Degree Reduction of Q-Bézier Curves via Squirrel Search Algorithm

Q-Bézier curves find extensive applications in shape design owing to their excellent geometric properties and good shape adjustability. In this article, a new method for the multiple-degree reduction of Q-Bézier curves by incorporating the swarm intelligence-based squirrel search algorithm (SSA) is proposed. We formulate the degree reduction as an optimization problem, in which the objective function is defined as the distance between the original curve and the approximate curve. By using the squirrel search algorithm, we search within a reasonable range for the optimal set of control points of the approximate curve to minimize the objective function. As a result, the optimal approximating Q-Bézier curve of lower degree can be found. The feasibility of the method is verified by several examples, which show that the method is easy to implement, and good degree reduction effect can be achieved using it.

Conversion Between Cubic Bezier Curves and Catmull–Rom Splines

Splines are one of the main methods of mathematically representing complicated shapes, which have become the primary technique in the fields of Computer Graphics (CG) and Computer-Aided Geometric Design (CAGD) for modeling complex surfaces. Among all, Bézier and Catmull–Rom splines are the most common in the sub-fields of engineering. In this paper, we focus on conversion between cubic Bézier and Catmull–Rom curve segments, rather than going through their properties. By deriving the conversion equations, we aim at converting the original set of the control points of either of the Catmull–Rom or Bézier cubic curves to a new set of control points, which corresponds to approximately the same shape as the original curve, when considered as the set of the control points of the other curve. Due to providing simple linear transformations of control points, the method is very simple, efficient, and easy to implement, which is further validated in this paper using some numerical and visual examples.

Interpolation with Specified Error of a Point Series Belonging to a Monotone Curve

The paper addresses the problem of modeling a smooth contour interpolating a point series belonging to a curve containing no special points, which represents the original curve with specified accuracy. The contour is formed within the area of possible location of the parts of the interpolated curve along which the curvature values are monotonously increased or decreased. The absolute interpolation error of the point series is estimated by the width of the area of possible location of the curve. As a result of assigning each intermediate point, the location of two new sections of the curve that lie within the area of the corresponding output section is obtained. When the interpolation error becomes less than the given value, the area of location of the curve is considered to be formed, and the resulting point series is interpolated by a contour that lies within the area. The possibility to shape the contours with arcs of circles specified by characteristics is investigated.

Intelligent Identification Method of Sedimentary Microfacies Based on DMC-BiLSTM

Sedimentary microfacies division is the basis of oil and gas exploration research. The traditional sedimentary microfacies division mainly depends on human experience, which is greatly influenced by human factor and is low in efficiency. Although deep learning has its advantage in solving complex nonlinear problems, there is no effective deep learning method to solve sedimentary microfacies division so far. Therefore, this paper proposes a deep learning method based on DMC-BiLSTM for intelligent division of well-logging—sedimentary microfacies. First, the original curve is reconstructed multi-dimensionally by trend decomposition and median filtering, and spatio-temporal correlation clustering features are extracted from the reconstructed matrix by Kmeans. Then, taking reconstructed features, original curve features and clustering features as input, the prediction types of sedimentary microfacies at current depth are obtained based on BiLSTM. Experimental results show that this method can effectively classify sedimentary microfacies with its recognition efficiency reaching 96.84%.

Generalized Birch lemma and the 2-part of the Birch and Swinnerton-Dyer conjecture for certain elliptic curves

In the present paper, we generalize the celebrated classical lemma of Birch and Heegner on quadratic twists of elliptic curves over ℚ {{\mathbb{Q}}} . We prove the existence of explicit infinite families of quadratic twists with analytic ranks 0 and 1 for a large class of elliptic curves, and use Heegner points to explicitly construct rational points of infinite order on the twists of rank 1. In addition, we show that these families of quadratic twists satisfy the 2-part of the Birch and Swinnerton-Dyer conjecture when the original curve does. We also prove a new result in the direction of the Goldfeld conjecture.

Sigmoidal water retention function with improved behaviour in dry and wet soils

A popular parameterized soil water retention curve (SWRC) has a hydraulic conductivity curve associated with it that can have a physically unacceptable infinite slope at saturation. The problem was eliminated before by giving the SWRC a non-zero air entry value. This improved version still has an asymptote at the dry end, which limits its usefulness for dry conditions and causes its integral to diverge for commonly occurring parameter values. We therefore joined the parameterizations' sigmoid midsection to a logarithmic dry section ending at zero water content for a finite matric potential, as was done previously for a power-law-type SWRC. We selected five SWRC parameterizations that had been proven to produce unproblematic near-saturation conductivities and fitted these and our new curve to data from 21 soils. The logarithmic dry branch gave more realistic extrapolations into the dry end of both the retention and the conductivity curves than an asymptotic dry branch. We tested the original curve, its first improvement, and our second improvement by feeding them into a numerical model that calculated evapotranspiration and deep drainage for nine combinations of soils and climates. The new curve was more robust than the other two. The new curve was better able to produce a conductivity curve with a substantial drop during the early stages of drying than the earlier improvement. It therefore generated smaller amounts of more evenly distributed deep drainage compared to the spiked response to rainfall produced by the earlier improvement.

Water level collapse effect of deep well in mining area

The use of Zaozhuang of Shandong province coal mine area is tao chong lu 15 Well and meteorological observation data, the three elements of observation Wells with tao chong effect comparing the dynamic characteristics of the mines. It is difficult to determine whether the stress state of the aquifer system in which the well was observed before the collapse has changed because of the water level in the deep well of Lu15 well on the original curve. Nakai fitting model is adopted to calculate the earth tide response amplitude factor ratio based on the water level data of Lu15 well, and the possibility of collapse caused by stress change is discussed in combination with non-natural seismic events in Taozhuang Coal mine

Sigmoidal Water Retention Function with Improved Behavior in Dry and Wet Soils

A popular parameterized soil water retention curve (SWRC) has a hydraulic conductivity curve associated with it that can have an infinite slope at saturation. The problem was eliminated before by giving the SWRC a non–zero air–entry value. This improved version still has an asymptote at the dry end, which limits its usefulness for dry conditions and causes its integral to diverge for commonly occurring parameter values. We therefore joined the parameterizations' sigmoid mid–section to a logarithmic dry section ending at zero water content for a finite matric potential, as was done previously for a power–law type SWRC. We selected five SWRC parameterizations that had been proven to produce unproblematic near–saturation conductivities and fitted these and our new curve to data from 21 soils. The logarithmic dry branch gave more realistic extrapolations into the dry end of both the retention and the conductivity curves than an asymptotic dry branch. We tested the original curve, its first improvement, and our second improvement by feeding them into a numerical model that calculated evapotranspiration and deep drainage for nine combinations of soils and climates. The new curve was more robust than the other two. The new curve was better able to produce a conductivity curve with a substantial drop during the early stages of drying than the earlier improvement. It therefore generated smaller amounts of more evenly distributed deep drainage compared to the spiked response to rainfall produced by the earlier improvement.

Analysis and Optimization of Interpolation Points for Quadruped Robots Joint Trajectory

Trajectory planning is the foundation of locomotion control for quadruped robots. This paper proposes a bionic foot-end trajectory which can adapt to many kinds of terrains and gaits based on the idea of trajectory planning combining Cartesian space with joint space. Trajectory points are picked for inverse kinematics solution, and then quintic polynomials are used to plan joint space trajectories. In order to ensure that the foot-end trajectory generated by the joint trajectory planning is closer to the original Cartesian trajectory, the distributions of the interpolation point are analyzed from the spatial domain to temporal domain. An evaluation function was established to assess the closeness degree between the actual trajectory and the original curve. Subsequently, the particle swarm optimization (PSO) algorithm and genetic algorithm (GA) for the points selection are used to obtain a more precise trajectory. Simulation and physical prototype experiments were included to support the correctness and effectiveness of the algorithms and the conclusions.

Curve Generalization Algorithm Based on Area of Bends using Head/tail Breaks

<p><strong>Abstract.</strong> For the early curve generalization algorithms, most of them only consider the reduction of the number of vertices, and do not take into consideration the important role of bends, especially the characteristic bends, on the shape of the curve. And the existing generalization methods based on the bends of the curve have complex algorithms and a large amount of calculation, focus on relationship between adjacent bends excessively and ignore the relationship among the overall bends. In addition, the threshold setting for filtering the bends is based on the unreasonable experience. Aiming at the problems above, a generalization algorithm based on the area of bends is proposed to achieve the purpose of simplifying the curve with the head/tail breaks classification method in this paper. Experiment shows that the algorithm is simple and efficient, and can iteratively take account of the overall bends with reasonable threshold, discarding the small bends and retaining the characteristic bends of the original curve to obtain generalization results which conform the natural law and is highly similar to the original graphics at different levels of detail.</p><p>Head/tail breaks is a classification method that is always applied to the classification of heavy-tailed data. Heavy-tailed data is universal in nature and human society. For example, there are more small towns than big cities in the world. However, small towns are less important than big cities in the field of economy and politics. Thus, cartographers will mark the big cities on the map and eliminate the small town. Map generalization is a progress of retaining important elements and delete unimportant elements. Head/tail breaks is able to extract significant data which can be retained as a generalization result by arithmetic mean.</p><p>Figure 1 shows the algorithm flow chart. First of all, we divide the curve into several bends with oblique-dividing-curve method. Secondly, we calculate the area of each bend, and then use head/tail breaks to complete the classification of the area of bends. If the percentage of bends in the head is less than 40%, it means the data conform to heavy-tailed distribution and can be classified with head/tail breaks. If the percentage of bends in the head is greater than 40%, the head/tail breaks is not applicable to this data. After classification, for the bends which is more important in the head, we reserve them directly. For the bends in the tail, we extract the feature points of each bend by retaining the point farthest from the axis so as to maintain the local shape of the original curve. Finally, we merge the bends in the head and the feature points as a generalization result.</p><p>The experimental result is shown in the Fig 2. The data of this experiment is administrative division map of Gansu Province extracted from a China map with a scale of 1&amp;thinsp;:&amp;thinsp;10,000,000. Because algorithm can be executed iteratively, it can generate results at different levels of detail. We can see that from the result in detail to concise result, the graphic changes progressively and there is no oversimplified result. With comparison of three algorithms in the Fig 3, the generalization results of both this paper and bend group division algorithm have better retention of characteristic bends than Douglas-Peuker algorithm. However, the algorithm of this paper has higher compression ratio and less execution time than bend group division algorithm, as shown in Table 1.</p><p>The algorithm of this paper is based on nature law rather than empirical threshold, and can generate progressive results at different levels of details by iteration. In addition, it takes overall relationship of bends into consideration, so the generalization result is unique. The experimental result shows this algorithm has not only better retention of characteristic bends than Douglas-Peuker algorithm but also higher compression ratio and less execution time than bend group division algorithm. To further optimize the algorithm, we will study how to evaluate the apparent extent of the curved feature better and how to extract and eliminate the small bend inside of the bend in the head in order to improve compression ratio in the future.</p>