Efficient Intersection Computation of the Bezier and Hermite Curves with Axis Aligned Bounding Box

Cubic parametric curves are used in many applications including the CAD/CAM systems. Especially the Hermite, Bezier and Coons formulations of a cubic parametric curve are used in E2 and E3 space. This paper presents efficient algorithm for the intersection computation of a cubic parametric curve with the Axis Aligned Bounding Box (AAB Box). Usual solution is to represent the cubic curve by a polyline, i.e. actually by sampled points of the given curve. However, this approach is dependent on the sampling frequency and can lead to problems especially in CAD/CAM systems and numerically controlled machines use.

Orthogonal Polynomials on Planar Cubic Curves

Orthogonal polynomials in two variables on cubic curves are considered. For an integral with respect to an appropriate weight function defined on a cubic curve, an explicit basis of orthogonal polynomials is constructed in terms of two families of orthogonal polynomials in one variable. We show that these orthogonal polynomials can be used to approximate functions with cubic and square root singularities, and demonstrate their usage for solving differential equations with singular solutions.

Cubic Curve Fitting Method in Reconstruction of Chinese Calligraphy Outline

Curve plays a significant role in CAGD and brings the good impact of computers to manufacturing industries in designing 2 and 3-dimensional shapes and objects. Reconstruction of Chinese calligraphy outline based on the actual character is presented in this paper. Chinese calligraphy is the stylized artistic writings of Chinese characters. It is believed that this writing may help to express the feelings and ideas of the writers, which are difficult to be described. The shapes, smooth lines, and perfect curves are among the important qualities which are particularly emphasized in selecting good Chinese calligraphy. The Cubic B-Spline, Cubic Trigonometric Spline, and Cubic Trigonometric Bezier were used to generate the curves. The factors that have influenced the effects of the curves modifications were examined based on the changes of control polygon and the values of shape parameter. The fastest approach was then chosen by measuring the processing time required to construct the complete design. Results show the Cubic Trigonometric Bezier curve produced the closest curves to the control polygon, accurate to the actual character with λ = 1 and CPU time taken is 2.032 seconds. This is followed by Cubic Trigonometric Spline and Cubic B-Spline.

The nonlinear effects of tourism on rural income inequality and urban–rural income inequality: Evidence from China

Tourism’s relationship with income inequality is a hot spot in the academic community; however, few studies focus on how tourism shapes rural income inequality, despite its significant role in rural poverty alleviation. This article investigates the nonlinear effects of tourism on urban–rural income inequality and rural income inequality based on data from China’s provincial regions over the period 1995–2018, using the dynamic panel data approach. This study further examines the regional heterogeneity of such effects. The results demonstrate that for the national sample, there exists a significant cubic curve relationship between domestic tourism and rural income inequality. However, inbound tourism’s relationship with rural income inequality is not significant. Besides, there exists an N-shaped Kuznets curve between domestic or inbound tourism and urban–rural income inequality. The effects of tourism on rural income inequality and urban–rural income inequality vary significantly across the eastern, middle, and western regions. From east to middle to west, the significance of inbound tourism’s influence on income inequality decreases gradually. This study also highlights some critical theoretical and practical implications. Unlike prior studies, this article builds the first comprehensive research framework consisting of tourism, rural income inequality, and urban–rural income inequality.

Effect of blending ratio on the hollow coffee carbon polyester/cotton blended yarn

The effect of blending ratio on the performance of hollow coffee carbon polyester/cotton blended compact siro-spun yarn was investigated. Five blend ratios of hollow coffee carbon polyester/cotton fiber (i.e. 20/80, 35/65, 50/50, 65/35, and 80/20) and polyester/cotton fiber 35/65 were designed, and six groups of 14.6 tex yarn were spun by compact siro. Indices for the performance of the yarn (surface morphology, evenness, hairiness, tensile property, and hydroscopicity) were tested and analyzed. The regression analysis showed the excellent linear relationship between the content of hollow coffee carbon polyester and each performance index was obtained. Cubic curve models were built to comprehensively evaluate the performance of the yarn. The blending effect in these yarns was evaluated using the Hamilton transfer index. The tests results show that with the increase of the content of hollow coffee carbon polyester in the blended yarn, the evenness and tensile properties of the hollow coffee carbon polyester/cotton blended yarn continue to increase, whereas the hairiness index and moisture regain of the yarn gradually decrease. With the change of blending ratio, the transfer index of each fiber in hollow coffee carbon polyester/cotton blended yarn will change. When the content of hollow coffee carbon polyester is more than or equal to 50%, it has the tendency to preferentially distribute inward, whereas the cotton fiber has the tendency to preferentially distribute outward. When the content of hollow coffee carbon polyester is less than 50%, the reverse is true. The comprehensive evaluation value of the yarn performance decreased first and then increased with the increase in the content of hollow coffee carbon polyester.

Constructing cubic curves with involutions

In 1888, Heinrich Schroeter provided a ruler construction for points on cubic curves based on line involutions. Using Chasles’ Theorem and the terminology of elliptic curves, we give a simple proof of Schroeter’s construction. In addition, we show how to construct tangents and additional points on the curve using another ruler construction which is also based on line involutions. As an application of Schroeter’s construction we provide a new parametrisation of elliptic curves with torsion group $$\mathbb {Z}/2\mathbb {Z}\times \mathbb {Z}/8\mathbb {Z}$$ Z / 2 Z × Z / 8 Z and give some configurations with all their points on a cubic curve.

Comprehensive Multi-Model Framework for establishing correlation between Particulate Matter in Residential and Commercial regions of Lucknow

Our Country India is a developing Country and with the onset of heavy production, urbanization and Industrialization in the recent years has imposed a heavy threat on the Environment, The direct impact of which could be seen in the deteriorating Air Quality. Since air pollution continues to rise at an alarming rate, it affects economies and inhabitants quality of life leading to health emergency situations. This paper presents the Air Quality data interpretation and modelling to study for correlation between different Air Quality Parameters specifically Particulate Matter in Residential and Commercial regions of Lucknow-The Second largest city of Uttar Pradesh. The levels of PM2.5 and PM10 was found to be exceeding than 24 Hrs avg as well as Annual Average value (as set by NAAQS ) in Commercial Region , while in Residential Region the levels of PM2.5 and PM10 were almost in acceptable range as set by NAAQS . Later on studying the comprehensive correlation between the PM2.5 and PM10 by applying various regression models, As per the levels of PM as obtained in commercial region , Compound Curve Regression Model seems to be of highest significance showing 92.1% relation response with highest Standardized Beta Coefficient having value of 2.611 . While for Residential region as per the levels of PM obtained, Cubic Curve Regression Model gave best suited result depicting 77.9% relation response amongst PM2.5 and PM10 with highest Standardized Beta Coefficient Value of 3.028 .

Research on Cracking Mechanism of Early-Age Restrained Concrete under High-Temperature and Low-Humidity Environment

How to prevent the cracking of tunnel lining concrete under a high-temperature and low-humidity environment has gradually become a challenge faced by the engineering community. Actually, the concrete structure will be restrained, which easily leads to cracking. Aiming at this problem, a self-restraint device of concrete specimens was designed in this paper, which aims to more realistically simulate the restrained state of concrete structures during construction. SEM, EDS and XRD detection methods were used to study the macroscopic and microscopic properties of an early-age restrained concrete specimen under a high-temperature and low-humidity environment, and the results were compared with those of a non-restrained concrete specimen. The results show that the change in the internal relative humidity of the concrete was an extremely slow process, and the response rate of the internal humidity of the concrete was much slower than that of the temperature. A cubic curve model was used to fit the measured concrete damage degree with the loading age, and the fitting effect was good. Under the environment of high temperature and low humidity, the loading age from the 0.6th day to the 1st day was the period of a relatively large fluctuation in the concrete temperature and humidity, and the restraint would aggravate the damage of the concrete. The damage degree increased with the increase in the loading age, the microcracks gradually increased and, finally, macrocracks were formed. The restraint effect was to intensify the formation of microcracks, affect the hydration of the cement at the micro level and, finally, increase the risk of concrete cracking perpendicular to the restrained direction at the macro level. The research results may provide guidance for research on the cracking mechanism of tunnel lining concrete constructed under a high-temperature and low-humidity environment.

AS-REGULARITY OF GEOMETRIC ALGEBRAS OF PLANE CUBIC CURVES

In noncommutative algebraic geometry an Artin–Schelter regular (AS-regular) algebra is one of the main interests, and every three-dimensional quadratic AS-regular algebra is a geometric algebra, introduced by Mori, whose point scheme is either $\mathbb {P}^{2}$ or a cubic curve in $\mathbb {P}^{2}$ by Artin et al. [‘Some algebras associated to automorphisms of elliptic curves’, in: The Grothendieck Festschrift, Vol. 1, Progress in Mathematics, 86 (Birkhäuser, Basel, 1990), 33–85]. In the preceding paper by the authors Itaba and Matsuno [‘Defining relations of 3-dimensional quadratic AS-regular algebras’, Math. J. Okayama Univ. 63 (2021), 61–86], we determined all possible defining relations for these geometric algebras. However, we did not check their AS-regularity. In this paper, by using twisted superpotentials and twists of superpotentials in the Mori–Smith sense, we check the AS-regularity of geometric algebras whose point schemes are not elliptic curves. For geometric algebras whose point schemes are elliptic curves, we give a simple condition for three-dimensional quadratic AS-regular algebras. As an application, we show that every three-dimensional quadratic AS-regular algebra is graded Morita equivalent to a Calabi–Yau AS-regular algebra.