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Journal articles on the topic 'Circle and point'
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1
Brown, Matthew R. "Laguerre geometries and some connections to generalized quadrangles." Journal of the Australian Mathematical Society 83, no. 3 (2007): 335–56. http://dx.doi.org/10.1017/s1446788700037964.
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AbstractA Laguerre plane is a geometry of points, lines and circles where three pairwise non-collinear points lie on a unique circle, any line and circle meet uniquely and finally, given a circle C and a point Q not on it for each point P on C there is a unique circle on Q and touching C at P. We generalise to a Laguerre geometry where three pairwise non-collinear points lie on a constant number of circles. Examples and conditions on the parameters of a Laguerre geometry are given.A generalized quadrangle (GQ) is a point, line geometry in which for a non-incident point, line pair (P. m) there exists a unique point on m collinear with P. In certain cases we construct a Laguerre geometry from a GQ and conversely. Using Laguerre geometries we show that a GQ of order (s. s2) satisfying Property (G) at a pair of points is equivalent to a configuration of ovoids in three-dimensional projective space.
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Shemyakina, E. "Pencils of circles with a straight line and circle as the basic elements." Differential Geometry of Manifolds of Figures, no. 50 (2019): 169–76. http://dx.doi.org/10.5922/0321-4796-2019-50-19.
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Pencils of circles with are a straight line and a circle as the basic elements are investigated. Three cases of arrangement of a basic straight line and a circle are considered: when the straight line does not intersect a circle, when the straight line and a circle have one generic point, and when the straight line intersects a circle in two points. A parameter is entered and the equations of new pencils of circles are registered. By means of mathematical manipulations the obtained equations are given to the initial equation of a circle. Different values are attached to the parameter and the circles belonging to new pencils are constructed. Based on the obtained graphs it is concluded that the pencil with not intersecting basic straight line and a circle forms a hyperbolic pencil of circles, a pencil with a basic straight line and a circle having one generic point forms a parabolic pencil, and a pencil with the intersecting basic straight line and a circle forms an elliptic pencil.
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Umbetov, N. "Demonstration of Common Elements of Involution on a Simple Example." Geometry & Graphics 10, no. 2 (2022): 27–34. http://dx.doi.org/10.12737/2308-4898-2022-10-2-27-34.
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The involution of projective rows with a common support, its geometric interpretation are considered. Taking the special case of the geometric interpretation of involution, the problem of constructing harmonically conjugate points is solved for given initial conditions, when one circle and a radical axis of this circle with a bundle of corresponding circles with a common radical axis are given. A proposal is given on the existence of a single circle in a bundle, the diametrical points of which on the lines of centers make up a harmonic four with diametral points of a given circle. It is shown that using the diametrical points of a given circle and points P, Q of the radical axis in elliptical involution, you can build double points X, Y and the radical axis of the PQ of circles in hyperbolic involution. And the tangent from the vertical diammetral point of the circle w1 to the circle passing through double points of hyperbolic involution - there is a point P(Q) of the radical axis of elliptical involution. The indicated properties make it possible to carry out a mutual transition from one involution to another. It was established that the diagonals of the quadrangles obtained when crossing all the circles of the bundle, orthogonal to the two given in elliptical involution, intersect in the center of the radical axis of the given circles in hyperbolic involution, and the diagonals of the quadrangles of all circles of the beam in hyperbolic involution are intersected in the center of the radical axis of the given circles in elliptical Involution. 
 The geometric place (GP) of each point of the harmonic four is constructed. In this case, the geometric place a pair of harmonic four in an elliptic involution turns out to be an ellipse that has a common tangent at points P with the circle of double points of the hyperbolic involution. And the GP pairs of the harmonic four for hyperbolic involution are two branches of the hyperbola that pass through the centers of the circles that define the elliptical involution.
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Кайынбаев, Жанбулат, та М. К. Нұрпейіс. "ШЕҢБЕР МЕН ДӨҢГЕЛЕККЕ БАЙЛАНЫСТЫ КҮРДЕЛІ ЕСЕПТЕРДІҢ БЕРІЛУ ТӘСІЛДЕРІ ЖӘНЕ ОЛАРДЫ ШЕШУ ЖОЛДАРЫ". Педагогика и методы обучения 60, № 3 (2022): 112–27. http://dx.doi.org/10.47344/sdu20bulletin.v60i3.822.
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The article is devoted to circles and circles, which are important issues in the content of elementary geometry, and ways to solve complex problems related to them. In general, difficult problems in this area include cases where one circle is inside another circle (these are two different things), when two circles have only one common point (there are two types of cases), when two circles intersect at two points and two circles do not have a ommon point. For each of these cases, solutions to difficult problems are discussed. The article also contains difficult tasks of drawing other shapes inside and outside the circle. The article is intended for specialists in the field of mathematics teaching methods, teachers, doctoral students and undergraduates
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Koval, Galyna, Margarita Lazarchuk, and Liudmila Ovsienko. "APPLICATION OF CIRCLES FOR CONJUGATION OF FLAT CONTOURS OF THE FIRST ORDER OF SMOOTHNESS." APPLIED GEOMETRY AND ENGINEERING GRAPHICS, no. 100 (May 24, 2021): 162–71. http://dx.doi.org/10.32347/0131-579x.2021.100.162-171.
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In geometric modeling of contours, especially for conjugation of sections of flat contours of the first order of smoothness, arcs of circles can be applied. The article proposes ways to determine the equations of a circle for two ways of its problem: the problem of a circle with a point and two tangents, none of which contains a given point, and the problem of a circle with three tangents. The equations of the circles were determined in both cases using a projective coordinate system.
 In the first case, when a circle is given by a point and two tangents, neither of which contains this point, the center of the conjugation circle is defined as the point of intersection of two locus of points - the bisector of the angle between the tangents and the parabola, the focus of which is a given point. given tangents. In the general case, there are 2 conjugation circles for which canonical equations are defined. Parametric equations of conjugate circles, the parameters of which are equal to 0 and ∞ on tangents and equal to one at a given point, with the help of affine and projective coordinates of points of contact are determined first in the projective coordinate system, and then translated into affine system.
 For the second case, when specifying a circle using three tangent lines, the equation of the second-order curve tangent to these lines is first determined in the projective coordinate system. The tangent lines are taken as the coordinate lines of the projective coordinate system. The unit point of the projective coordinate system is selected in the metacenter of the thus obtained base triangle. The equation of the tangent to the base lines of the second order contains two unknown variables, positive or negative values which determine the location of four possible tangents of the second order. After writing the vector-parametric equation of the tangent curve of the second order in the affine coordinate system, the equation is written to determine the parameters of cyclic points. In order for the equation of the tangent curve of the second order obtained in the projective plane to be an equation of a circle, it must satisfy the coordinates of the cyclic points of the plane, which allows to write the second equation to determine the parameters of cyclic points. By solving a system of two equations, we obtain the required equations of circles tangent to three given lines.
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Kimberling, Clark. "Twenty-one points on the nine-point circle." Mathematical Gazette 92, no. 523 (2008): 29–38. http://dx.doi.org/10.1017/s002555720018249x.
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The nine points for which the nine-point circle is named are the vertices of three triangles: the medial, orthic, and Euler triangles of a reference triangle ABC. All nine of the vertices lie on the circle, which Dan Pedoe [1] proclaims ‘the first really exciting one to appear in any course on elementary geometry’. An online summary of properties of the circle, which is also called the Euler circle, is given at MathWorld [2].
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WANG, LIANG, HONGXUN YAO, and H. D. CHENG. "EFFECTIVE AND AUTOMATIC CALIBRATION USING CONCENTRIC CIRCLES." International Journal of Pattern Recognition and Artificial Intelligence 22, no. 07 (2008): 1379–401. http://dx.doi.org/10.1142/s0218001408006831.
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In this paper, we present an effective, flexible and completely automated camera calibration approach using only one pair of concentric circles. This approach utilizes the characteristics of concentric circles' tangent lines to locate the center of these circles, and finds the geometric constraints for calibration based on the orthogonality formed by a point on the circle and the two intersected points of the circle with the line through the center of the circle. The entire process requires no conic equation fitting and no metric measurement of the test pattern, which is very flexible to implement.
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Wijaya, I. P. Hendra, and I. P. Pasek Suryawan. "MENGKONSTRUKSI NINE POINT CIRCLE DAN HUBUNGAN NINE POINT CIRCLE DENGAN CIRCUM CIRCLE SEGITIGA." Jurnal IKA 17, no. 1 (2019): 1. http://dx.doi.org/10.23887/ika.v17i1.19836.
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Pada makalah ini dibahas mengenai cara dalam mengkonstruksi nine point circle dan hubungan antara nine point circle dengan circumcircle segitiga yang ditinjau dari jari-jari dan jarak titik pusatnya ke orthocentre. Diketahui bahwa melalui tiga buah titik yang tidak segaris selalu dapat dikonstruksi sebuah lingkaran, yang dapat juga disebut sebagai circumcircle dari segitiga yang titik-titik sudutnya merupakan tiga buah titik yang tidak segaris tersebut. Namun melalui empat buah titik yang tidak segaris sangatlah jarang dapat dikonstruksi lingkaran, kecuali keempat titik tersebut dapat membentuk segiempat tali busur. Mengingat untuk mengkonstruksi lingkaran yang melalui empat titik yang tidak segaris sangatlah susah, tidak terbayang bagaimana sulitnya untuk mengkonstruksi lingkaran yang melalui sembilan titik yang tidak segaris. Walaupun sangat sulit untuk menemukannya, tetapi ada kondisi khusus dari titik-titik tersebut sehingga kesembilan titik tersebut dapat dilalui oleh sebuah lingkaran, dan lingkaran yang terbentuk disebut dengan “nine point circle”. Untuk dapat mengkonstruksi nine point circle dengan baik dan cepat, diperlukan tahapan-tahapan khusus yang nantinya menjadi panduan dalam mengkonstruksi lingkaran dari sembilan titik yang tidak segaris. Selain itu, jika nine point circle dibandingkan dengan circumcircle segitiga, maka didapatkan hubungan: (1) jari-jari nine point circlesama dengan setengah jari-jari circumcircle segitiga tersebut; (2) jarak titik pusat nine point circle ke orthocentre sama dengan setengah jarak titik pusat circumcircle ke orthocentre. Untuk kajian lebih lanjut terkait hubungan antara nine point circle dengan circumcircle dari segitiga dapat dilanjutkan oleh pembaca.
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Silvester, John R. "Reflected circles, and congruent perspective triangles." Mathematical Gazette 93, no. 526 (2009): 10–26. http://dx.doi.org/10.1017/s0025557200184141.
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For any three points X, Y, Z, let ⊙XYZ denote the circle through X, Y, Z (the circumcircle of ∆XYZ) or, if X, Y, Z happen to be collinear, the line XYZ. (We shall often regard lines as special circles, circles of infinite radius.) This paper is about the following theorem, and extensions of it:Theorem 1: Given ∆ABC and a point P, reflect ⊙PBC, ⊙APC, ⊙ABP in the lines BC, AC, AB respectively. Then the three reflected circles have a common point, Q (see Figure 1).I do not know if this theorem is new, but I have not come across it in the literature. The reader is invited to prove it by angle-chasing, using circle theorems: let two of the reflected circles meet at Q and then prove that this point lies on the third reflected circle. This method is rather diagram-dependent, and does not seem to lead to the extensions of Theorem 1 referred to above in any very obvious manner, so we shall adopt a different approach.
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Calvo, R., A. Arteaga, and R. Domingo. "A comparison of fitting criteria for circle arc measurement applications." IOP Conference Series: Materials Science and Engineering 1193, no. 1 (2021): 012073. http://dx.doi.org/10.1088/1757-899x/1193/1/012073.
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Abstract Measuring circular shape is a main task of dimensional metrology, characterized by circle diameter and its roundness, both for full circles and circle arcs. Point coordinates allows measuring both arcs and full circle, by fitting to substitution geometry, in many cases by least-squares criteria and fitting. Nevertheless, circle shape can be also characterized by the minimum zone, minimum circumscribed and maximum inscribed circles. This research presents a systematic experimental analysis of results of normal distributed points around the substitution circle through simulation, for different circle arc angles to the full circle and for the four mentioned fitting criteria. The results show the influence of arc angle in the variability of the results across criteria and the different behaviour depending on the arc amplitude. The results confirm the good stability and behaviour of least squares and minimum zone criteria, while warns the use of minimum circumscribed and maximum inscribed circles over half circumference. Experimental regression facilitates estimation of the minimum zone criteria from the least squares fitting that are independently verified with literature datasets with good results.
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Ou, Yun, Honggui Deng, Yang Liu, et al. "A Fast Circle Detection Algorithm Based on Information Compression." Sensors 22, no. 19 (2022): 7267. http://dx.doi.org/10.3390/s22197267.
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Circle detection is a fundamental problem in computer vision. However, conventional circle detection algorithms are usually time-consuming and sensitive to noise. In order to solve these shortcomings, we propose a fast circle detection algorithm based on information compression. First, we introduce the idea of information compression, which compresses the circular information on the image into a small number of points while removing some of the noise through sharpness estimation and orientation filtering. Then, the circle parameters stored in the information point are obtained by the average sampling algorithm with a time complexity of O(1) to obtain candidate circles. Finally, we set different constraints on the complete circle and the defective circle according to the sampling results and find the true circle from the candidate circles. The experimental results on the three datasets show that our method can compress the circular information in the image into 1% of the information points, and compared to RHT, RCD, Jiang, Wang and CACD, Precision, Recall, Time and F-measure are greatly improved.
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KIM, DONGUK, DEOK-SOO KIM, and KOKICHI SUGIHARA. "EUCLIDEAN VORONOI DIAGRAM FOR CIRCLES IN A CIRCLE." International Journal of Computational Geometry & Applications 15, no. 02 (2005): 209–28. http://dx.doi.org/10.1142/s021819590500166x.
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Presented in this paper is an algorithm to compute a Euclidean Voronoi diagram for circles contained in a large circle. The radii of circles are not necessarily equal and no circle inside the large circle wholly contains another circle. The proposed algorithm uses the ordinary point Voronoi diagram for the centers of inner circles as a seed. Then, we apply a series of edge-flip operations to the seed topology to obtain the correct topology for the desired one. Lastly, the equations of edges are represented in a rational quadratic Bézier curve form.
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Betyayev, S. K., A. M. Gaifullin, and S. V. Gordeyev. "The point-circle vortex." Journal of Applied Mathematics and Mechanics 58, no. 4 (1994): 749–54. http://dx.doi.org/10.1016/0021-8928(94)90149-x.
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Fraivert, David. "New points that belong to the nine-point circle." Mathematical Gazette 103, no. 557 (2019): 222–32. http://dx.doi.org/10.1017/mag.2019.53.
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In the present paper, we show that the point of intersection of the bimedians of a cyclic quadrilateral belongs to the nine-point circle (Euler’s circle) of the triangle with one vertex at the point of intersection of the quadrilateral’s diagonals and the other vertices at the points of intersection of the extensions of its pairs of opposite sides.
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BABAEI, Ali. "The Boiling Circle, the Rotating Circle (Two kinds of symbols of the Circle in Islamic Tradition)." WISDOM 18, no. 2 (2021): 162–75. http://dx.doi.org/10.24234/wisdom.v18i2.484.
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In theological sources, many symbols are used to explain the transcendent truths of existence. Among the shapes, the circle has the most use of a symbol which is important for Religious, philosophers, and mystics. However, what is refer mostly to the shape of a circle is the rotation of a circular line that begins at a point on a surface and ends at the same point; then, the most superficial and intermediate symbols of facts are explained with it. Contrary, the present article proposes a novel way of drawing a circle, and with this approach, examines some philosophical concepts. We call this drawing "Boiling Circle", because, the rays are coming out boiling from the center. We also have analyzed and introduced a unique example of a mystical-philosophical-religious Architectural building, during which a circular spring has been built. Its water comes out boiling of the center and fills the five circles within itself and twelve eyes around it. This article begins with the drawing of a boiling circle, continues with explaining the philosophical symbols of the boiling circle and boiling spring according to Islamic mysticism, and ends with comparing and expressing the differences between the symbols of the two circles.
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Zhao, Yue, Yuyang Chen, and Liping Yang. "Calibration of Double-Plane-Mirror Catadioptric Camera Based on Coaxial Parallel Circles." Journal of Sensors 2022 (August 25, 2022): 1–15. http://dx.doi.org/10.1155/2022/7145400.
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A catadioptric camera with a double-mirror system, composed of a pinhole camera and two planar mirrors, can capture multiple catadioptric views of an object. The catadioptric point sets (CPSs) formed by the contour points on the object lie on circles, all of which are coaxial parallel. Based on the property of the polar line of the infinity point with respect to a circle, the infinity points in orthogonal directions can be obtained using any two CPSs, and a pole and polar pair with respect to the image of the absolute conic (IAC) can be obtained through inference of the Laguerre theorem. Thus, the camera intrinsic parameters can be solved. Furthermore, as the five points needed to fit the image of a circle are not easy to obtain accurately, only sets in which five points can be located can be obtained, whereas the points on the line of intersection between the two plane mirrors and the ground plane can easily be obtained accurately. An optimization method based on the analysis of neighboring point sets to compare the intersection points with an image of the center of multiple circle images fitted using the point sets is proposed. Bundle adjustment is then applied to further optimize the camera intrinsic parameters. The feasibility and validity of the proposed calibration methods and their optimization were confirmed through simulation and experiments. Two primary innovations were obtained from the results of this study: (1) by applying coaxial parallel circles to the double-plane-mirror catadioptric camera model, a variety of calibration methods were derived, and (2) we found that the overall model could be optimized by analyzing the features of the neighboring point set and bundle adjustment.
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Li, Dayong, Chuanping Sun, and Yukun Zhang. "Principle of pole point method of Mohr’s circle of strain and its applications in geotechnical engineering." Soils and Rocks 48, no. 1 (2025): e2025009823. https://doi.org/10.28927/sr.2025.009823.
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Mohr’s circle is a convenient geometric method to solve two-dimensional stress and strain problems in geotechnical engineering and materials engineering. The pole point is such a special point that can help readily find stresses and strains on any specified plane by using a diagram instead of complex computations. This paper first presents two conventional pole point methods of a Mohr’s strain circle, i.e., the parallel line method and the normal line method. A new method of determining the pole point of strain, called the ray method, is then proposed. It was found that the parallel line method and the normal line method are two special cases of the ray method; however, the parallel line method was proved the most efficient way to determine the strains for a specified plane. The uniqueness of the pole point was proved by using an indirect proof; and the pole point method was verified by a theoretical method. Results also show that Mohr’s strain and stress circles can be drawn in a concentric circle. Based on the relationship between the pole points of stress and strain, any relatively complex stress and strain states can be determined by using the pole point method rather than using the theoretical method.
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WOLFF, ALEXANDER, MICHAEL THON, and YINFENG XU. "A SIMPLE FACTOR-2/3 APPROXIMATION ALGORITHM FOR TWO-CIRCLE POINT LABELING." International Journal of Computational Geometry & Applications 12, no. 04 (2002): 269–81. http://dx.doi.org/10.1142/s0218195902000888.
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Given a set P of n points in the plane, the two-circle point-labeling problem consists of placing 2n uniform, non-intersecting, maximum-size open circles such that each point touches exactly two circles. It is known that this problem is NP-hard to approximate. In this paper we give a simple algorithm that improves the best previously known approximation factor from [Formula: see text] to 2/3. The main steps of our algorithm are as follows. We first compute the Voronoi diagram, then label each point optimally within its cell, compute the smallest label diameter over all points and finally shrink all labels to this size. We keep the O(n log n) time and O(n) space bounds of the previously best algorithm.
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Графский, O. Grafskiy, Пономарчук, and Yu Ponomarchuk. "On One Property of a Circle on the Coordinate Plane." Geometry & Graphics 5, no. 2 (2017): 13–24. http://dx.doi.org/10.12737/article_5953f2af770c35.65774157.
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Descartes’ and Fermat's method allowed to define many geometrical forms, including circles, on the coordinate plane by means of the arithmetic equations and to make necessary analytical operations in order to solve many problems of theoretical and applied research in various scientific areas, for example. However, the equations of a circle and other conics in the majority of research topics are used in the subsequent analysis of applied problems, or for analytical confirmation of constructive solutions in geometrical research, according to Russian geometrician G. Monge and others, including. It is natural to consider a circle as a locus of points, equidistant from a given point — a center of the circle, with a constant distance R. There is another definition of a circle: a set of points from which a given segment is visible under constant directed angle. Besides, a circle is accepted to model the Euclid plane in the known scheme of non-Euclidean geometry of Cayley-Klein, it is the absolute which was given by A. Cayley for the first time in his memoirs. It is possible to list various applications of this geometrical form, especially for harmonism definition of the corresponding points, where the diametral opposite points of a circle are accepted as basic, and also for construction of involutive compliances. The construction of tangents to a circle can be considered as a classical example. Their constructive definition is simple, but also constructions on the basis of known projective geometry postulates are possible (a hexagon when modeling a series of the second order, Pascal's lines). These postulates can be applied to construction of tangents to a circle (to an ellipse and hyperboles to determination of imaginary points of intersection of a circle and a line. This paper considers the construction of tangents to a circle without the use of arches of auxiliary circles, which was applied in order to determine the imaginary points of intersection of a circle and a line (an axis of coordinates). Besides, various dependences of parameter p2, which is equal to the product of the values of the intersection points’ coordinates of a circle and coordinate axes, are analytically determined.
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Anglesio, Jean, Jiro Fukuta, Shinsei-cho, et al. "A Six-Point Circle: 10469." American Mathematical Monthly 105, no. 9 (1998): 862. http://dx.doi.org/10.2307/2589228.
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Suceava, Bogdan, and Achilleas Sinefakopoulos. "A Six Point Circle: 10710." American Mathematical Monthly 107, no. 6 (2000): 572. http://dx.doi.org/10.2307/2589366.
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Margetson, J., and R. Buckingham. "73.11 The Ten-Point Circle." Mathematical Gazette 73, no. 463 (1989): 41. http://dx.doi.org/10.2307/3618210.
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Scott, J. A. "86.55 An Eight-Point Circle." Mathematical Gazette 86, no. 506 (2002): 326. http://dx.doi.org/10.2307/3621878.
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Maehara, Hiroshi. "Circle lattice point problem, revisited." Discrete Mathematics 338, no. 3 (2015): 164–67. http://dx.doi.org/10.1016/j.disc.2014.11.004.
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Baura, Gail. "The Circle [Point of View]." IEEE Engineering in Medicine and Biology Magazine 26, no. 5 (2007): 69. http://dx.doi.org/10.1109/emb.2007.901782.
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Rahav, S., O. Richman, and S. Fishman. "Point perturbations of circle billiards." Journal of Physics A: Mathematical and General 36, no. 40 (2003): L529—L536. http://dx.doi.org/10.1088/0305-4470/36/40/l02.
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Nastro, Vincenzo, and Urbano Tancredi. "Great Circle Navigation with Vectorial Methods." Journal of Navigation 63, no. 3 (2010): 557–63. http://dx.doi.org/10.1017/s0373463310000044.
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The present paper is concerned with the solution of a series of practical problems relevant to great circle navigation, including the determination of the true course at any point on the great circle route and the determination of the lateral deviation from a desired great circle route. Intersection between two great circles or between a great circle and a parallel is also analyzed. These problems are approached by means of vector analysis, which yields solutions in a very compact form that can be computed numerically in a very straightforward manner. This approach is thus particularly appealing for performing computer-aided great circle navigation.
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Ge, Bao Zhen, Qi Jun Luo, Bin Ma, Yong Jie Wei, Bo Chen, and Sheng Zhao Jiang. "The Algorithm to Measure Crack Width with Incircle." Advanced Materials Research 684 (April 2013): 481–85. http://dx.doi.org/10.4028/www.scientific.net/amr.684.481.
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Crack is a major defect of buildings. Digital image methods are often used to detect cracks. But incorrect or un-unique results may be inverted with an inappropriate algorithm. An image processing way is presented to obtain the sole width value. Meanwhile, the crack with several branches can be measured. In the processing, the crack skeleton is first calculated. Then each of the points on the skeleton is served as a center of a group of circles, one by one. The radius of the circles is increased step by step. The iterations will not stop until any point in the circle goes out of the crack. Thus the last circle in the iteration is served as an incircle of the crack. The diameter of the incircle is a crack width in a given skeleton point. The maximal and average width of the crack will be calculated after all the incircles with all the skeleton point are traversed. The experimental results show the proposed method can extract the width of cracks in a complex context.
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Natchev, Vladislav. "Apollonian Sphere and Properties of Stereographic Projection around the Lemoine Point." Mathematics and Informatics 68, no. 1 (2025): 35–50. https://doi.org/10.53656/math2025-1-3-apo.
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The definition and some of the properties of the Apollonian circle in the plane find their analogies in the Euclidean three-dimensional space. Thus, we manage to introduce a new concept in solid geometry that we call an “Apollonian sphere”. It appears that the Apollonian sphere not only possesses classical properties similar to the Apollonian circle such as orthogonality and coaxiality, but also analogies of its lesser-known connection with the Lemoine point and the circumcenter. We also discover two notable properties of stereographic projection that we prove with an Apollonian sphere. They include collinearity of the projection point with the Lemoine points of the projection and the projected triangles or with the centers of their Apollonian circles. Moreover, we connect the newly introduced concepts and the rich configurations they generate with Olympiad geometry.
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AHN, SUNG JOON, and WOLFGANG RAUH. "GEOMETRIC LEAST SQUARES FITTING OF CIRCLE AND ELLIPSE." International Journal of Pattern Recognition and Artificial Intelligence 13, no. 07 (1999): 987–96. http://dx.doi.org/10.1142/s0218001499000549.
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The least squares fitting of geometric features to given points minimizes the squares sum of error-of-fit in predefined measures. By the geometric fitting, the error distances are defined with the orthogonal, or shortest, distances from the given points to the geometric feature to be fitted. For the geometric fitting of circle and ellipse, robust algorithms are proposed which are based on the coordinate descriptions of the corresponding point on the circle/ellipse for the given point, where the connecting line of the two points is the shortest path from the given point to the circle/ellipse.
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Hüsler, Jürg. "On point processes on the circle." Journal of Applied Probability 23, no. 2 (1986): 322–31. http://dx.doi.org/10.2307/3214176.
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Point processes on the circle with circumference 1 are considered, which are related to the coverage problem of the circle by n randomly placed arcs of a fixed length. The anticlockwise endpoint of each arc is assumed to be uniformly distributed on the circle. We deal with a general limit result on the convergence of these point processes to a Poisson process on the circle. This result is then applied to several cases of the coverage problem, giving improved limit results in these cases. The proof uses a new convergence result of general point processes.
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Hüsler, Jürg. "On point processes on the circle." Journal of Applied Probability 23, no. 02 (1986): 322–31. http://dx.doi.org/10.1017/s0021900200029636.
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Point processes on the circle with circumference 1 are considered, which are related to the coverage problem of the circle by n randomly placed arcs of a fixed length. The anticlockwise endpoint of each arc is assumed to be uniformly distributed on the circle. We deal with a general limit result on the convergence of these point processes to a Poisson process on the circle. This result is then applied to several cases of the coverage problem, giving improved limit results in these cases. The proof uses a new convergence result of general point processes.
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Ploom, Ülar. "On the Poetic of the Double Point and Circle in Dante’s Paradiso 30 and in Desmond Hogan’s Short Story “The Last Time”." Studia Metrica et Poetica 6, no. 1 (2019): 79–93. http://dx.doi.org/10.12697/smp.2019.6.1.03.
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This essay discusses the interaction of the divine point of light and Beatrice as the unattainable point of revelation for Dante in Paradiso 30. The two points with their respective circles of understanding and expression form a powerful figure which calls for conceptualisation both in the context of Canto 30 but also the whole of the Divina Commedia. Despite the different epochs, ideologies and contexts, a striking similarity as to the poetic of the double point and circle may be found also in Hogan’s text.
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Cotar, Codina, and Stanislav Volkov. "A note on the lilypond model." Advances in Applied Probability 36, no. 2 (2004): 325–39. http://dx.doi.org/10.1239/aap/1086957574.
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We consider some generalizations of the germ-grain growing model studied by Daley, Mallows and Shepp (2000). In this model, a realization of a Poisson process on a line with points Xi is fixed. At time zero, simultaneously at each Xi, a circle (grain) starts growing at the same speed. It grows until it touches another grain, and then it stops. The question is whether the point zero is eventually covered by some circle. In our note we expand this model in the following three directions. We study: a one-sided growth model with a fixed number of circles; a grain-growth model on a regular tree; and a grain-growth model on a line with non-Poisson distributed centres of the circles.
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Cotar, Codina, and Stanislav Volkov. "A note on the lilypond model." Advances in Applied Probability 36, no. 02 (2004): 325–39. http://dx.doi.org/10.1017/s0001867800013495.
Full textAbstract:
We consider some generalizations of the germ-grain growing model studied by Daley, Mallows and Shepp (2000). In this model, a realization of a Poisson process on a line with points X i is fixed. At time zero, simultaneously at each X i , a circle (grain) starts growing at the same speed. It grows until it touches another grain, and then it stops. The question is whether the point zero is eventually covered by some circle. In our note we expand this model in the following three directions. We study: a one-sided growth model with a fixed number of circles; a grain-growth model on a regular tree; and a grain-growth model on a line with non-Poisson distributed centres of the circles.
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Lei, Xian Qing, and Feng Qin Ding. "Evaluating Method of Roundness Error Based on the Maximum Inscribed Circle and Minimum Circumscribed Circle." Advanced Materials Research 655-657 (January 2013): 847–50. http://dx.doi.org/10.4028/www.scientific.net/amr.655-657.847.
Full textAbstract:
Based on the geometric characteristics of maximum inscribed circle (MIC) and Minimum circumscribed circle (MCC), a novel method for roundness error evaluation using Geometric Approximation Searching Algorithm (GASA) has been presented in this paper. First, a square is allocated by taking the predetermined reference point as initial reference point, and the radius value of all the measured points are calculated by regarding each vertex of the square and the initial reference point as the ideal centres respectively. Second, the minimum and maximum radius are obtained when each vertex of the square and the initial reference point as the ideal centre point. Final, compare these radius and arrange the square repeatedly. Repeat the steps, the roundness error of the maximum inscribed circle can be obtained complies with ANSI and ISO standards. The experimental results show that this algorithm is effective and accurate.
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Roy, Kushal, Debashis Dey та Mantu Saha. "Certain Fixed Point Results On 𝔄-Metric Space Using Banach Orbital Contraction and Asymptotic Regularity". Mathematica Slovaca 73, № 2 (2023): 485–500. http://dx.doi.org/10.1515/ms-2023-0036.
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ABSTRACT In this paper, we investigate the existence of φ-fixed point for Banach orbital contraction over 𝔄-metric space. Also a fixed point result has been established via asymptotic regularity property over such generalized metric space. Our fixed point theorems have also been applied to the fixed circle problem. Moreover, we give some new solutions to the open problem raised by Özgür and Taş on the geometric properties of φ-fixed points of self-mappings and the existence and uniqueness of φ-fixed circles and φ-fixed discs for various classes of self-mappings.
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Tomšič, Neja. "Around Circle." Maska 39, no. 219 (2024): 46–61. http://dx.doi.org/10.1386/maska_00184_1.
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The starting point of the performance Circle, created in collaboration with Nonument Group, is a negligible point in time and space: a former railroad workers’ park in Cluj, Romania. When we started creating Circle, this particular spot was just a patch of land, which is precisely how we approached it: as travellers walking around a seemingly blank slate, an empty lot. This construction site was a vision announcing the present and the future of the city, while simultaneously erasing the old.
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Silvester, John R. "The trisectrix and Langley's problem." Mathematical Gazette 106, no. 565 (2022): 21–27. http://dx.doi.org/10.1017/mag.2022.5.
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If a circle rolls without slipping around an equal fixed circle, then a point carried by the rolling circle traces out a limaçon of Pascal. (This is Etienne Pascal, father of Blaise. The word limaçon is derived from the Latin limax, a snail.) If the fixed and rolling circles have radius 1, and the point P carried by the rolling wheel is distant a from its centre, then for a > 1 the limaçon has an inner and an outer loop, joining up at a node. For a = 1 it has a cusp, and is then a cardioid, so-called because it is heart-shaped. See Figure 1, where we have plotted the cases a = $${3 \over 4}$$ , a = 1 and a = $${3 \over 2}$$ .
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Popescu, Iulian, Ludmila Sass, and Alina Elena Romanescu. "Rolling with and without Slipping, during Epicycloids Generation." Applied Mechanics and Materials 880 (March 2018): 63–68. http://dx.doi.org/10.4028/www.scientific.net/amm.880.63.
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The starting point consists in the modality to generate epicycloids when two external circles are considered. The mobile circle is rolling on the fix circle without slipping, such as two arcs belonging to these circles are equal. The specialty literature presents an example with a simple planetary gear in which the “satellite porting” arm provides the rolling of the mobile circle on the fix circle. Our original idea, not approached in the specialty literature, considers the rolling with slipping of the mobile circle on the fix circle. Instead of the gears providing the rolling without slipping, two wheels with smooth surfaces are used now. The case when the two involved arc are no longer equal is analysed. Between them appear either frictions generating braking or “skating like” rolling when the lubricant layer is too thick. An analysis of the theoretical case when the slipping has a sense opposite to that of a normal rolling is also performer. A significant class of curves was obtained. Some are even epicycloids obtained with slipping, with other parameters.
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Беглов, И., I. Beglov, Вячеслав Рустамян, and Vyacheslav Rustamyan. "Method of Rotation of Geometrical Objects Around the Curvilinear Axis." Geometry & Graphics 5, no. 3 (2017): 45–50. http://dx.doi.org/10.12737/article_59bfa4eb0bf488.99866490.
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Rotation is the motion of geometric objects along a circle. This is one of geometric techniques used to form lines and surfaces. In this paper has been considered the rotation of objects in a three-dimensional space around a straight axis. It is known that a straight line can be considered as a particular case of a circle with a radius equal to infinity. Such circle’s center is at infinite distance from the considered straight line segment. Then in the general case, the rotation axis is a closed curve, for example, a circle with a radius of finite magnitude. Rotation of a point around a straight axis now splits into two trajectories. One of them is a circle with a radius, the second is a straight line crossing with the axis, and the center of this trajectory is at an infinite distance from the point. The method of point rotation about an axis of finite radius was considered. Note that a circle is a special case of an ellipse. When the actual focus of the circle is stratified into two, the line itself loses its curvature constancy, and is called an ellipse. The point, rotating around the elliptical axis, is stratified into four ones, forming four circles (trajectories). Axis foci appearing in turn in the role of the main one determine two trajectories by each with a trivial and nontrivial center of rotation. We have considered the variant for arrangement of the generating circle so that its center coincided with one of the elliptic axis’s foci. The obtained surfaces are a pair of co-axial Dupin cyclides, since they have identical properties. Changing the circle generatrix radius, other things being equal, we get different types of closed cyclides.
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Li, Yuanzhen, and Yue Zhao. "Paracatadioptric camera calibration based on properties of polar line of infinity point with respect to circle and line." International Journal of Advanced Robotic Systems 15, no. 5 (2018): 172988141880385. http://dx.doi.org/10.1177/1729881418803856.
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Two linear calibration methods based on space-line projection properties for a paracatadioptric camera are presented. Considering the central catadioptric system, a straight line is projected into a circle on the viewing spherical surface for the first projection. The tangent lines in a group at antipode point pairs with respect to the circle are parallel, with the infinity point being the intersection point; therefore, the infinity line can be obtained from two groups of antipode point pairs. Further, the direction of the polar line of an infinity point with respect to the circle is orthogonal to the direction of its infinity point. Hence, on the imaging plane, images of the circular points or orthogonal vanishing points are used to determine the intrinsic parameters. On the basis of the properties of the antipodal point pairs and a least-squares fitting, a corresponding optimization algorithm for line image fitting is proposed. Experimental results demonstrate the robustness of the two calibration methods, that is, for images of the circular points and orthogonal vanishing points.
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Zadorozhnyi, Bohdan, Tetyana Romanova, Petro Stetsyuk, Stanislav Tyvodar, and Sergiy Shekhovtsov. "Packing Unequal Circles into a Minimum-Radius Circle Using r-Algorithm." Cybernetics and Computer Technologies, no. 4 (December 18, 2024): 5–21. https://doi.org/10.34229/2707-451x.24.4.1.
Full textAbstract:
Two approaches to employ the Shor’s r-algorithm for solving a problem of packing unequal circles into a minimum-radius circle are studied. The first approach uses multistart of the r-algorithm with a step dichotomy from a set of feasible starting points. Each feasible point is taken as the best solution found by a heuristic algorithm. Two versions of the algorithm are considered. For the first version, the step value is halved during the iteration process. The second version provides an option that allows to restore the maximum value of the r-algorithm step value. The algorithm is implemented using Rust 1.70.0 programming language and nalgebra 0.32.3 library. Both versions of the algorithm are tested for 50 test problems of the international competition “Dense packing of circles in a circle of minimum radius” to improve the results found by the heuristic. In most cases, the second version showed better solutions. The second approach employs the r-algorithm with an adaptive step to find the best local minimum of a multiextremal nonsmooth function using multistart strategy from a set of randomly chosen starting points. It is implemented using Julia programming language and uses large numbers (128 and 256 bits). Computational experiments are tested for a benchmark problem with five circles. These results are compared to the problem solutions provided on the website http://www.packomania.com/. It is shown that increasing the bit depth leads to decreasing the number of the r-algorithm iterations while increasing the accuracy of the objective function value. With correctly chosen parameters, the r-algorithm finds all 28 digits after the decimal point, which are presented on the website http://www.packomania.com/. Keywords: circle packing, r-algorithm, heuristic algorithm, Rust, Julia.
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Silvester, John R. "The seven circles theorem revisited." Mathematical Gazette 102, no. 554 (2018): 280–301. http://dx.doi.org/10.1017/mag.2018.59.
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The circles C1, & , Cn form a chain of length n if Ci touches Ci + 1, for i = 1, & , n − 1, and the chain is closed if also Cn touches C1. A cyclic chain is a chain for which all the circles touch another circle S, the base circle of the chain. If Ci touches S at Pi, then P1, & , Pn are the base points of the chain. Sometimes there may be coincidences among the base points; in particular, if Pi = Pj, then the line PiPj should be interpreted as the tangent S to at Pi.The seven circles theorem first appeared in [1, §3.1], and some historical details of its genesis can be found in John Tyrrell's obituary [2]. The theorem concerns a closed cyclic chain of length 6, and says that, if a certain extra condition is satisfied, then the lines P1P4, P2P5, P3P6 joining opposite base points are concurrent. Here and throughout, ‘concurrent’ should be read as ‘concurrent or all parallel’, that is, the point of concurrency might be at infinity.
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Khyade, Vitthalrao Bhimasha, and Avram Hershko. "Attempt on Magnification of the Mechanism of Enzyme Catalyzed Reaction through Bio-geometric Model for the Five Points Circle in the Triangular Form of Lineweaver-Burk Plot." International Journal of Emerging Scientific Research 1 (December 31, 2020): 1–19. http://dx.doi.org/10.37121/ijesr.vol1.120.
Full textAbstract:
The bio-geometrical model is dealing with correlation between the “five events for enzyme catalyzed reaction” and “triple point event serving groups on the circle” in triangle obtained for the graphical presentation of the double reciprocal for magnification of the mechanism of enzyme catalyzed reaction. This model is based on the nine point circle in triangle of the double reciprocal plot. The five significant points (B, D, E, F and G) resulted for the circle with x – and y – coordinates. The present attempt is considering interactions among enzymes and substrates for the successful release of product through each and every point on the circle in triangle. The controlling role of the point, “O”, center of circle in each and every event of the biochemical reaction is obligatory. The model is allotting specific role for the significant events in the biochemical reaction catalyzed by the enzymes. The enzymatic catalysis is supposed to be completed through five events, which may be named as, “Bio-geometrical events of enzyme catalyzed reaction”. These five events for enzyme catalyzed reaction include: (1) Initial event of enzymatic interaction with the substrates; (2) Event of the first transition state for the formation of “enzyme-substrate” complex; (3) Event of the second transition state for the formation of “enzyme-product” complex; (4) Event of release of the product and relieve enzyme and (5) The event of directing the enzyme to continue the reaction. The model utilizes the “triple point serving group on the circle” for the success of each and every event in the biochemical reaction. Thus, there is involvement of the three points including the point “O” for each event in the enzyme catalyzed reaction. The group of points serving for carrying out the event may be classified into five conic sections like: B-O-E; E-O-G; G-O-D; D-O-F and F-O-B. The bio-geometrical model is correlation between the “five events for enzyme catalyzed reaction” and “triple point event serving groups on the circle” in a triangle of the double reciprocal plot.
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Khyade, Vitthalrao Bhimasha, and Avram Hershko. "Attempt on Magnification of the Mechanism of Enzyme Catalyzed Reaction through Bio-geometric Model for the Five Points Circle in the Triangular Form of Lineweaver-Burk Plot." International Journal of Emerging Scientific Research 1 (December 31, 2020): 1–19. http://dx.doi.org/10.37121/ijesr.vol1.120.
Full textAbstract:
The bio-geometrical model is dealing with correlation between the “five events for enzyme catalyzed reaction” and “triple point event serving groups on the circle” in triangle obtained for the graphical presentation of the double reciprocal for magnification of the mechanism of enzyme catalyzed reaction. This model is based on the nine point circle in triangle of the double reciprocal plot. The five significant points (B, D, E, F and G) resulted for the circle with x – and y – coordinates. The present attempt is considering interactions among enzymes and substrates for the successful release of product through each and every point on the circle in triangle. The controlling role of the point, “O”, center of circle in each and every event of the biochemical reaction is obligatory. The model is allotting specific role for the significant events in the biochemical reaction catalyzed by the enzymes. The enzymatic catalysis is supposed to be completed through five events, which may be named as, “Bio-geometrical events of enzyme catalyzed reaction”. These five events for enzyme catalyzed reaction include: (1) Initial event of enzymatic interaction with the substrates; (2) Event of the first transition state for the formation of “enzyme-substrate” complex; (3) Event of the second transition state for the formation of “enzyme-product” complex; (4) Event of release of the product and relieve enzyme and (5) The event of directing the enzyme to continue the reaction. The model utilizes the “triple point serving group on the circle” for the success of each and every event in the biochemical reaction. Thus, there is involvement of the three points including the point “O” for each event in the enzyme catalyzed reaction. The group of points serving for carrying out the event may be classified into five conic sections like: B-O-E; E-O-G; G-O-D; D-O-F and F-O-B. The bio-geometrical model is correlation between the “five events for enzyme catalyzed reaction” and “triple point event serving groups on the circle” in a triangle of the double reciprocal plot.
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Scott, J. A. "103.10 Around the nine-point circle." Mathematical Gazette 103, no. 556 (2019): 150–52. http://dx.doi.org/10.1017/mag.2019.25.
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ANDRADE, MARCUS VINÍCIUS ALVIM, and JORGE STOLFI. "EXACT ALGORITHMS FOR CIRCLES ON THE SPHERE." International Journal of Computational Geometry & Applications 11, no. 03 (2001): 267–90. http://dx.doi.org/10.1142/s021819590100050x.
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We describe exact representations and algorithms for geometric operations on general circles and circular arcs on the sphere, using integer homogeneous coordinates. The algorithms include testing a point against a circle, computing the intersection of two circles, and ordering three arcs out of the same point. These tools support robust and efficient operations on maps overs the sphere, such as point location and map overlay, and provide a reliable framework for robotics, geographic information systems, and other geometric applications.
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Avelanda. "Properties of Objects and Their Transformation on the Plane." Indian Journal of Advanced Mathematics 4, no. 1 (2024): 19–21. http://dx.doi.org/10.54105/ijam.a1166.04010424.
Full textAbstract:
Every function has its properties relative to that of the mechanisms of a circle, since cycles repeat. Such that all equations factors so well within its definition, that clearly, every circle has a radius that is greater than zero. If the radius expands by any set of numbers, then it is undergoing transformation. Hence, every object under it expands at a certain ratio. To a point that it is quite natural for phenomena to repeat; given that it is within the area of its circumference. Although without events being the same. Functions tends to be in approximation with mathematical constants, merely for the periodicity of their behaviour- since it is cyclic under certain conditions. The centre of a black hole is a point; so much like that of a circle. Since therefore, objects alters the state of dimensions they occupy, so that relative to the point of reference: they either appear as paraboloids, ellipses, hyperbolas, or circles; depending on the context.
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Avelanda. "Properties of Objects and Their Transformation on the Plane." Indian Journal of Advanced Mathematics (IJAM) 4, no. 1 (2024): 19–21. https://doi.org/10.54105/ijam.A1166.04010424.
Full textAbstract:
<strong>Abstract:</strong> Every function has its properties relative to that of the mechanisms of a circle, since cycles repeat. Such that all equations factors so well within its definition, that clearly, every circle has a radius that is greater than zero. If the radius expands by any set of numbers, then it is undergoing transformation. Hence, every object under it expands at a certain ratio. To a point that it is quite natural for phenomena to repeat; given that it is within the area of its circumference. Although without events being the same. Functions tends to be in approximation with mathematical constants, merely for the periodicity of their behaviour- since it is cyclic under certain conditions. The centre of a black hole is a point; so much like that of a circle. Since therefore, objects alters the state of dimensions they occupy, so that relative to the point of reference: they either appear as paraboloids, ellipses, hyperbolas, or circles; depending on the context.