enhance the imaging effects and the precision and accuracy of knot area and shape predictions.

Three-dimensional inversion of knot defects recognition in timber cutting

The comprehensive utilization of wood is the main goal of log cutting, but knot defects increase the difficulty of rationally optimizing cutting. Due to the lack of real shape data of knot defects in logs, it is difficult for detection methods to establish a correlation between signal and defect morphology. An image-processing method is proposed for knot inversion based on distance regularized level set segmentation (DRLSE) and spatial vertex clustering, and with the inversion of the defects existing relative board position in the log, an inversion model of the knot defect is established. First, the defect edges of the top and bottom images of the boards are extracted by DRLSE and ellipse fitting, and the major axes of the ellipses made coplanar by angle correction; second, the coordinate points of the top and bottom ellipse edges are extracted to form a spatial straight line; third, to solve the intersection dispersion of spatial straight lines and the major axis plane, K-medoids clustering is used to locate the vertex. Finally, with the vertex and the large ellipse, a 3D cone model is constructed which can be used to invert the shape of knots in the board. The experiment was conducted on ten defective larch boards, and the experimental results showed that this method can accurately invert the shapes of defects in solid wood boards with the advantages of low cost and easy operation.

Prediction models of knot volume inside the stem for Korean pine plantation

Based on 1207 knots from 49 sample trees of 29 standard plots of Korean pine plantations in Linkou and Dongjingcheng Forest Bureau of Heilongjiang Province, China, we extracted longitudinal sections of knots using the image processing software Digimizer and represented the shape of knots using two-dimensional scatter plots. According to the two-dimensional scatter plots, knots of Korean pine plantation were divided into three types: 1) alive knots (whole knot contained only sound knot portion); 2) non-occluded dead knots (whole knot contained both sound and loose knot portions); 3) occluded dead knots (the sound and loose portion of the knot were partially occluded by the bark). For all the three types of knots, the volume of sound knot was calculated by mathematical integral of the sound knot shape equation. The volume of loose knot was calculated using the volume equation of a cylinder. The total volume of knots was calculated as the sum of sound and loose knot volume. Finally, based on knot variables (diameter, relative height and total length of knots) and tree variable (diameter at breast height), we established the prediction models for sound knot volume, loose knot volume, and total volume of knot using the linear mixed model at plot level and tree level. Compared with fixed-effects model, the mixed effects models of the volume of sound knot, loose knot, and total knots provided more accurate parameter estimation, more uniform residual distribution, and higher model fitting precision. The validation results showed that prediction precision of each fixed-effect model was higher than 90%, while that of the mixed models with plot and tree effect was above 93%, indicating that the established model could well predict the volume of knot for Korean pine plantation.

Kelvin wave and knot dynamics on entangled vortices

In this paper, starting from Biot–Savart mechanics for entangled vortex-membranes, a new theory — knot physics — is developed to explore the underlying physics of quantum mechanics. Owning to the conservation conditions of the volume of knots on vortices in incompressible fluid, the shape of knots will never be changed and the corresponding Kelvin waves cannot evolve smoothly. Instead, the knot can only be split. The knot-pieces evolve following the equation of motion of Biot–Savart equation that becomes Schrödinger equation for probability waves of knots. The classical functions for Kelvin waves become wave-functions for knots. The effective theory of perturbative entangled vortex-membranes becomes a traditional model of relativistic quantum field theory — a massive Dirac model. As a result, this work would help researchers to understand the mystery in quantum mechanics.

The average shape of the closed trefoil knot fluctuating on a floppy rope

The average shape of the trefoil knot tied on a floppy, hard rope subject to thermal fluctuations has been determined. The fluctuations of the shape of knots were performed by random bending. As a result of the changing shape procedure large sets of deformed conformations of the initial knot were obtained. Afterwards, these sets were subject to the shape-fitting procedure. It has been found that the conformation is different from the ideal conformation of the knot.

A Modified Synchrotron Model for Knots in the M87 Jet

To explain the broadband spectral shape of knots in the M87 jet from radio through optical to X-rays, we propose a modified synchrotron model that considers the integrated effect of particle injection from different acceleration sources in the thin acceleration region. This results in two break frequencies at two sides of which the spectral index of knots in the M87 jet changes. We discuss the possible implications of these results for the physical properties in the M87 jet. The observed flux of the knots in the M87 jet from radio to X-rays can be satisfactorily explained by the model, and the predicted spectra from ultraviolet to X-rays could be further tested by future observations. The model implies that knots D, E, F, A, B, and C1 are unlikely to be candidates for the TeV emission recently detected in M87.

Local structure of ideal shapes of knots

Relatively extremal knots are the relative minima of the ropelength functional in the C 1 topology. They are the relative maxima of the thickness (normal injectivity radius) functional on the set of curves of fixed length, and they include the ideal knots. We prove that a C 1 , 1 relatively extremal knot in R n either has constant maximal (generalized) curvature, or its thickness is equal to half of the double critical self distance. This local result also applies to the links. Our main approach is to show that the shortest curves with bounded curvature and C 1 boundary conditions in R n contain CLC (circle–line–circle) curves, if they do not have constant maximal curvature.

Dynamics of a rigid body in a Stokes fluid

We demonstrate that the dynamics of a rigid body falling in an infinite viscous fluid can, in the Stokes limit, be reduced to the study of a three-dimensional system of ordinary differential equations can be approximated using the Rotne–Prager theory, and we present various examples corresponding to certain ideal shapes of knots which illustrate the various possible multiplicities of steady states. Our simulations of rigid ideal knots in a Stokes fluid predict an approximate linear relation between sedimentation speed and average crossing number, as has been observed experimentally for the much more complicated system of real DNA knots in gel electrophoresis.

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.

Global curvature, thickness, and the ideal shapes of knots.

The global radius of curvature of a space curve is introduced. This function is related to, but distinct from, the standard local radius of curvature and is connected to various physically appealing properties of a curve. In particular, the global radius of curvature function provides a concise characterization of the thickness of a curve, and of certain ideal shapes of knots as have been investigated within the context of DNA.

Modelling of knots in logs

A geometrical model was derived to describe knots in logs and on the surface of lumber beams sawn from those logs. Each knot is defined by 7 parameters related to the shape and position in the stem. A computer simulation program was written to study knot shapes on a variety of observation planes. An initial investigation on the shape of knots in Scots pine logs and lumber beams suggests that the model is sufficiently accurate to describe knottiness in this species. Potential applications of this model include automated lumber grading, computerized log reconstitution and yield optimization studies.