Curve Interpolation Research Articles

A Hybrid Parametrization Method for B‐Spline Curve Interpolation via Supervised Learning

AbstractB‐spline curve interpolation is a fundamental algorithm in computer‐aided geometric design. Determining suitable parameters based on data points distribution has always been an important issue for high‐quality interpolation curves generation. Various parameterization methods have been proposed. However, there is no universally satisfactory method that is applicable to data points with diverse distributions. In this work, a hybrid parametrization method is proposed to overcome the problem. For a given set of data points, a classifier via supervised learning identifies an optimal local parameterization method based on the local geometric distribution of four adjacent data points, and the optimal local parameters are computed using the selected optimal local parameterization method for the four adjacent data points. Then a merging method is employed to calculate global parameters which align closely with the local parameters. Experiments demonstrate that the proposed hybrid parameterization method well adapts the different distributions of data points statistically. The proposed method has a flexible and scalable framework, which can includes current and potential new parameterization methods as its components.

Jerk limited continuous tool path generation for flexible systems in non-cartesian coordinate systems

Inspired by computer numerical control (CNC) technology and drastic changes in the world of electronics and microcontrollers, the idea of using non-cartesian coordinate system for CNC machines became a topic of interest. Non-cartesian coordinate systems provide freedom of movement in systems with large working spaces where high dimensional accuracy is not a prime concern. Applications of such systems can be found in murals, where a painting is applied to a large wall of any arbitrary build material. In the present paper, a new design was introduced to automate the process of drawing murals on large surfaces. The proposed mechanism utilized chain and sprocket to overcome the size limitations of traditionally available x-y tables. In addition to that, the developed mechanism does not utilize the common robust structures used in the conventional CNC systems and the tool motion is created using flexible arms that are not heavily constrained. The effect of systems unwanted vibration has been eliminated using continuous trajectory and kinematic profiles. Parametric curve interpolation algorithms for trajectory planning were implemented to increase the flexibility of drawing complex curves. Parametric curves such as B-spline and non-uniform rational B-spline (NURBS) were employed to generate a jerk limited trajectory for motion control and extraction of kinematic profiles. Such trajectory constrains the jerk of the system and guarantees a smooth motion free of feed-rate fluctuations. Compared to other methods available for extracting the kinematic profiles, jerk limited method decreases the travel time and trajectory error. Ultimately, motion commands were produced by using kinematic profiles and the second order Taylor interpolator. STM32746ZG card was used as the main processor and two stepper motors controlled by a Bit Pattern interpolation algorithm were utilized to drive the proposed mechanism. The input pulses of stepper motors were stored as binary bits and transmitted to drivers in real-time. The feedback was also evaluated by two rotary encoders, placed at the end of the motors’ shafts. Encoder data were acquired and transmitted using a TCP/IP protocol to guarantee zero data loss during the transmission.

Generation of Picture Fuzzy B-Spline Curve Interpolation with Open Uniform Knot Vector

Generation of Picture Fuzzy B-Spline Curve Interpolation with Open Uniform Knot Vector

An efficient and automatic method based on monocular camera and GNSS for collecting and updating geographical coordinates of mileage pile in highway digital twin map

Abstract The high-precision positioning of mileage piles on a digital map ensures accurate data for high-speed event releases, toll audits, and road condition monitoring. This paper introduces an efficient and automatic system for positioning highway mileage piles. The self-developed data acquisition system collects mileage pile images and road trajectory. Considering the limited variety of mileage piles, a Simplified-YOLOv5m (S-YOLOv5m) is proposed. Then a high-precision character detection network of S-YOLOv5m is proposed to enhance feature extraction and improve the accuracy of character detection on mileage piles. Thirdly, the end-to-end monocular distance measurement combines the target detection with the distance estimation, enabling simultaneous object detection and distance measurement. Fourthly, by combining with the geographical coordinates of the acquisition point, the direct solution to geodetic problems is applied to calculate the spatial coordinates of the mileage piles. Finally, the missing mileage piles are compensated for using the road curve and equal distance interpolation. The complete mileage piles and geographical information list of the inspection trajectory are output. Through a series of verification tests, the average positioning error of the mileage pile system is 1.265 m. The contribution of the automatic positioning system of mileage piles is to construct the relationship between mileage piles and geographical coordinates, match mileage piles with digital maps to realize the full-featured, real-scene, and high-dynamic management of road attributes.

The Magnetic Anomaly Map of Paraná State: An Integration of Airborne Surveys Performed Over the Years

Over the last decades, several aerial survey campaigns have been conducted in the State of Paraná. However, we did not have a complete magnetic data integration and analysis focused on the Paraná’s geological context. Therefore, the main objective of this work is to provide the integration of all available data and realize a general analysis to make the Integrated Magnetic Map of the Paraná State, in collaboration with the Geological Survey of Brazil (SGB-CPRM). After integrate the data we have applied processing techniques on public airborne magnetic data, available from the Geological Survey of Brazil database (GeoSGB). For this integration, it was used the minimum curvature interpolation, with cell sizes according to each survey. All gridded data was extrapolated upward from the original height to the altitude of 1800 m. Afterwards, we applied the upward continuation for 2700 m and the following filters: vertical gradient (VDR), total horizontal derivative (THDR), analytic signal (AS), or tilt derivative (TDR), tilt angle of the horizontal derivative (TAHG) and vertical integral of the analytic signal (VIAS). The final results provide valuable maps (and grids) that can be used by the community for future works. In this work we tried to do a simple overview of the structural framework of Paraná State, representing the main magnetic structures of the Paraná State basement and compare them with the most important structures already mapped in surface over the years. All the data present in this work and other filters tests are available and can be access in the supplementary material (https://acervodigital.ufpr.br/ https://hdl.handle.net/1884/89075 and also at zenodo repository https://doi.org/10.5281/zenodo.12820642).

Polishing Machine Control System Based on B-Spline Curve Trajectory

In this study, a B-spline trajectory control algorithm is presented, showcasing its ability to achieve smooth trajectory control within an embedded motion control system. Traditional surface machining methods involve converting interpolated trajectories into G-code using numerical control (NC) software, which is then downloaded and executed on the numerical control system. This research introduces a specialized third-degree B-spline interpolation method tailored for curved surface trajectories, aiming to enhance machining efficiency. A significant strength of this project lies in the holistic design of an embedded system, encompassing both hardware circuitry and software logic. Utilizing the cost-effective STM32 architecture, a B-spline discrete velocity interpolation algorithm is implemented, validated through experiments conducted on a polishing machine system. The project involves MATLAB software for data simulations and experiments on a polishing machine using B-spline curve interpolation, confirming the practicality and effectiveness of the third-degree B-spline algorithm within an embedded system.

FRKVF: High-accuracy motion interpolation for polishing operations using fourth-order Runge-Kutta and velocity flexibility planning

Traditional interpolation algorithms often fail to meet the precision requirements of ultra-precision machining when applied to super-precision polishing machines. Moreover, the processing of fused silica glass optical components is frequently threatened by mechanical impacts due to their fragility. In order to address this issue, this paper proposes a high-precision motion interpolation method based on fourth-order Runge-Kutta and velocity-flexible planning. This method is designed for open-architecture small multi-axis optical polishing machines to polish quartz glass. The algorithm initially employs composite Simpson's rule to calculate the lengths of sub-paths within the polishing trajectory. Based on these length values, flexible velocity planning is executed to ensure the smooth continuity of velocity, acceleration, and jerk during motion interpolation. This reduces the risk of mechanical impacts that could damage the components during the machining process. The introduction of the adaptive fourth-order Runge-Kutta method significantly enhances the parameter point calculation accuracy of NURBS curves. The incorporation of adaptive principles also maintains a higher processing speed, thereby greatly improving processing efficiency. This method comprehensively addresses both the precision of curve interpolation and execution efficiency. Finally, experimental validation is conducted on an open-architecture small multi-axis optical polishing machine. The proposed method based on FRKVF not only mitigates mechanical impacts resulting from discontinuous acceleration, thereby ensuring the machining quality of optical components, but also satisfies the high-precision requirements for processing fused silica optical elements.

Aquifer system deformation in the San Luis Valley: A new framework for modeling subsidence in agricultural regions

The San Luis Valley (SLV), Colorado, is challenged with implementing sustainable groundwater management in the face of increasing surface water scarcity due to climate change. Groundwater extraction in unconsolidated aquifers such as the SLV can cause cm-scale subsidence and rebound. This study utilizes Interferometric Synthetic Aperture Radar (InSAR) data, validated by Global Navigation Satellite System (GNSS) measurements, to measure subsidence and analyze the groundwater dynamics that cause it. Addressing the challenges of phase decorrelation and data gaps, notably from September 2018 to April 2019, we adopted a modified DS-interpolation algorithm, alongside a Singular Spectrum Analysis (SSA)-based gap filling technique. Furthermore, we enhanced the temporal resolution of groundwater level data through the Theis curve interpolation. These methodologies enabled the integration of observational well data with satellite measurements to calibrate a one-dimensional deformation model, capturing both the elastic and inelastic responses of the aquifer system. Our investigation, spanning 2015 to 2021, reveals both seasonal and long-term subsidence, with the confined aquifer section experiencing up to 1 cm/year of subsidence alongside notable seasonal fluctuations. The methodology presented here provides a path to model subsidence in regions with sparse groundwater level and noisy InSAR data. It also provides valuable insights for developing effective water management strategies in the SLV.

Could Road Structures Impact the Avian Community? A Study Case from the South Coast Remained Forest in Malang Region, East Java Province, Indonesia

The remaining tropical forest on the south coast of the Malang region is one of the secluded areas that served as an important habitat for its biodiversity, particularly for avians. Nevertheless, the presence of the road structure that crosses over the forest might impact avian communities, which needs to be investigated. The avian survey was conducted to investigate the diversity, community profiles, abundance, and feeding guild based on two different ecosystem patches (on the roads versus outside the road structures) during January–April 2022. The audiovisual encounter methods were performed during the surveys on a total of ten sampling points. Avian species were identified using field guides where the conservation status was based on the national regulation (P106 KLHK), CITES, and IUCN Redlist. The comparison through the avian community based on species richness was analyzed using Venn Diagrams and predicted using rarefaction and interpolation curves on INEXT packages. Additionally, the comparison of sizes of avian abundance was investigated using the α-diversity parameter index, and the feeding guild was determined by five diet guilds. In total, 2536 individuals from 67 species, 34 families, and 13 orders were found during the survey. The avian communities were richer outside the road structure, but their abundance qualitatively reveals that both different ecosystem patches showed relatively good condition. Overall, road structures generate both positive (unique habitat patches) and negative (physical barriers that could disrupt natural ecological processes) impacts on avian diversity. In light of these, sustainable management and conservation action plans were urgently needed to prevent the negative effects on avian communities posed by the road structure.

The Application of MATLAB in the Mathematics Teaching of Computer Majors

Under the background of new engineering courses, professional mathematics teaching needs to keep pace with the times and improve the progressiveness and scientific nature of teaching methods.In order to improve the mathematics teaching effect of computer majors, This article constructs a new teaching environment through simulation system construction methods, analyzes the shortcomings of traditional teaching models through comparative analysis methods, and improves the teaching platform based on the actual needs of mathematics professionals in current society. The curve interpolation technology is improved, and a forward-looking real-time Matlab simulation algorithm is designed. Through the evaluation and analysis, it can be seen that the mathematics teaching system for computer majors based on MATLAB can effectively improve the intuitive effect of advanced mathematics teaching, and help to improve the teaching quality of advanced mathematics.

Path Planning of a Mobile Robot Based on the Improved Rapidly Exploring Random Trees Star Algorithm

With the increasing utilization of sampling-based path planning methods in the field of mobile robots, the RRT* algorithm faces challenges in complex indoor scenes, including high sampling randomness and slow convergence speed. To tackle these issues, this paper presents an improved RRT* path-planning algorithm based on the generalized Voronoi diagram with an adaptive bias strategy. Firstly, the algorithm leverages the properties of the generalized Voronoi diagram (GVD) to obtain heuristic paths, and a sampling region with target bias is constructed, increasing the purposefulness of the sampling process. Secondly, the node expansion process incorporates an adaptive bias strategy, dynamically adjusting the step size and expanding direction. This strategy allows the algorithm to adapt to the local environment leading to improved convergence speed. To ensure the generation of smooth paths, the paper employs the cubic spline curve interpolation algorithm for trajectory optimization to ensure that the mobile robotic can obtain the best trajectory. Finally, the proposed algorithm is experimentally compared with existing algorithms, including the RRT* and Informed-RRT* algorithms, to verify the feasibility and stability.

Spatial interpolation techniques comparison and evaluation: The case of ground-based gravity and elevation datasets of the central Main Ethiopian rift

The ground-based gravity data reveals diverse anomaly signatures in areas of the Main Ethiopian rift where active volcanic and tectonic activities are dominant. In such a region ground-based data collection is restricted to existing roads and relies on accessible stations. These resulted in gaps in data, either missing, uneven, or insufficient spatial coverage that must be estimated with proper interpolation techniques. Comparison and evaluations of the spatial interpolation methods that are commonly used in potential field geophysical data analysis were made for the terrestrial gravity and elevation data of the central Main Ethiopian rift. In this research, two widely used interpolation techniques, minimum curvature interpolation, and Ordinary Kriging were compared and assessed. A 10 % hold-out validation was employed, where 90 % of the data points were used to generate interpolated surfaces, which were then evaluated against the remaining 10 %. Following interpolation with each technique, the generated grid was converted into discrete data points (estimated values). These are then compared with the available gravity data, which were deliberately excluded from the gridding process (10 % remaining dataset). The accuracy of each method was assessed by evaluation metrics such as mean value, variance, Mean Absolute Error (MAE), Root Mean Square Error (RMSE), correlation coefficient (r), and R-squared. The results showed that the ordinary Kriging interpolation method outperformed the minimum curvature interpolants for gravity data with all performance metrics, while both interpolants seem to perform equally well for the elevation dataset. Therefore, it is proposed to use the Kriging interpolation method for potential field gravity studies conducted in the central Main Ethiopia rift.

CUBIC SPLINE INTERPOLATION TO APPROXIMATE SEA DEPTH BETWEEN TELUK SUAK AND LEMUKUTAN ISLAND

Interpolation is a technique that determines the value of a function at a point located between several known data points. Various interpolation methods include polynomial interpolation, Lagrange interpolation, Newton interpolation, and spline interpolation. Cubic spline interpolation aims to produce accurate approximations characterized by minimal oscillations in the resulting curve. This research uses the cubic spline interpolation method to estimate the sea depth in the waters between Teluk Suak and Pulau Lemukutan. The sea depth measurements are conducted to acquire information regarding the underwater topography. We collected data points using remote sensing techniques through Google Earth Pro. These points are selected based on the sea depth profile considerations, including steep areas and points of depth variation to maintain the seabed conditions within the interpolation curve, with a total of 31 data points. Subsequently, we subjected these data points to the conditions necessary for cubic spline interpolation, resulting in a system of 120 linear equations. After solving this system of linear equations, we obtained cubic spline interpolation polynomials for each subinterval and then estimated the sea depth at other points. Based on the cubic spline interpolation results, we achieved a Mean Average Percentage Error (MAPE) of 1.58%, indicating a highly accurate interpolation outcome.

Vision-based algorithm for online TIG welding deviation detection of stainless steel welded pipes

Tungsten inert gas (TIG) welding is the main welding process in the production of stainless steel welded pipe. According to the morphological characteristics of the welding molten pool image during the TIG welding process of stainless steel welded pipes, the exact position of the tungsten needle tip is calculated using image moments. Extract the weld region in the contour of the molten pool, interpolate the contour curve based on the cubic B-spline curve interpolation method, utilize the characteristics of the S-G filter, remove the interference coordinates in the contour curve through the detrending of the contour curve, extract the weld feature points, and realize the accurate identification of weld seams. The experimental results show that the method can accurately calculate the welding deviation in the welding process.

Small-sample linear profile error uncertainty assessment based on grey system

The uncertainty assessment of the profile error of the cam profile, as defined in the national standard method, is difficult to carry out under conditions of small sample size and absence of probability distribution assumptions. This paper proposes a small-sample assessment model for the uncertainty of the profile error based on grey system. Firstly, the coordinate transformation is conducted using Vector Alignment Method to reduce systematic errors, and the non-uniform rational B-splines curve interpolation is utilized to fit the cam profile curve and perform error assessment. Subsequently, based on the error assessment results, Grey Information Measurement Model (GIMM) for the uncertainty of the profile error in small samples is established. This model employs Grey Relational Analysis to eliminate outliers and evaluates the uncertainty of the profile error by solving grey correlation coefficients. Maximum-Minimum Information Measure Method is used to assess the optimal sample size. Finally, numerical experiments and experimental tests were conducted on the uncertainty of camshaft profile error in automobiles. A total of 15 sets of profile data were compared with Guide to the Representation of Uncertainty in Measurement (GUM) and Monte Carlo Method (MCM) under different sample sizes. The results showed that GIMM achieved evaluation with only 8 sets of data samples under small sample and poor information conditions, with an uncertainty of 0.6338 μm, compared to 0.6346 μm for GUM and 0.6391 μm for MCM. The acceptance rate of GIMM reached 95.2%. This model outperforms other methods, providing a simplified and reliable assessment of cam profile error uncertainty.

Pκ-Curves: Interpolatory curves with curvature approximating a parabola

This paper introduces a novel class of fair and interpolatory planar curves called pκ-curves. These curves are comprised of smoothly stitched Bézier curve segments, where the curvature distribution of each segment is made to closely resemble a parabola, resulting in an aesthetically pleasing shape. Moreover, each segment passes through an interpolated point at a parameter where the parabola has an extremum, encouraging the alignment of interpolated points with curvature extrema. To achieve these properties, we tailor an energy functional that guides the optimization process to obtain the desired curve characteristics. Additionally, we develop an efficient algorithm and an initialization method, enabling interactive modeling of the pκ-curves without the need for global optimization. We provide various examples and comparisons with existing state-of-the-art methods to demonstrate the curve modeling capabilities and visually pleasing appearance of pκ-curves.

Using MATLAB to Make Population Prediction Analysis

Through the curve fitting and interpolation function of matlab software, based on the demographic data in the Statistical Yearbook of Hunan Province from 2000 to 2020, the total population, birth rate and death rate of Hunan Province were predicted, and the data and images were obtained, showing intuitive results. This study uses the two forecasting methods to obtain images and compare them to enrich the population forecasting methods, which has great theoretical and research significance for the population development trend of the future society and the formulation of correct and reasonable population planning schemes.

Three-dimensional simulation of heat and moisture transfer in woven fabric structures

Fabric structure parameters have a significant impact on the comfort of heat and moisture transfer in garments. Previous numerical simulations required extensive mathematical calculations and mostly investigated one- or two-dimensional models of fiber assembly without considering the weave structure, which is a key parameter, that significantly influences the porosity of the woven structure and consequently its heat and moisture management. While the finite element method supports the simulation of the optimal shape and material properties with better visibility, previous finite element models focused on heat transfer and neglected water vapor transfer in fabrics. In this article, the finite element simulation of heat and moisture transfer in woven fabrics is established based on the testing principle of thermal resistance and moisture resistance tester using COMSOL Multiphysics software. In this simulation, three-dimensional parametric geometrical models of the fabric are created using curve interpolation methods by acquiring the control point coordinates of different weaves (plain, 2/2 balanced twill, and 4/1 unbalanced twill weaves). Heat and moisture transfer properties of fabric models in the horizontal and vertical directions were analyzed, including the heat flux, moisture resistance, water vapor permeability, and water vapor concentration. The article also deals with the effects of weave structure and fabric cover in a range of 72.1–85.1% on the fabric heat flux and water vapor concentration. Comparison between model and experimental results revealed that the three-dimensional simulation can accurately predict the impact of weave pattern and fabric cover on the fabric heat and moisture transfer performance. In addition, this model can be utilized to study the distribution of heat and water vapor within fabrics, providing a theoretical foundation for optimizing heat and moisture comfort in woven fabrics.

Computing nodes for plane data points by constructing cubic polynomial with constraints

To construct a parametric polynomial curve for interpolating a set of data points, the interpolation accuracy and shape of the constructed curve are influenced by two principal factors: parameterization of the data points (computing a node for each data point) and interpolation method. A new method of computing nodes for a set of data points was proposed. In this paper, the functional relationship between data points and corresponding nodes in cubic polynomials was established. Using this functional relationship, a functional cubic polynomial with one degree of freedom can pass through four adjacent data points. The degree of the freedom can be represented by two adjacent node intervals can be obtained by minimizing the cubic terms of the cubic polynomial. Since each node is computed in different node spaces, a method for constructing a quadratic curve is presented, which transforms all the quadratic curves into a unified form to compute nodes. Nodes computed using the new method exhibit quadratic polynomial precision, i.e., if the set of data point is taken from a quadratic polynomial F(t), the nodes by the new method are used to construct a interpolation curve, an interpolation method reproducing quadratic polynomial gives quadratic polynomial F(t). The primary advantage of the proposed method is that the constructed curve has a shape described by data points. Another advantage of the new method is that the nodes computed by it have affine invariance. The experimental results indicate that the curve constructed by the nodes using the new method has a better interpolation accuracy and shape compared to that constructed using other methods.

Complete CMC-1 surfaces in hyperbolic space with arbitrary complex structure

We prove that every open Riemann surface [Formula: see text] is the complex structure of a complete surface of constant mean curvature [Formula: see text] ([Formula: see text]) in the three-dimensional hyperbolic space [Formula: see text]. We go further and establish a jet interpolation theorem for complete conformal [Formula: see text] immersions [Formula: see text]. As a consequence, we show the existence of complete densely immersed [Formula: see text] surfaces in [Formula: see text] with arbitrary complex structure. We obtain these results as application of a uniform approximation theorem with jet interpolation for holomorphic null curves in [Formula: see text] which is also established in this paper.