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Journal articles on the topic 'Area and perimeter of plane figures'

Author: Grafiati

Published: 4 June 2021

Last updated: 12 February 2022

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1

Shahin, Louis. "An Interesting Solid." Mathematics Teacher 79, no. 5 (May 1986): 378–79. http://dx.doi.org/10.5951/mt.79.5.0378.

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Some plane figures are interesting because their perimeter and area are numerically equal. For example, a square with a side of 4 cm has a perimeter and area numerically equal to sixteen. A further study of plane figures with this interesting property can be found in Bates (1979) and Markowitz (1981).

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2

Kellerer, A. M. "Counting figures in planar random configurations." Journal of Applied Probability 22, no. 1 (March 1985): 68–81. http://dx.doi.org/10.2307/3213749.

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Random configurations are considered that are generated by a Poisson process of figures in the plane, and a recent result is used to derive formulae for the estimation of the number of figures, and their mean area and perimeter. The formulae require merely the determination of the area, the perimeter, and the Euler–Poincaré characteristic of the random configurations in a fixed field of view. There are no similar formulae for the standard deviations of the estimates; their magnitudes in typical cases are therefore assessed by Monte Carlo simulations.

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3

Kellerer, A. M. "Counting figures in planar random configurations." Journal of Applied Probability 22, no. 01 (March 1985): 68–81. http://dx.doi.org/10.1017/s0021900200029028.

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Random configurations are considered that are generated by a Poisson process of figures in the plane, and a recent result is used to derive formulae for the estimation of the number of figures, and their mean area and perimeter. The formulae require merely the determination of the area, the perimeter, and the Euler–Poincaré characteristic of the random configurations in a fixed field of view. There are no similar formulae for the standard deviations of the estimates; their magnitudes in typical cases are therefore assessed by Monte Carlo simulations.

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4

Kesarkar, Kashmira, Avinash Tamgadge, Treville Peirera, Sandhya Tamgadge, Swati Gotmare, and Pooja Kamat. "Evaluation of Mitotic Figures and Cellular and Nuclear Morphometry of Various Histopathological Grades of Oral Squamous Cell Carcinoma: Comparative study using crystal violet and Feulgen stains." Sultan Qaboos University Medical Journal [SQUMJ] 18, no. 2 (September 9, 2018): 149. http://dx.doi.org/10.18295/squmj.2018.18.02.005.

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Objectives: The objectives of this study were to quantitatively estimate the number of mitotic figures (MFs) and evaluate the cellular and nuclear features of various histological grades of oral squamous cell carcinoma (OSCC) using Feulgen and 1% crystal violet stains. Methods: This case-control study took place at the Dr D. Y. Patil Dental College & Hospital in Mumbai, Maharashtra, India, between June and December 2016. A total of 51 samples were retrieved from the hospital archives. Of these, 15 well-differentiated, 15 moderately-differentiated and six poorly-differentiated OSCC samples formed the case group while 15 samples of normal gingival mucosa constituted the control group. Each sample was dyed using Feulgen and 1% crystal violet stains and the mitotic count, nuclear area (NA), cellular area (CA), nuclear perimeter (NP), cellular perimeter (CP) and nuclear-to-cytoplasmic (N/C) ratio was calculated using computeraided morphometry techniques. Results: The number of MFs visible per field was significantly higher in Feulgen-stained sections as compared to those stained with crystal violet (P = 0.050). In addition, the NA, NP, CA and CP values and N/C ratios of samples in the experimental group increased significantly in accordance with an increase in OSCC grade (P <0.001). Conclusion: The Feulgen stain is more reliable than 1% crystal violet in terms of the selective staining of MFs. Moreover, the findings of this study indicate that computer-based morphometric analysis is an effective tool for differentiating between various grades of OSCC.Keywords: Crystal Violet; Feulgen Stain; Mitotic Index; Image Cytometry; Squamous Cell Carcinoma; Oral Cancers.

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5

Lince Loaiza, Yuliana Andrea. "Material educativo computarizado (MEC) en el proceso de enseñanza y de aprendizaje de la geometría / Computerized Educational Material (MEC) in the Teaching and Learning Process of Geometry." Revista Internacional de Aprendizaje en Ciencia, Matemáticas y Tecnología 6, no. 1 (March 25, 2019): 29–33. http://dx.doi.org/10.37467/gka-revedumat.v6.1908.

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ABSTRACTIn this work we report results of the research that references the title. The project was developed with fifth grade students; starting from the importance of mathematics in the training of primary school students, the great interest they have in ICT, the difficulties detected during the exercise of educational work. Important information was collected through a survey that was applied to the other teachers in charge of the orientation of the area of mathematics, which was reflected in the creation and application of a tool that allowed the student to assimilate assertively and motivating the learning of concepts Basic geometry in topics such as plane figures, angles, perimeters, area and volume.RESUMENEn este trabajo se reportan resultados de la investigación que referencia el título. El proyecto se desarrolló con estudiantes del grado quinto; partiendo de la importancia que tiene la matemática en la formación de los estudiantes de la básica primaria, del gran interés que ellos presentan hacia las TIC, las dificultades detectadas durante el ejercicio de la labor educativa. Se recolectó información importante mediante una encuesta que se aplicó a los demás docentes encargados de la orientación del área de matemáticas, lo que se vio reflejado en la creación y aplicación de una herramienta que permitió al estudiante asimilar de forma asertiva y motivadora el aprendizaje de conceptos básicos de la geometría en temas como las figuras planas, ángulos, perímetros, área y volumen.

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6

Ahn, Hee-Kap, and Otfried Cheong. "Aligning Two Convex Figures to Minimize Area or Perimeter." Algorithmica 62, no. 1-2 (November 3, 2010): 464–79. http://dx.doi.org/10.1007/s00453-010-9466-1.

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7

Sandefur, James T. "Using Similarity to Find Length and Area." Mathematics Teacher 87, no. 5 (May 1994): 319–25. http://dx.doi.org/10.5951/mt.87.5.0319.

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Generally in high school, students learn how to find the length of straight lines and can use that information to find the perimeter and the length of the diagonal of simple polygons. They also learn the formula relating the diameter and the perimeter of a circle, and they know how to find the areas of circles, rectangles, triangles, and, possibly, regular n-gons. But the curves whose length and the figures whose area can be calculated by students are relatively mundane, and the methods are straightforward. The techniques used do not lend themselves easily to combining algebra and geometry

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8

Lester, J. A. "Euclidean plane point-transformations preserving unit area or unit perimeter." Archiv der Mathematik 45, no. 6 (December 1985): 561–64. http://dx.doi.org/10.1007/bf01194898.

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9

Gallego, Eduardo, and Gil Solanes. "Perimeter, Diameter and Area of Convex Sets in the Hyperbolic Plane." Journal of the London Mathematical Society 64, no. 1 (August 2001): 161–78. http://dx.doi.org/10.1112/s002461070100223x.

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10

Tsintsifas, George. "Some Inequalities for Convex Sets." Axioms 9, no. 3 (September 17, 2020): 111. http://dx.doi.org/10.3390/axioms9030111.

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11

MAO, YUEYUE, SHENGLIANG PAN, and YILING WANG. "AN AREA-PRESERVING FLOW FOR CLOSED CONVEX PLANE CURVES." International Journal of Mathematics 24, no. 04 (April 2013): 1350029. http://dx.doi.org/10.1142/s0129167x13500298.

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Motivated by Gage [On an area-preserving evolution equation for plane curves, in Nonlinear Problems in Geometry, ed. D. M. DeTurck, Contemporary Mathematics, Vol. 51 (American Mathematical Society, Providence, RI, 1986), pp. 51–62] and Ma–Cheng [A non-local area preserving curve flow, preprint (2009), arXiv:0907.1430v2, [math.DG]], in this paper, an area-preserving flow for convex plane curves is presented. This flow will decrease the perimeter of the evolving curve and make the curve more and more circular during the evolution process. And finally, as t goes to infinity, the limiting curve will be a finite circle in the C∞ metric.

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12

Franceschi, Valentina, and Giorgio Stefani. "Symmetric double bubbles in the Grushin plane." ESAIM: Control, Optimisation and Calculus of Variations 25 (2019): 77. http://dx.doi.org/10.1051/cocv/2018055.

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We address the double bubble problem for the anisotropic Grushin perimeter Pα, α ≥ 0, and the Lebesgue measure in ℝ2, in the case of two equal volumes. We assume that the contact interface between the bubbles lies on either the vertical or the horizontal axis. We first prove existence of minimizers via the direct method by symmetrization arguments and then characterize them in terms of the given area by first variation techniques. Even though no regularity theory is available in this setting, we prove that angles at which minimal boundaries intersect satisfy the standard 120-degree rule up to a suitable change of coordinates. While for α = 0 the Grushin perimeter reduces to the Euclidean one and both minimizers coincide with the symmetric double bubble found in Foisy et al. [Pacific J. Math. 159 (1993) 47–59], for α = 1 vertical interface minimizers have Grushin perimeter strictly greater than horizontal interface minimizers. As the latter ones are obtained by translating and dilating the Grushin isoperimetric set found in Monti and Morbidelli [J. Geom. Anal. 14 (2004) 355–368], we conjecture that they solve the double bubble problem with no assumptions on the contact interface.

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13

Grechuk, Bogdan, and Sittichoke Som-am. "A Convex Cover for Closed Unit Curves has Area at Least 0.0975." International Journal of Computational Geometry & Applications 30, no. 02 (June 2020): 121–39. http://dx.doi.org/10.1142/s0218195920500065.

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We combine geometric methods with a numerical box search algorithm to show that the minimal area of a convex set in the plane which can cover every closed plane curve of unit length is at least [Formula: see text]. This improves the best previous lower bound of [Formula: see text]. In fact, we show that the minimal area of the convex hull of circle, equilateral triangle, and rectangle of perimeter [Formula: see text] is between [Formula: see text] and [Formula: see text].

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14

Szetela, Walter, and Douglas T. Owens. "Finding the Area of a Circle: Use a Cake Pan and Leave Out the Pi." Arithmetic Teacher 33, no. 9 (May 1986): 12–18. http://dx.doi.org/10.5951/at.33.9.0012.

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The fact that many students have misconceptions and misunde rstandings of the concept of area has been clearly documented (e.g., Carpenter et al. 1980; Jamski 1978). For example, one of the most common misconceptions held by students in grades seven and eight and even preservice teachers is that polygons with the same perimeter have the same area. Szetela (1980) found that fifty-four of ninety-four (57 percent) stude nts in grades seven and eight, when presented with diagrams of a regular hexagon and an equilateral triangle known to have the same perimeter, stated that the areas of the two regions were the same despite the fact that the hexagon contained 50 percent more area than the triangle. Even more astonishing are the results of a similar experiment by Woodward and Byrd (1983). They found that 157 of 258 (61 percent) students in grade eight stated that the areas of five different rectangles with the same perimeters were the same even though one of the rectangles was almost four times as large as another! If students have difficulty with the concept of area of polygonal figures, they can be expected to have even more difficulty with areas of irregular figures, “blobs,” and circular regions.

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15

YANG, YUNLONG, and DEYAN ZHANG. "DEFORMING A CONVEX DOMAIN INTO A DISK BY KLAIN’S CYCLIC REARRANGEMENT." Bulletin of the Australian Mathematical Society 97, no. 2 (February 20, 2018): 313–19. http://dx.doi.org/10.1017/s0004972717001113.

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For a convex domain, we use Klain’s cyclic rearrangement to obtain a sequence of convex domains with increasing area and the same perimeter which converges to a disk. As a byproduct, we give a proof of the classical isoperimetric inequality in the plane.

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16

Miles, Roger E., and Margaret S. Mackisack. "Further random tessellations with the classic Poisson polygon distributions." Advances in Applied Probability 28, no. 2 (June 1996): 338–39. http://dx.doi.org/10.2307/1428052.

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It is well-known that Poisson lines in the plane, with orientation distribution Θ on [0, π), generate a (random) tessellation M0 of (random) convex polygons whose characteristics (area, perimeter, etc.) conform, in an ergodic sense, to a certain class {DΘ} of distributions.

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17

Miles, Roger E., and Margaret S. Mackisack. "Further random tessellations with the classic Poisson polygon distributions." Advances in Applied Probability 28, no. 02 (June 1996): 338–39. http://dx.doi.org/10.1017/s0001867800048370.

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It is well-known that Poisson lines in the plane, with orientation distribution Θ on [0, π), generate a (random) tessellation M 0 of (random) convex polygons whose characteristics (area, perimeter, etc.) conform, in an ergodic sense, to a certain class {D Θ} of distributions.

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18

Jati, Yael Narwastu, and Dylmoon Hidayat. "The Effect of Using Origami Paper to Teach the Perimeter of Plane Figures on Cognitive Achievement of Students Grade IX." Polyglot: Jurnal Ilmiah 13, no. 1 (March 13, 2017): 35. http://dx.doi.org/10.19166/pji.v13i1.337.

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<p>This pre-experimental research design aims to know whether there is an effect of using origami paper to teach the perimeter of plane figures on cognitive achievement of students in grade IX and how the use of origami paper can affect students’ cognitive achievement. The subject was taken from 16 students of class IX-A as an experimental class that were going to study using origami paper as the teaching aid. The data obtained was students’ pre-test and post-test. The gained mean of students’ score between pre-test and post-test was significantly greater than the expected score 0.4, in fact the gain reached 0.82 which is categorized high. Thus, it can be concluded that there was an effect of using origami paper to teach the perimeter of plane figures on students’ cognitive achievement.</p><em>BAHASA INDONESIA ABSTRAK: Desain penelitian eksperimen ini dilakukan untuk melihat apakah terdapat pengaruh penggunaan kertas origami untuk mengajar keliling dari suatu bidang datar terhadap hasil belajar kognitif siswa kelas IX dan bagaimana penggunaan tersebut mempengaruhi hsil belajar. Sampel penelitian adalah 16 siswa kelas IX-A sebagai kelompok ekperimen yang akan menggunakan kertas origami. Data diperolehh dari hasil pre-tests dan post-tests. Hasil penelitian menunjukkan ada perbedaan rata-rata skor yang signifikan antara hasil pre-tests dan post-tests yang di duga, yaitu 4.0 (normalized gain), bahkan mencapai 0.8 yang termasuk golongan tinggi. Sehingga, dapat disimpulkan bahwa terdapat pengaruh pada hasil belajar kognitif pada pengajaran keliling suatu bidang datar dengan menggunakan kertas origami</em><p><em> </em></p>

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19

BOSE, PROSENJIT, MERCÈ MORA, CARLOS SEARA, and SAURABH SETHIA. "ON COMPUTING ENCLOSING ISOSCELES TRIANGLES AND RELATED PROBLEMS." International Journal of Computational Geometry & Applications 21, no. 01 (February 2011): 25–45. http://dx.doi.org/10.1142/s0218195911003536.

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Given a set of n points in the plane, we show how to compute various enclosing isosceles triangles where different parameters such as area or perimeter are optimized. We then study a 3-dimensional version of the problem where we enclose a point set with a cone of fixed apex angle α.

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20

Young, Elaine. "Mathematical Exploration: Trapezoids to Triangles." Mathematics Teaching in the Middle School 15, no. 7 (March 2010): 414–19. http://dx.doi.org/10.5951/mtms.15.7.0414.

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Pattern blocks are often used in math class during geometry lessons to support concepts of shape, angle, tessellations, and symmetry. They are also used with fraction study (e.g., Lanius 2007; NCTM 2009a). Using pattern blocks to construct other larger geometric shapes can help students understand the relationships among various shapes. Algebraic relationships and functions emerge when students count blocks and measure figures. The activities in this article support the development of such geometric vocabulary as perimeter, area, sequence, and nonstandard measurement.

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21

Astudillo, Patricio, Peter Mortier, Johan Bosmans, Ole De Backer, Peter de Jaegere, Matthieu De Beule, and Joni Dambre. "Enabling Automated Device Size Selection for Transcatheter Aortic Valve Implantation." Journal of Interventional Cardiology 2019 (November 3, 2019): 1–7. http://dx.doi.org/10.1155/2019/3591314.

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The number of transcatheter aortic valve implantation (TAVI) procedures is expected to increase significantly in the coming years. Improving efficiency will become essential for experienced operators performing large TAVI volumes, while new operators will require training and may benefit from accurate support. In this work, we present a fast deep learning method that can predict aortic annulus perimeter and area automatically from aortic annular plane images. We propose a method combining two deep convolutional neural networks followed by a postprocessing step. The models were trained with 355 patients using modern deep learning techniques, and the method was evaluated on another 118 patients. The method was validated against an interoperator variability study of the same 118 patients. The differences between the manually obtained aortic annulus measurements and the automatic predictions were similar to the differences between two independent observers (paired diff. of 3.3 ± 16.8 mm2 vs. 1.3 ± 21.1 mm2 for the area and a paired diff. of 0.6 ± 1.7 mm vs. 0.2 ± 2.5 mm for the perimeter). The area and perimeter were used to retrieve the suggested prosthesis sizes for the Edwards Sapien 3 and the Medtronic Evolut device retrospectively. The automatically obtained device size selections accorded well with the device sizes selected by operator 1. The total analysis time from aortic annular plane to prosthesis size was below one second. This study showed that automated TAVI device size selection using the proposed method is fast, accurate, and reproducible. Comparison with the interobserver variability has shown the reliability of the strategy, and embedding this tool based on deep learning in the preoperative planning routine has the potential to increase the efficiency while ensuring accuracy.

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22

박혜경, Jun,Pyung-Kuk, and 김영희. "An analysis of the knowledge state related to area measurement of plane figures." SECONDARY EDUCATION RESEARCH 56, no. 2 (August 2008): 169–96. http://dx.doi.org/10.25152/ser.2008.56.2.169.

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23

Westegaard, Susanne K. "Using Quilt Blocks to Construct Understanding." Mathematics Teaching in the Middle School 13, no. 6 (February 2008): 361–65. http://dx.doi.org/10.5951/mtms.13.6.0361.

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A quilting magazine's thirteen-part series titled “Taking the Mathematics Out” prompted the mathematics teacher and the quilter in me to think, “Say it isn't so.” Give up the chance to use a four-patch quilt block to work with writing and plotting coordinate pairs and writing equations for lines? Not use nine-patch quilt blocks to look for examples of reflections, translations, rotations, and changes in scale? Never use quilt blocks to classify triangles and quadrilaterals, find perimeter and area, look for angle relationships, or practice with similar figures?

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24

COX, B. L., and J. S. Y. WANG. "FRACTAL ANALYSES OF ANISOTROPIC FRACTURE SURFACES." Fractals 01, no. 03 (September 1993): 547–59. http://dx.doi.org/10.1142/s0218348x93000575.

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Natural surfaces of rock fractures often have anisotropic asperity distributions, especially for shear fractures or faults. The asperity distributions could be treated as self-affine fractals with directional dependent scaling in the plane of the rock surfaces. Different fractal analyses (divider, slit-island, variogram) are applied to surface distributions of asperity data (topography): (1) a granitic fracture from the Stripa mine in Sweden; (2) a faulted and geothermally altered fracture from Dixie Valley, Nevada, USA. The cutoff patterns (indicator maps) of the granitic fracture show a radial pattern, while those of the faulted fracture show a very anisotropic stretched pattern of shapes. Different cutoff patterns of the same surface generally yield the same fractal dimension with the slit-island technique. The slit-island technique assumes that the cut-off patterns are self-similar in the plane of the surface, with the perimeter versus area analyzed for the entire population of contours, regardless of aspect ratio. We measure the variance in the two coordinate directions as a function of perimeter/area ratio for the anisotropic fracture from Dixie Valley to determine a self-affine scaling ratio for the slit-island analysis. We compare this ratio with anisotropy ratios obtained from simulated flow models based on channeling of flow through the largest openings. The possible applications of fractal analyses to both the geometry and flow are evaluated.

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25

Zimmerman, R. W. "Compressibility of Two-Dimensional Cavities of Various Shapes." Journal of Applied Mechanics 53, no. 3 (September 1, 1986): 500–504. http://dx.doi.org/10.1115/1.3171802.

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Muskhelishvili-Kolosov complex stress functions are used to find the stresses and displacements around two-dimensional cavities under plane strain or plane stress. The boundary conditions considered are either uniform pressure at the cavity surface with vanishing stresses at infinity, or a traction-free cavity surface with uniform biaxial compression at infinity. A closed-form solution is obtained for the case where the mapping function from the interior of the unit circle to the region outside of the cavity has a finite number of terms. The area change of the cavity due to hydrostatic compression at infinity is examined for a variety of shapes, and is found to correlate closely with the square of the perimeter of the hole.

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26

DEVILLERS, OLIVIER, and MATTHEW J. KATZ. "OPTIMAL LINE BIPARTITIONS OF POINT SETS." International Journal of Computational Geometry & Applications 09, no. 01 (February 1999): 39–51. http://dx.doi.org/10.1142/s0218195999000042.

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Let S be a set of n points in the plane. We study the following problem: Partition S by a line into two subsets Sa and Sb such that max{f(Sa), f(Sb)} is minimal, where f is any monotone function defined over 2S. We first present a solution to the case where the points in S are the vertices of a convex polygon and apply it to some common cases — f(S′) is the perimeter, area, or width of the convex hull of S′⊆S — to obtain linear solutions (or O(n log n) solutions if the convex hull of S is not given) to the corresponding problems. This solution is based on an efficient procedure for finding a minimal entry in matrices of some special type, which we believe is of independent interest. For the general case we present a linear space solution which is in some sense output sensitive. It yields solutions to the perimeter and area cases that are never slower and often faster than the best previous solutions.

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27

Groemer, H. "Stability Theorems for Convex Domains of Constant Width." Canadian Mathematical Bulletin 31, no. 3 (September 1, 1988): 328–37. http://dx.doi.org/10.4153/cmb-1988-048-3.

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AbstractIt is known that among all plane convex domains of given constant width Reuleaux triangles have minimal and circular discs have maximal area. Some estimates are given concerning the following associated stability problem: If K is a convex domain of constant width w and if the area of K differs at most ∊ from the area of a Reuleaux triangle or a circular disc of width w, how close (in terms of the Hausdorff distance) is K to a Reuleaux triangle or a circular disc? Another result concerns the deviation of a convex domain M of diameter d from a convex domain of constant width if the perimeter of M is close to πd.

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28

Donkoh, Elvis K., and Alex A. Opoku. "Optimal Geometric Disks Covering using Tessellable Regular Polygons." Journal of Mathematics Research 8, no. 2 (March 10, 2016): 25. http://dx.doi.org/10.5539/jmr.v8n2p25.

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<p>Geometric Disks Covering (GDC) is one of the most typical and well studied problems in computational geometry. Geometric disks are well known 2-D objects which have surface area with circular boundaries but differ from polygons whose surfaces area are bounded by straight line segments. Unlike polygons covering with disks is a rigorous task because of the circular boundaries that do not tessellate. In this paper, we investigate an area approximate polygon to disks that facilitate tiling as a guide to disks covering with least overlap difference. Our study uses geometry of tessellable regular polygons to show that hexagonal tiling is the most efficient way to tessellate the plane in terms of the total perimeter per area coverage.</p>

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29

Miles, R. E. "A heuristic proof of a long-standing conjecture of D. G. Kendall concerning the shapes of certain large random polygons." Advances in Applied Probability 27, no. 02 (June 1995): 397–417. http://dx.doi.org/10.1017/s0001867800026938.

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In the early 1940s David Kendall conjectured that the shapes of the ‘large' (i.e. large area A) convex polygons determined by a standard Poisson line process in the plane tend to circularity (as A increases). Subject only to one heuristic argument, this conjecture and the corresponding two results with A replaced in turn by number of sides N and perimeter S, are proved. Two further similar limiting distributions are considered and, finally, corresponding limiting non-deterministic shape distributions for the small polygons are determined.

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30

Miles, R. E. "A heuristic proof of a long-standing conjecture of D. G. Kendall concerning the shapes of certain large random polygons." Advances in Applied Probability 27, no. 2 (June 1995): 397–417. http://dx.doi.org/10.2307/1427833.

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In the early 1940s David Kendall conjectured that the shapes of the ‘large' (i.e. large area A) convex polygons determined by a standard Poisson line process in the plane tend to circularity (as A increases). Subject only to one heuristic argument, this conjecture and the corresponding two results with A replaced in turn by number of sides N and perimeter S, are proved. Two further similar limiting distributions are considered and, finally, corresponding limiting non-deterministic shape distributions for the small polygons are determined.

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31

Rizcallah, Joseph A. "Isoperimetric Triangles." Mathematics Teacher 111, no. 1 (September 2017): 70–74. http://dx.doi.org/10.5951/mathteacher.111.1.0070.

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The isoperimetric problem is a well-known problem in geometry, and it has a long and rich history (Blasjo 2005). In the plane, the isoperimetric problem consists of finding the simple closed curve of a given perimeter that encloses the greatest area, with the circle being the famous solution. Attempts to solve the isoperimetric problem, as well as other analogous problems in calculus and physics, were undertaken by many great mathematicians in the past whose work ultimately laid the foundation for the elegant branch of analysis known today as the calculus of variations.

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32

SADJADI, H. MOHSENI, and M. ALIMOHAMMADI. "CONFINEMENT AND SCREENING OF THE SCHWINGER MODEL ON THE POINCARÉ HALF PLANE." International Journal of Modern Physics A 16, no. 09 (April 10, 2001): 1631–44. http://dx.doi.org/10.1142/s0217751x01003536.

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We discuss the confining features of the Schwinger model on the Poincaré half plane. We show that despite the fact that the expectation value of the large Wilson loop of massless Schwinger model displays perimeter behavior, the system can be in confining phase due to the singularity of the metric at horizontal axis. It is also shown that in the quenched Schwinger model, the area dependence of the Wilson loop, in contrast to the flat case, is a not a sign of confinement and the model has a finite energy even for large external charges separation. The presence of dynamical fermions cannot modify the screening or the confining behavior of the system. Finally we show that in the massive Schwinger model, the system is again in screening phase. The zero curvature limit of the solutions is also discussed.

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33

Kjelgren, Roger, and Larry Rupp. "RESPONSE OF LONDON PLANE AND CORKSCREW WILLOW TO VARIABLE IRRIGATION." HortScience 29, no. 7 (July 1994): 741a—741. http://dx.doi.org/10.21273/hortsci.29.7.741a.

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We investigated growth and water relations of London plane and corkscrew willow irrigated at 80% and 0% replacement of potential evapotranspiration (ETo). In Spring 1991, whips were planted in a randomized complete-block design in a silt loam soil that was clean-cultivated through two seasons. In 1992, tree response was measured in water relations [water potential (ψ)] at predawn and midday and dawn-to-dusk stomatal conductance (gs), trunk growth, and total leaf area. Soil-water depletion was monitored with a neutron probe. Measured ETo was 98.6 mm, and actual water applied based on final leaf area was 92% and 38% of ETo for plane trees and willows, respectively. Nonirrigated trees received 4% of ETo from rain. Soil water content at the 0.90-m depth was lower in the 0% ETo treatment. There were, however, no differences in predawn ψ through the season. Plane trees had consistently higher dawn-to-dusk gs than the willows, but there were no differences in gs or midday ψ between irrigation treatments for either species. Despite lower gs, willows had greater total leaf area and trunk growth than the plane trees, but again, there were no differences among irrigation treatments. Lack of detectable water-stress effects suggests that, in the absence of competition from other species, an expanding perimeter of root growth explored new soil and allowed nonirrigated trees to exploit soil water ahead of moisture depletion within the root zone.

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Petrovic, Maja, Bojan Banjac, and Branko Malesevic. "The geometry of trifocal curves with applications in architecture, urban and spatial planning." Spatium, no. 32 (2014): 28–33. http://dx.doi.org/10.2298/spat1432028p.

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In this paper we consider historical genesis of trifocal curve as an optimal curve for solving the Fermat?s problem (minimizing the sum of distance of one point to three given points in the plane). Trifocal curves are basic plane geometric forms which appear in location problems. We also analyze algebraic equation of these curves and some of their applications in architecture, urbanism and spatial planning. The area and perimeter of trifocal curves are calculated using a Java application. The Java applet is developed for determining numerical value for the Fermat-Torricelli-Weber point and optimal curve with three foci, when starting points are given on an urban map. We also present an application of trifocal curves through the analysis of one specific solution in South Stream gas pipeline project.

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Easterday, Kenneth, and Tommy Smith. "A Monte Carlo Application to Approximate Pi." Mathematics Teacher 84, no. 5 (May 1991): 387–90. http://dx.doi.org/10.5951/mt.84.5.0387.

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Instruction in the concept of area begins in elementary school and continues through college calculus. Students learn to apply formulas for certain plane figures and later, integration techniques. For figures that are not “nice” shapes or for “irregular” functions, students are taught approximation techniques, which range from counting squares in middle school to trigonometric approximations in high school and various methods of calculus. The purpose of this article is to offer the use of the Monte Carlo procedure on the microcomputer as an alternative tool for approximating areas. One of the introductory applications of the technique is to find the area of a circle, which permits us to approximate pi.

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Do, Jong Hoon, and Yun Beom Park. "An axiomatic analysis on contents about the area of plane figures in the elementary school mathematics." Education of Primary School Mathematics 17, no. 3 (December 31, 2014): 253–63. http://dx.doi.org/10.7468/jksmec.2014.17.3.253.

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Ying, Xiaoyu, and Wenzhe Li. "Effect of Floor Shape Optimization on Energy Consumption for U-Shaped Office Buildings in the Hot-Summer and Cold-Winter Area of China." Sustainability 12, no. 5 (March 8, 2020): 2079. http://dx.doi.org/10.3390/su12052079.

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This paper explored the effects of the side proportion of building floor shape on building energy consumption. It is based on the analysis of regression models that were developed in the present study. The simplified building models can be used to conduct a parametric study to investigate the effect of building plane shape parameters on total heating and cooling load. DesignBuilder was used to build and simulate individual building configuration. Energy consumption simulations for forty-eight U-shaped buildings with different plane layouts were performed to create a comprehensive dataset covering general ranges of side proportions of U-shaped buildings and building orientations. Statistical analysis was performed using MATLAB to develop a set of regression equations predicting energy consumption and optimizing floor shapes. Furthermore, perimeter-area ratio (PAR), width ratio, and depth ratio were considered as three factors to characterize the quantitative relationship between floor shape and energy consumption. It is envisioned that the binary quadratic polynomial regression models, visualized as a smooth surface in space and mapped to a vortex image on the plane, can be used to estimate the energy consumption in the early stages of the design when different building schemes and design concepts are being considered.

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Bell, Carol J. "Measuring Tangrams on a Geoboard." Mathematics Teaching in the Middle School 22, no. 6 (February 2017): 374–78. http://dx.doi.org/10.5951/mathteacmiddscho.22.6.0374.

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many of us are familiar with the tangram and the geoboard. A tangram is a dissection puzzle consisting of seven flat shapes. One objective of the puzzle is to form a specified shape using all seven pieces, with no pieces overlapping. Geoboards are generally used to explore basic concepts found in plane geometry, such as perimeter and area, or basic shapes, such as polygons. Using tangrams in combination with a geoboard provides a rich activity for students that will help them further develop their skills and understanding of operations and algebraic thinking, number and operations, measurement, and geometry, also mentioned in the Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010).

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Zhang, Qian, Ning Shan, Feng Chen Qian, and Ya Lin Ye. "A New Method of Expressing Point Model Based on Kd-Tree for Plane Area." Advanced Materials Research 842 (November 2013): 658–61. http://dx.doi.org/10.4028/www.scientific.net/amr.842.658.

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According to the space equal features of midpoint segmentation KD-tree, anisotropic quantitative method is proposed. After having used this method, the coordinate figures of sampling points in point model were quantified again; and the quantitative results have been represented as encoding of space partition methods and results in the process of constructing KD-tree. Meantime, the effectiveness that the method was used to express point model have been simulated and studied. It turned out that the quantitative method is very suitable for large flat area, because it is not only directly reduce the amount of point model geometry data, but also greatly reduce sampling points in case of keeping the numerical precision of the model geometry data unchanged, so as to further reduce the amount of point model data.

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Hueter, Irene. "The convex hull of a normal sample." Advances in Applied Probability 26, no. 04 (December 1994): 855–75. http://dx.doi.org/10.1017/s0001867800026653.

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Consider the convex hull of n independent, identically distributed points in the plane. Functionals of interest are the number of vertices Nn , the perimeter Ln and the area An of the convex hull. We study the asymptotic behaviour of these three quantities when the points are standard normally distributed. In particular, we derive the variances of Nn, Ln and An for large n and prove a central limit theorem for each of these random variables. We enlarge on a method developed by Groeneboom (1988) for uniformly distributed points supported on a bounded planar region. The process of vertices of the convex hull is of central importance. Poisson approximation and martingale techniques are used.

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Hueter, Irene. "The convex hull of a normal sample." Advances in Applied Probability 26, no. 4 (December 1994): 855–75. http://dx.doi.org/10.2307/1427894.

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Consider the convex hull of n independent, identically distributed points in the plane. Functionals of interest are the number of vertices Nn, the perimeter Ln and the area An of the convex hull. We study the asymptotic behaviour of these three quantities when the points are standard normally distributed. In particular, we derive the variances of Nn, Ln and An for large n and prove a central limit theorem for each of these random variables. We enlarge on a method developed by Groeneboom (1988) for uniformly distributed points supported on a bounded planar region. The process of vertices of the convex hull is of central importance. Poisson approximation and martingale techniques are used.

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Silva, Angelica da Fontoura Garcia, Susana Maris França da Silva, and Maria Elisa Esteves Lopes Galvão. "Reflexões e conhecimentos evidenciados por professores que estudam área de figuras planas." Zetetike 28 (December 18, 2020): e020029. http://dx.doi.org/10.20396/zet.v28i0.8652733.

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This study aims to discuss and understand the process of (re)significance of teachers’ professional knowledge based on discussions and reflections carried out by a group of teachers who teach mathematics for the initial years of the elementary education. The teachers studied how to calculate the area of plane figures using squared arrays. This study is based on Zeichner’s and Serrazina’s research to discuss the reflection on the practice and, in Ball, Thames, and Phelps, to analyse the (re) significance of the knowledge of the professionals involved. The data analysed were collected during the group meetings at the school where the participants taught. Through the analyses of the discussions held, we could observe that the teachers expanded their common and specialised content knowledge, mainly in relation to the strategies to calculate area, and started using reconfiguration of the figures and formulas of area more assertively. Mutual support allowed them to identify their own needs and then modify their analyses of the strategies for area calculation to be adopted.

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OPPENHEIM, A. C., R. BRAK, and A. L. OWCZAREK. "ANISOTROPIC STEP, SURFACE CONTACT, AND AREA WEIGHTED DIRECTED WALKS ON THE TRIANGULAR LATTICE." International Journal of Modern Physics B 16, no. 09 (April 10, 2002): 1269–99. http://dx.doi.org/10.1142/s0217979202010087.

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We present results for the generating functions of single fully-directed walks on the triangular lattice, enumerated according to each type of step and weighted proportional to the area between the walk and the surface of a half-plane (wall), and the number of contacts made with the wall. We also give explicit formulae for total area generating functions, that is when the area is summed over all configurations with a given perimeter, and the generating function of the moments of heights above the wall (the first of which is the total area). These results generalise and summarise nearly all known results on the square lattice: all the square lattice results can be obtaining by setting one of the step weights to zero. Our results also contain as special cases those that already exist for the triangular lattice. In deriving some of the new results we utilise the Enumerating Combinatorial Objects (ECO) and marked area methods of combinatorics for obtaining functional equations in the most general cases. In several cases we give our results both in terms of ratios of infinite q-series and as continued fractions.

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Korenovskyi, A. A. "Calculations for one illustration of the Pythagorean Theorem." BULLETIN of L.N. Gumilyov Eurasian National University. MATHEMATICS. COMPUTER SCIENCE. MECHANICS Series 133, no. 4 (2020): 40–53. http://dx.doi.org/10.32523/2616-7182/2020-133-4-40-53.

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On the sides of a right-angled isosceles triangle, squares external to this triangle are built. The task is to construct a straight line on a plane that divides a figure consisting of these three squares into two equal-sized (that is, equal area) figures. Some parameters determined by this line are calculated, in particular, the areas of parts of the original squares in half-planes. The corresponding results are illustrated by various drawings, tables and graphs.

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Girsh, A. "Circles on the Complex Plane." Geometry & Graphics 8, no. 4 (March 4, 2021): 3–12. http://dx.doi.org/10.12737/2308-4898-2021-8-4-3-12.

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The Euclidean plane and Euclidean space themselves do not contain imaginary elements by definition, but are inextricably linked with them through special cases, and this leads to the need to propagate geometry into the area of imaginary values. Such propagation, that is adding a plane or space, a field of imaginary coordinates to the field of real coordinates leads to various variants of spaces of different dimensions, depending on the given axiomatics. Earlier, in a number of papers, were shown examples for solving some urgent problems of geometry using imaginary geometric images [2, 9, 11, 13, 15]. In this paper are considered constructions of orthogonal and diametrical positions of circles on a complex plane. A generalization has been made of the proposition about a circle on the complex plane orthogonally intersecting three given spheres on the proposition about a sphere in the complex space orthogonally intersecting four given spheres. Studies have shown that the diametrical position of circles on the Euclidean E-plane is an attribute of the orthogonal position of the circles’ imaginary components on the pseudo-Euclidean M-plane. Real, imaginary and degenerated to a point circles have been involved in structures and considered, have been demonstrated these circles’ forms, properties and attributes of their orthogonal position. Has been presented the construction of radical axes and a radical center for circles of the same and different types. A propagation of 2D mutual orthogonal position of circles on 3D spheres has been made. In figures, dashed lines indicate imaginary elements.

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Chen, Samuel L., Mayil S. Krishnam, Thangavijayan Bosemani, Sumudu Dissayanake, Michael D. Sgroi, John S. Lane, and Roy M. Fujitani. "Geometric changes of the inferior vena cava in trauma patients subjected to volume resuscitation." Vascular 23, no. 5 (October 8, 2014): 459–67. http://dx.doi.org/10.1177/1708538114552665.

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ObjectiveDynamic changes in anatomic geometry of the inferior vena cava from changes in intravascular volume may cause passive stresses on inferior vena cava filters. In this study, we aim to quantify variability in inferior vena cava dimensions and anatomic orientation to determine how intravascular volume changes may impact complications of inferior vena cava filter placement, such as migration, tilting, perforation, and thrombosis.MethodsRetrospective computed tomography measurements of major axis, minor axis, and horizontal diameters of the inferior vena cava at 1 and 5 cm below the lowest renal vein in 58 adult trauma patients in pre-resuscitative (hypovolemic) and post-resuscitative (euvolemic) states were assessed in a blinded fashion by two independent readers. Inferior vena cava perimeter, area, and volume were calculated and correlated with caval orientation.ResultsMean volumes of the inferior vena cava segment on pre- and post-resuscitation scans were 9.0 cm3and 11.0 cm3, respectively, with mean percentage increase of 48.6% ( P < 0.001). At 1 cm and 5 cm below the lowest renal vein, the inferior vena cava expanded anisotropically, with the minor axis expanding by an average of 48.7% ( P < 0.001) and 30.0% ( P = 0.01), respectively, while the major axis changed by only 4.2% ( P = 0.11) and 6.6% ( P = 0.017), respectively. Cross-sectional area and perimeter at 1 cm below the lowest renal vein expanded by 61.6% ( P < 0.001) and 10.7% ( P < 0.01), respectively. At 5 cm below the lowest renal vein, the expansion of cross-sectional area and perimeter were 43.9% ( P < 0.01) and 10.7% ( P = 0.002), respectively. The major axis of the inferior vena cava was oriented in a left-anterior oblique position in all patients, averaging 20° from the horizontal plane. There was significant underestimation of inferior vena cava maximal diameter by horizontal measurement. In pre-resuscitation scans, at 1 cm and 5 cm below the lowest renal vein, the discrepancy between the horizontal and major axis diameter was 2.1 ± 1.2 mm ( P < 0.001) and 1.7 ± 1.0 mm ( P < 0.001), respectively, while post-resuscitation studies showed the same underestimation at 1 cm and 5 cm below the lowest renal vein to be 2.2 ± 1.2 mm ( P < 0.01) and 1.9 ± 1.0 mm ( P < 0.01), respectively.ConclusionsThere is significant anisotropic variability of infrarenal inferior vena cava geometry with significantly greater expansive and compressive forces in the minor axis. There can be significant volumetric changes in the inferior vena cava with associated perimeter changes but the major axis left-anterior oblique caval configuration is always maintained. These significant dynamic forces may impact inferior vena cava filter stability after implantation. The consistent major axis left-anterior oblique obliquity may lead to underestimation of the inferior vena cava diameter used in standard anteroposterior venography, which may influence initial filter selection.

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Vyzhva, Z., V. Demidov, and A. Vyzhva. "STATISTICAL SIMULATION OF RANDOM FIELD ON 2D AREA WITH WHITTLE-MATERN TYPECORRELATION FUNCTION IN THE GEOPHYSICAL PROBLEM OF ENVIRONMENT MONITORING." Visnyk of Taras Shevchenko National University of Kyiv. Geology, no. 3 (86) (2019): 55–61. http://dx.doi.org/10.17721/1728-2713.86.08.

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Due to the increasing number of natural and technogenic disasters the development of geological environment monitoring system is actual one using modern mathematical tools and information technology. The local monitoring of potentially dangerous objects is an important part of the overall environment monitoring system. The complex geophysical research was conducted on Rivne NPP area. The monitoring observations radioisotope study of soil density and humidity near the perimeter of buildings is of the greatest interest among these. In this case a problem occurred to supplement simulated data that were received at the control of chalky strata density changes at the research industrial area with use of radioisotope methods on a grid that included 29 wells. This problem was solved in this work by statistical simulation method that provides the ability to display values (the random field of a research object on a plane) in any point of the monitoring area. The chalk strata averaged density at the industrial area was simulated using the built model and the involvement optimal in the mean square sense Whittle-Matern type correlation function. In this paper the method is used and the model and procedure were developed with enough adequate data for Whittle-Matern type correlation function. The model and algorithm were developed and examples of karst-suffusion phenomena statistical simulation were given in the problem of density chalk strata monitoring at the Rivne NPP area. The statistical model of averaged density chalk strata distribution was built in the plane and statistical simulation algorithm was developed using Whittle-Matern type correlation function on the basis of spectral decomposition. The research subject realizations were obtained with required detail and regularity at the observation grid based on the developed software. Statistical analysis of the numerical simulation results was done and tested for its adequacy.

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Papadopoulos, Ioannis. "“REINVENTING” TECHNIQUES FOR THE ESTIMATION OF THE AREA OF IRREGULAR PLANE FIGURES: FROM THE EIGHTEENTH CENTURY TO THE MODERN CLASSROOM." International Journal of Science and Mathematics Education 8, no. 5 (December 15, 2009): 869–90. http://dx.doi.org/10.1007/s10763-009-9190-y.

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49

Porto, Anderson Luiz Pedrosa, Douglas Frederico Guimarães Santiago, Leonardo Gomes, and Márcio Henrique Marques Macedo. "Do círculo ao quadrado, um estudo sobre a convergência de uma sequência de curvas." Ciência e Natura 41 (July 16, 2019): 23. http://dx.doi.org/10.5902/2179460x32231.

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In this paper we present convergence concepts applications of numerical and functions sequences to a geometric problem which raised from an analysis using GeoGebra. This question involves the traces’s family of plane curves behavior, whose initial trace of the curve is given by a circle of radius k and the whose others, intuitively, approach to a square with side measuring 2k. A proof of this convergence is done, as well a demonstration that the areas bounded by the traces of the curves and their lengths converge, respectively, to the area and length of the boundary square. We also present a brief historical setting of convergence concepts and the ideas of infinity behind these concepts. In order to be developed to graduation level, this paper goes some way meet what historically was done to determine properties of curved figures through approximations by rectilinear figures.

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Σφέικος, A. Ν., and Π. Γ. Μαρίνος. "BEHAVIOR OF PINDOS FLYSCH DURING TUNNELLING THROUGH A THRUST ZONE. DEFORMATION AND ROCK BEHAVIOR. EXPIERENCE FROM THE ACHELOOS RIVER DIVERSION TUNNEL TO THESSALY." Bulletin of the Geological Society of Greece 36, no. 4 (January 1, 2004): 1843. http://dx.doi.org/10.12681/bgsg.16658.

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The Acheloos diversion project consists of a series of reservoirs and a diversion tunnel. The tunnel has a designed internal diameter of 6m and a total length of ca. 17 400 m. Through the tunnel waters from the upper Achellos will be transferred to Thessaly. The project area belongs entirely to the Pindos zone. To the west (Mouzaki - Drakotrypa area) limestone, Jurassic chert and transitional strata (limestone and siltstone interchanges) overthrust sandstone and siltstone of the Pindos Flysch association (Paleogene). The thrust plane is well exposed and its geometrical features are clearly defined on the surface. Within silt- and sand- stones of the flysch, the developement of shear zones borders the thrust plane. Tunneling through limestone and chert advanced without specific problems. Tunneling through the flysch sequence slowed down the advancing rate. This was partially due to the composition and structure of the formation. The stand-up time was reduced due to compositional changes and the throughout development of shear zones. Heavy support measures were insalled and immediate monitoring began in the area where the thrust zone was developed. Data analysis and its results show that the rock formation remained into a dynamically active status. Four months after the excavation forces acting at the tunnel perimeter, exceeded the support measures bearing abilty causing tunnel radial convergence and the development of damages became visible. In this paper we describe tunneling conditions, the geology and the response of formations during excavation, as this was interpreted by monitoring data. We describe the damages caused as well as the counter measures applied in order to control and terminate the tunnel convergence.

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