Common Knot Research Articles

Tangled Physics: Knots as a challenge for physical scene understanding

A resurgence of interest in intuitive physics has revealed a remarkable capacity in humans to make common-sense predictions about the unfolding of physical scenes. For example, recent work has shown that observers can correctly judge properties such as stability, weight distribution, gravity and collision physics, especially in rich naturalistic images. These results have suggested that physical scene understanding recruits a general-purpose “physics engine” that reliably simulates how scenes will unfold. Here, we complicate this picture by introducing knots to the study of intuitive physics. Knots are naturalistic stimuli that appear across cultures and time periods, and are widely used in both mundane scenarios (e.g., closing a bag or tying one’s shoes) and more technical applications (e.g., securing a boat or even supporting a rock-climber). Yet, here we show that even basic judgments about knots strain human physical reasoning. Observers viewed photographs of simple “bends” (i.e., knots joining two lengths of string) that share strong visual similarity but greatly differ in structural integrity. For example, observers saw not only “reef” knots (a common knot used for millennia), but also “thief” knots (which differ only in the position of a single strand but are significantly less secure than “reefs”), along with “granny” and “grief” knots (which share a similar relationship). In a two-alternative forced-choice task, observers judged each knot’s stability relative to every other knot. Strikingly, observers reliably ranked weaker knots as strong and stronger knots as weak, both across knot-families (e.g., incorrectly judging granny knots as stronger than reef knots) and within a given family (e.g., failing to judge “thiefs” as weaker than “reefs”). These failures challenge a general-purpose “physics engine” hypothesis, and perhaps suggest that knots and other examples of soft-body physics recruit different cognitive processes than rigid-body physics.

A Knot Invariant Defined Based on the Skein Relation with Two Equations

Knot theory is a branch of the geometric topology, the core question of knot theory is to explore the equivalence classification of knots; In other words, for a knot, how to determine whether the knot is an unknot; giving two knots, how to determine whether the two knots are equivalent. To prove that two knots are equivalent, it is necessary to turn one knot into another through the same mark transformation, but to show that two knots are unequal, the problem is not as simple as people think. We cannot say that they are unequal because we can't see the deformation between them. For the equivalence classification problem of knots, we mainly find equivalent invariants between knots. Currently, scholars have also defined multiple knot invariants, but they also have certain limitations, and even more difficult to understand. In this paper, based on existing theoretical results, we define a knot invariant through the skein relation with two equations. To prove this knot invariant, we define a function <i>f</i>(<I>L</I>), and to prove <i>f</i>(<I>L</I>) to be a homology invariant of a non-directed link, we need to show that it remains constant under the Reideminster moves. This article first defines the <i>f_k</i>(<I>L</I>), the property of <i>f</i>(<I>L</I>) is obtained by using the properties of <i>f_k</i>(<I>L</I>). In the process of proof, the induction method has been used many times. The proof process is somewhat complicated, but it is easier to understand. And the common knot invariant is defined by one equation, which defining the knot invariant with two equations in this paper.

Modello per Pirandello: Capuana e gli spiriti

Il saggio analizza l’influenza di Capuana sull’opera di Luigi Pirandello. Dalle riflessioni sullo spiritismo, e dalle esperienze mediatiche: di scrittura automatica, Capuana elabora la teoria dell’allucinazione artistica e proclama il principio di un’arte che crea personaggi vivi, capaci di agire nella realtà come esseri autonomi. Tra il 1904 e il 1915, il giovane Pirandello raccoglierà alcuni spunti di quelle riflessioni, com’è evidente ne Il Fu Mattia Pascal, per la scena della seduta spiritica, e soprattutto nel racconto Personaggi e poi in Sei personaggi in cerca d’autore. Il mistero della perce- zione sensibile è il nodo comune dei due scrittori siciliani che, alle soglie del Novecen- to, sembrano passarsi il testimone sulle domande relative al principio della duplicità dell’essere.

A moray’s many knots: knot tying behaviour around bait in two species of Gymnothorax moray eel

Being able to tie their body into a knot is beneficial to some moray eels and other cylindrical vertebrates for various reasons, although not all morays’ tie knots. Knot tying can generate leverage for feeding activity by moving the knot up the body, and this leverage may be affected by knot type. Three types of knots are known to have been used by morays’ and this study further elucidates this behaviour and ability. Baited Remote Underwater Videos (BRUVs) were used to compare knot tying between the moray eels Gymnothorax prasinus and Gymnothorax prionodon in the Solitary Islands region, eastern Australia. Gymnothorax prasinus tied more knots overall and more types of knot than G. prionodon (five types vs one type, respectively). Both tied the overhand knot and this was the most common knot used by G. prasinus. The figure of eight knot was used two times more than the double overhand knot, being relatively wider and likely easier to move up the body. Gymnothorax prasinus also tied two undescribed knots, now called the ‘moray knot’ and the ‘morays banana knot’, which have more crossings and are bulkier, increasing potential for generating leverage. However, their uncommon use indicates costs may outweigh benefits of these more complex knots. Lower habitat complexity in deeper waters may influence differences in knot tying as G. prasinus were only recorded shallower than 40 m and G. prionodon mainly deeper. These findings further reveal the utility and complexity of knot tying in moray species and cylindrical vertebrates generally.

B-Spline Surface Approximation to Scanned Data Using Least Square Approximation

Lofting is widely used to approximate the scanned data in row-wise fashion, but this method is prone to result an astonishing number of control points in the process of making the rows curve compatible. A novel algorithm of B-spline surface approximation to the scanned data is presented in this paper to solve this problem. Firstly, the scanned data are interpolated by rows of curves; then these curves are approximated by other curves using least square approximation. In this process, all curves are approximated by a common knot vector, and it is different form the traditional method that each curve is approximated by a different knot vector, so we needn’t insert many knots in each curve to make curves compatible. We also can meet high accuracy without losing the shape of lofting surface because we firstly interpolate the data, the best least square approximation substitute insertion of knots in lofting. Numerical example shows that the proposed method is efficient in reducing control points of the lofting surface.

Automated Feature Extraction from Profiles with Application to a Batch Fermentation Process

SummaryAn automated approach to extract interpretable features of univariate or multivariate profiles (functional data) is proposed. A landmark alignment algorithm is modified and the alignment is combined with piecewise linear approximations. Least absolute shrinkage and selection operator (lasso) regression is used for selecting the most important intercepts and slopes and yields an alternative to partial least squares to model a response associated with the profiles. Latent variables can be difficult to interpret but our extracted features simply correspond to slopes and intercepts of particular parts of the profiles. Also, features that relate to the degree of warping between a given profile and a reference can be extracted as predictors. Selection criteria for the number of knots and common knot locations between profiles are developed. We apply our proposed method to batch fermentation data where the profiles consist of on-line measurements of process variables and the corresponding yield of the process. The extracted features have good interpretability (with large dimensional reduction) and in combination with the lasso have prediction accuracy which is comparable with that of partial least squares applied to the original profiles. Also our proposed feature extraction method is applied to publicly available data where near infrared spectra define the profiles and the prediction accuracy of our feature lasso method is comparable with those of more complicated alternatives.

Application of a real-coded genetic algorithm for the fitting of a ship hull surface through a single non-uniform B-spline surface

In digital ship-design processes, surface modeling needs to be as accurate as possible for effectiveness in ship production as well as numerical analysis of the performance. Traditionally, the form of a ship hull is constructed from a set of cross-sectional data. This approach entails difficulties in the cross-sectional spacing and accuracy of the characteristic curves, such as the stern and bow profiles, deck side line, bottom tangential line, and unconnected curves. Genetic algorithms (GAs) have attracted increasing attention as a multimodal optimization solution for surface reconstruction that enable construction of a single non-uniform B-spline (NUB) surface at the initial stage of ship design with constraints such as knuckles, discontinuity conditions, and bulbous bows with high curvatures, . The first, simultaneous multi-fitting GA determines the boundary curves, such as the stem and stern profiles, and finds the common knot values for both curves. Similarly, the same GA technique is applied for other boundary curves at the bottom and the deck. The second GA is employed to fit the interior data points after the boundary curves are fitted. The encoded design variables for surface construction are the locations of the vertices and the knot values. Those variables are modified for improving the surface quality until a predefined degree of precision is attained. In four instances of application, the GA technique developed in this research has been shown to provide good, single, NUB surfaces with high efficiency. In the early design stage, a single NUB surface is more convenient for performance visualization and finite-element methods. It can be readily translated into many CAD/CAM packages, which facilitate the smooth transition of data across the different design stages.

B-spline surface fitting based on adaptive knot placement using dominant columns

By expanding the idea of B-spline curve fitting using dominant points (Park and Lee 2007 [13]), we propose a new approach to B-spline surface fitting to rectangular grid points, which is based on adaptive knot placement using dominant columns along u - and v -directions. The approach basically takes approximate B-spline surface lofting which performs adaptive multiple B-spline curve fitting along and across rows of the grid points to construct a resulting B-spline surface. In multiple B-spline curve fitting, rows of points are fitted by compatible B-spline curves with a common knot vector whose knots are computed by averaging the parameter values of dominant columns selected from the points. We address how to select dominant columns which play a key role in realizing adaptive knot placement and thereby yielding better surface fitting. Some examples demonstrate the usefulness and quality of the proposed approach.

Reducing control points in lofted B-spline surface interpolation using common knot vector determination

A new algorithm for reducing control points in lofted surface interpolation to rows of data points is presented in this paper. The key step of surface lofting is to obtain a set of compatible B-spline curves interpolating each row. Given a set of points and their parameterization, a necessary and sufficient condition is proposed to determine the existence of interpolating B-spline curves defined on a given knot vector. Based on this condition, we first properly construct a common knot vector that guarantees the existence of interpolating B-spline curves to each row of points. Then we calculate a set of interpolating B-spline curves defined on the common knot vector by energy minimization. Using this method, fewer control points are employed while maintaining a visually pleasing shape of the lofted surface. Several experimental results demonstrate the usability and quality of the proposed method.

Dominant point detecting based non-uniform B-spline approximation for grain contour

Three-dimension reconstruction from serial sections has been used in the last decade to obtain information concerning three-dimensional microstructural geometry. One of the crucial steps of three-dimension reconstruction is getting compact and fairing grain contours. Based on the achievement of closed raw contours of ceramic composite grains by using wavelet and level set, an adaptive method is adopted for the polygonal approximation of the digitized raw contours. Instead of setting a fixed length of support region in advance, the novel method computes the suitable length of support region for each point to find the best estimated curvature. The dominant points are identified as the points with local maximum estimated curvatures. Periodic closed B-spline approximation is used to find the most compact B-spline grain boundary contours within the given tolerance. A flexible distance selection approach is adopted to obtain the common knot vector of serial contours consisting of less knots that contain enough degrees of freedom to guarantee the existence of a B-spline curve interpolating each contour. Finally, a B-spline surface interpolating the serial contours is generated via B-spline surface skinning.

Lofted B-spline surface interpolation by linearly constrained energy minimization

This paper proposes a new approach for lofted B-spline surface interpolation to serial contours, where the number of points varies from contour to contour. The approach first finds a common knot vector consisting of fewer knots that contain enough degrees of freedom to guarantee the existence of a B-spline curve interpolating each contour. Then, it computes from the contours a set of compatible B-spline curves defined on the knot vector by adopting B-spline curve interpolation based on linearly constrained energy minimization. Finally, it generates a B-spline surface interpolating the curves via B-spline surface lofting. As the energy functional is quadratic, the energy minimization problem leads to that of solving a linear system. The proposed approach is efficient in computation and can realize more efficient data reduction than previous approaches while providing visually pleasing B-spline surfaces. Moreover, the approach works well on measured data with noise. Some experimental results demonstrate its usefulness and quality.

Free-form surface reconstruction based on NURBS to serial cross-sections

A new method for recovering shape from cross-sectional contours with complex branching structures is presented. First, each branching problem by providing an intermediate contour using distance function and image processing technology is solved. Then, all contours are divided into several groups of simple contours. For each group, a NURBS curve is fitted to contour points in each section within a given accuracy on a common knot vector. Finally, the NURBS surface skinning of these contours is performed for providing a smooth geometric model. The method is suitable to reproduce the object by NC machining or rapid prototyping. Some results demonstrate its usefulness and feasibility.

A method for approximate NURBS curve compatibility based on multiple curve refitting

This paper presents an approximate method for making a set of NURBS curves of various types compatible. The method is based on refitting input curves simultaneously on a common knot vector while making the refitted curves have parametrically regular transition across curves. It adopts a concept of reducing the number of interior knots of the common knot vector to acquire more compact representation for the resultant curves. It also incorporates energy minimization in the curve refitting process to stabilize the solution of the curve fitting problem and to get smooth curves. The proposed method can be utilized in several construction tasks including surface skinning and surface creation from a mesh of curves. It plays an important role in improving the quality of surfaces and realizing efficient reduction of their control points especially when the parameterization and the compatibility of input curves are not good enough. Some experimental results demonstrate its usefulness and quality.

B-Spline Surface Approximation to Cross-Sections Using Distance Maps

The shape reconstruction of a 3D object from its 2D crosssections is important for reproducing it by NC machining or rapid prototyping. In this paper, we present a method of surface approximation to cross-sections with multiple branching problems. In this method, we first decompose each multiple branching problem into a set of single branching problems by providing a set of intermediate contours using distance maps. For each single branching region, a procedure then performs the skinning of contour curves represented by cubic B-spline curves on a common knot vector, each of which is fitted to its contour points within a given accuracy. In order to acquire a more compact representation for the surface, the method includes an algorithm for reducing the number of knots in the common knot vector. The approximation surface to the crosssections is represented by a set of bicubic B-spline surfaces. This method provides a smooth surface model, yet realises efficient data reduction.

Smooth surface approximation to serial cross-sections

The reconstruction of the surface model of an object from 2D cross-sections plays an important role in many applications. In this paper, we present a method for surface approximation to a given set of 2D contours. The resulting surface is represented by a bicubic closed B-spline surface with C 2 continuity. The method performs the skinning of intermediate contour curves represented by cubic B-spline curves on a common knot vector, each of which is fitted to its contour points within a given accuracy. In order to acquire more compact representation for the surface, the method includes an algorithm for reducing the number of knots in the common knot vector. The proposed method provides a smooth and accurate surface model, yet realizes efficient data reduction. Some experimental results are given using synthetic and MRI data.

Multiresponse spline regression

In this paper we propose using free-knot splines with a common knot vector to approximate multiresponse model functions and their associated error variance-covariance matrix. A “loose coupling” type of algorithm for minimizing the determinant parameter estimation criterion is constructed for multiresponse spline models. The algorithm evaluates the increment on the parameters of the common knot positions first, then evaluates the increment on spline coefficients individually. One constraint on using the determinant criterion for multiresponse estimation is that the number of design points has to be greater than the number of parameters. This makes spline models infeasible when a large number of responses are measured. We loosen the constraint on the number of parameters allowed and propose a more appropriate residual degrees of freedom for the spline models.

Représentation mathématique des noeuds dans le tronc des arbres

This study is aimed at developing equations describing the shape of any particular knot in the trunk of a tree. The present model differs from previous models in that it is not limited by the assumption of a straight conical knot. The knot is represented by an assemblage of elliptical cone sections with the axis defined by a Hermite-type cubic equation or a straight line, and ends in a half ellipsoid symbolizing the enclosure of an intergrown knot. Simulations of knot shapes are presented to demonstrate the versatility of the model. The model seems flexible enough to adjust to common knot shapes presented in the literature. This model is intended to be incorporated into tree felling, bucking, and sawing simulation softwares.

Ammi visnaga(L.) Lamarck (Apiaceae): Associated Beneficial Insects and Implications for Biological Control, with Emphasis on the Bell-Pepper Agroecosystem

ABSTRACT Toothpick ammi (Ammi visnaga (L.) Lamarck, Apiaceae), a summer annual of Eurasian origin, commonly occurs along agricultural field margins in the Sacramento Valley of northern California, typically flowering from June to August. Observations during 1979–1984 yielded 65 distinguishable insect taxa attending the flowers, including 60 groups that were at least in part entomophagous. Peak attendances occurred for Braconidae, Ichneumonidae, Sarcophagidae, Sphecidae, Tachinidae, and Vespidae during early morning hours. Hay alfalfa (Medicago sativa L., Fabaceae) was grown in replicated plots with borders of either: (1) toothpick ammi, (2) common knot weed (Polygonum aviculare L., Polygonaceae), (3) prostrate pigweed (Amaranthus graecizans L., Amaranthaceae), or (4) Clean-cultivated edges. Borders of flowering toothpick ammi attracted numerous entomophaga, but had no effect on densities of minute pirate bug (Orius tristicolor [White], Hemiptera: Anthocoridae) occurring in alfalfa. Pepper (Capsicum annuum...

Degree elevation of B-spline curves

An algorithm is presented which expresses any given linear combination of B-splines of order k as a linear combination of B-splines of order k + 1. This algorithm consists of two parts: (1) A simple representation of the curve by a sum of B-splines of order k + 1, and (2) a representation of these splines on a common knot vector by inserting new knots.