Relevant bibliographies by topics / Similar Triangles

Contents

  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers
  6. Reports

Academic literature on the topic 'Similar Triangles'

Author: Grafiati
Published: 4 June 2021
Last updated: 9 February 2022

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Journal articles on the topic "Similar Triangles"

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Zak, Andrzej. "Dissection of a Triangle into Similar Triangles." Discrete & Computational Geometry 34, no. 2 (March 18, 2005): 295–312. http://dx.doi.org/10.1007/s00454-005-1167-1.

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Biber, Abdullah Çağrı. "Students' difficulties in similar triangle questions." Cypriot Journal of Educational Sciences 15, no. 5 (October 29, 2020): 1146–59. http://dx.doi.org/10.18844/cjes.v15i5.5161.

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Similar triangles in questions are usually given as separate, adjacent or overlapped. Furthermore, similarity types such as Side-Angle-Side (S.A.S.), Side-Side-Side (S.S.S.) and Angle-Angle (A.A.) are requested in the questions. Students have more trouble in other types of questions. The purpose of this study is to investigate the difficulties of students about similar triangles and the reasons for these difficulties. This research was carried out with the case study method, which is one of the qualitative research approaches. The study was conducted with 55 Science High School 9th grade students and 9 open-ended questions were used to examine students' knowledge about “similarity in triangles”. Furthermore, 5 students were interviewed to find out the reasons for their solutions. Descriptive analysis method was used to analyze the data. As a result, it can be concluded that students have difficulties mostly in overlapped triangles and Angle-Angle type questions. On the other hand, it can be concluded that students are quite successful where similar triangles are given separately. In the light of the findings obtained in this study, it can be advised for lecturers to focus on the questions where similar triangles are overlapped while explaining the similarity in the triangle. Keywords: Similarity, Triangles, Difficulties, High School Students.
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Odehnal, Boris. "Two Convergent Triangle Tunnels." KoG, no. 22 (2018): 3–11. http://dx.doi.org/10.31896/k.22.1.

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A semi-orthogonal path is a polygon inscribed into a given polygon such that the $i$-th side of the path is orthogonal to the $i$-th side of the given polygon. Especially in the case of triangles, the closed semi-orthogonal paths are triangles which turn out to be similar to the given triangle. The iteration of the construction of semi-orthogonal paths in triangles yields infinite sequences of nested and similar triangles. We show that these two different sequences converge towards the bicentric pair of the triangle's Brocard points. Furthermore, the relation to discrete logarithmic spirals allows us to give a very simple, elementary, and new constructions of the sequences' limits, the Brocard points. We also add some remarks on semi-orthogonal paths in non-Euclidean geometries and in $n$-gons.
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Jones, C. A., P. Jones, and A. B. Bolt. "Dissections of Triangles into Five Similar Triangles." Mathematical Gazette 82, no. 494 (July 1998): 225. http://dx.doi.org/10.2307/3620405.

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Okumura, Hiroshi. "79.56 Two Similar Triangles." Mathematical Gazette 79, no. 486 (November 1995): 569. http://dx.doi.org/10.2307/3618096.

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Johnson, Gwen. "Mathematical Exploration: Similar Triangles." Mathematics Teaching in the Middle School 16, no. 4 (November 2010): 248–53. http://dx.doi.org/10.5951/mtms.16.4.0248.

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Within school mathematics, similarity is a topic that has many connections to real life. Similarity is related to scale drawings, such as those used in architecture, and models, including models of trains, cars, and boats. It is also used to solve problems that involve similar figures, which are common in middle-grades textbooks.
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Richardson, Thomas J. "Optimal packing of similar triangles." Journal of Combinatorial Theory, Series A 69, no. 2 (February 1995): 288–300. http://dx.doi.org/10.1016/0097-3165(95)90054-3.

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Crilly, Tony, and Colin R. Fletcher. "Triangles meeting triangles." Mathematical Gazette 98, no. 543 (November 2014): 432–51. http://dx.doi.org/10.1017/s0025557200008135.

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We consider two connected problems: •For a given but otherwise arbitrary triangle in the plane, to construct similar triangles which ‘meet’ this triangle.•To find the triangle so formed which has least area.1. Constructing a triangle which meets anotherThese problems beg the question of what is meant by ‘meet’ and we now aim to make this precise: Definition: A triangle XYZ will meet a given triangle ABC if on the triangle ABC, the vertex X lies on a line through AB, the vertex Y lies on a line through BC, and the vertex Z lies on a line through CA.When triangle XYZ is actually ‘in’ the triangle ABC, ‘meet’ is synonymous with the traditional ‘inscribe’ (such as in case (1) below). For ‘inscribe’ we understand that some of X, Y, Z may coincide with the vertices of ABC (such as case (2) below).More generally we use ‘meet’ to extend these possibilities by allowing XYZ to meet triangle ABC with its sides produced externally (such as case (3) below).
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Rešić, Sead, Alma Šehanović, and Amila Osmić. "ISOSCELES TRIANGLES ON THE SIDES OF A TRIANGLE." Journal Human Research in Rehabilitation 9, no. 1 (April 2019): 123–30. http://dx.doi.org/10.21554/hrr.041915.

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Famous construction of Fermat-Toricelly point of a triangle leads to the question is there a similar way to construct other isogonic centers of a triangle in a similar way. For a purpose we remember that Fermat-Torricelli point of a triangle ΔABC is obtained by constructing equilateral triangles outwardly on the sides AB,BC and CA. If we denote thirth vertices of those triangles by C1 ,A1 and B1 respectively, then the lines AA1 ,BB1 and CC1 concurr at the Fermat-Torricelli point of a triangle ΔABC (Van Lamoen, 2003). In this work we present the condition for the concurrence, of the lines AA1 ,BB1 and C1 , where C1 ,A1 and B1 are the vertices of an isosceles triangles constructed on the sides AB,BC and CA (not necessarily outwordly) of a triangle ΔABC. The angles at this work are strictly positive directed so we recommend the reader to pay attention to this fact.
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Ramachandra, K. "86.54 Pythagoras' Theorem and Similar Triangles." Mathematical Gazette 86, no. 506 (July 2002): 324. http://dx.doi.org/10.2307/3621877.

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Dissertations / Theses on the topic "Similar Triangles"

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Cruz, Josinaldo dos Santos. "O uso de investigações matemáticas na abordagem da semelhança de triângulos e aplicações." Universidade Federal de Sergipe, 2015. https://ri.ufs.br/handle/riufs/6509.

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES
This study aims to understand the importance of mathematical investigations in the study of similar triangles and their applications. Considering the importance of research activities for teaching and learning, this paper presents some de nitions , aspects and considerations of this teaching methodology . We speak also of the teacher's role and possible obstacles in carrying out investigative tasks. To this end, data was collected through the implementation of three investigative activities in a class of 9th grade of elementary school of a college of the state of Sergipe. The results show that the inclusion of research activities in the classroom everyday , at any level of education , indicates a better learning of the content studied.
Este trabalho tem como objetivo compreender a import^ancia das investiga c~oes matem aticas no estudo da semelhan ca de tri^angulos e suas aplica c~oes. Levando em considera c~ao a import^ ancia das atividades investigativas para o ensino e a aprendizagem, este trabalho apresenta algumas de ni c~oes, aspectos e considera c~oes acerca desta metodologia de ensino. Falamos, tamb em, do papel do professor e dos poss veis obst aculos na realiza c~ao das tarefas investigativas. Para tanto, os dados foram coletados por meio da realiza c~ao de tr^es atividades investigativas em uma turma do 9 ano do Ensino Fundamental de um col egio da rede estadual de Sergipe. Os resultados apontam que a inser c~ao de atividades investigativas no cotidiano de sala de aula, em qualquer n vel de ensino, indica um melhor aprendizado do conte udo estudado.
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Tseng, Po-Yu, and 曾伯宇. "Exploring the Ninth-Grade Students to Solve Similar Triangle Problems: Case Study." Thesis, 2017. http://ndltd.ncl.edu.tw/handle/eg5hhn.

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碩士
臺北市立大學
數學系數學資訊教育教學碩士學位班
105
This research studies the case studies of ninth grade students in solving similar triangle problems. The research objects are two students in the same class study at a public secondary school in the new North City, Taiwan. They have different degrees of achiever, say, in high and middle. The research was first conducted by a pre-test question, which assesses how these two participants will approach a similar triangle question. And then through the main questions to observe the high achiever and the middle achiever in the use of similar nature to solve similar triangle problem performance and differences. During the test, we have recorded our participant’s written notes, process of tackling the problem and finally given them an individual interview in order to study the reason that causes performance difference. The result shows the high achiever are biased towards procedural understanding rather than conceptual understanding when approaching similar triangle problems, whereas the middle achiever cannot remember the nature of meaning correctly. Not only the latter lacks of the conceptual understanding but also procedural understanding. In terms of different approaches use by two participants and performance differences in solving triangle similar problems, the high achiever can use her own concepts for the similarity of triangle and understand the integration of key information with the subject, in a structured and contextual way to solve problems. But sometimes she tries to solve the problem in a wrong way; The middle achiever, however, tends to rely on the subject information and past memories. He struggles in explaining the process clearly. If the question has high complexity, it is relatively difficult for the middle achiever, and sometimes even fails to be solved. This paper suggests lecture can be delivered by explaining the principle details, reasoning concepts and give more examples, which help students develop their problem-solving strategies and skills. Due to the reason that this study does not adopt ZPD to help students solve the problem, future works can be done in this area or studying students in different grades and themes.
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Chang, Chih-Cheng, and 張志成. "The Study of Gifted Mathematics in Some Courses of Similar Triangels in Junior High School." Thesis, 2014. http://ndltd.ncl.edu.tw/handle/42ch7y.

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碩士
中原大學
應用數學研究所
102
The purpose of this research is to understand the myth of learning similar triangles and explore what is the appropriate courses for the mathematical gifted students. This research was based on the analysis of teaching materials, which was begun to sort out after collecting the related questions of similar triangles. After demonstrated literature review and item analysis has come to the conclusion as the followings: Ⅰ)Suitable auxiliary lines helps to logical thinking and analytical graphic capabilities in learning similar triangles and even geometry problem in junior high school . Ⅱ)In this study, the geometry problem-solving techniques are induced to solve difficult problems. Based on those conclusions stated above, has raised the following recommendations as the reference for educational administrative authorities: 1) It is important to teachers to have professional competence about image thinking, problem-solving diversity and implementation.2) Dynamic geometry software is used to simulated and observed the graph transformation to verify the geometric conceptions. 3) There is a need of enriching the teaching materials and increasing the remaining geometry textbooks.
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Books on the topic "Similar Triangles"

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J, Shaw Dennis, and United States. Forest Service. Southern Research Station., eds. A tree taper model based on similar triangles and use of crown ratio as a measure of form in taper equations for longleaf pine. Asheville, NC: U.S. Dept. of Agriculture, Forest Service, Southern Research Station, 2003.

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Erdem, Uğur Murat, Nicholas Roy, John J. Leonard, and Michael E. Hasselmo. Spatial and episodic memory. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780199674923.003.0029.

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The neuroscience of spatial memory is one of the most promising areas for developing biomimetic solutions to complex engineering challenges. Grid cells are neurons recorded in the medial entorhinal cortex that fire when rats are in an array of locations in the environment falling on the vertices of tightly packed equilateral triangles. Grid cells suggest an exciting new approach for enhancing robot simultaneous localization and mapping (SLAM) in changing environments and could provide a common map for situational awareness between human and robotic teammates. Current models of grid cells are well suited to robotics, as they utilize input from self-motion and sensory flow similar to inertial sensors and visual odometry in robots. Computational models, supported by in vivo neural activity data, demonstrate how grid cell representations could provide a substrate for goal-directed behavior using hierarchical forward planning that finds novel shortcut trajectories in changing environments.
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Guitton, Clement. Conclusion. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780190699994.003.0008.

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The conclusion starts by summarizing the contributions of this book, before considering what the future of attribution may look like. It briefly analyzes the consequences for privacy of large metadata collection, and the effects on Internet governance of attribution mechanisms being revealed. More specifically, it contends that the potential breaking up of the Internet into multiple ‘Internets’ will reshape the practice of cyber security to some extent, but that the political limitations of attribution will remain very similar. This last chapter also briefly outlines the basis for future work by showing that attribution is not only limited to cyber attacks but also emerges in many other situations, involving the need to strike a "right" balance between anonymity, privacy and security, which is one of the key challenges of the twenty-first century. Hence, the approach to attribution will reflect how society wishes to address the anonymity-privacy-security triangle, and how it has the potential to shape the balance between these three questions for years to come.
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Wittman, David M. Spacetime Geometry. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780199658633.003.0011.

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This chapter shows that the counterintuitive aspects of special relativity are due to the geometry of spacetime. We begin by showing, in the familiar context of plane geometry, how a metric equation separates frame‐dependent quantities from invariant ones. The components of a displacement vector depend on the coordinate system you choose, but its magnitude (the distance between two points, which is more physically meaningful) is invariant. Similarly, space and time components of a spacetime displacement are frame‐dependent, but the magnitude (proper time) is invariant and more physically meaningful. In plane geometry displacements in both x and y contribute positively to the distance, but in spacetime geometry the spatial displacement contributes negatively to the proper time. This is the source of counterintuitive aspects of special relativity. We develop spacetime intuition by practicing with a graphic stretching‐triangle representation of spacetime displacement vectors.
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Burdmann, Emmanuel A., and Vivekanad Jha. Rickettsiosis. Edited by Vivekanand Jha. Oxford University Press, 2015. http://dx.doi.org/10.1093/med/9780199592548.003.0193.

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Rickettsiae are obligate intracellular bacteria transmitted by arthropods to a vertebrate host. Clinically relevant rickettsioses have a similar clinical pattern, manifesting as an acute febrile disease accompanied by headache, articular and muscle pain, and malaise.Epidemic typhus is a worldwide distributed disease caused by the Rickettsia prowazekii, with a human louse as a vector. Data on epidemic typhus-related renal injury is extremely scarce.Murine typhus is caused by the Rickettsia typhi and has a rodent flea as the vector. It is one of the most frequent rickettsioses, and is usually a self-limited febrile illness. Proteinuria, haematuria, elevations in serum creatinine (SCr) and/or blood urea nitrogen (BUN) and AKI have been reported. The real frequency of renal involvement in murine typhus is unknown. Renal abnormalities recover after the infectious disease resolution.Scrub typhus, caused by the Orientia tsutsugamushi, has the Leptotrombidium mite larva as vector. It is endemic in the Tsutsugamushi triangle delimited by Japan, Australia, India, and Siberia. It can manifest either as a self-limiting disease or as a severe, life-threatening multiorgan illness. Early administration of adequate antibiotics is essential to prevent adverse outcomes. Proteinuria, haematuria, and acute kidney injury (AKI) are frequent.Tick-borne rickettsioses are caused by bacteria from the spotted fever group and have ticks as vectors. Rocky Mountain spotted fever (RMSF) is caused by Rickettsia rickettsii. It is the most severe of the spotted fever rickettsial diseases, causing significant morbidity and lethality. RMSF occurs in North, Central, and South America. Renal impairment is frequent in severe forms of RMSF. Mediterranean spotted fever is caused by Rickettsia conorii, and is endemic in the Mediterranean area. It is usually a benign disease, but may have a severe course, clinically similar to RMSF. Haematuria, proteinuria, increased serum creatinine, and AKI may occur. Japanese spotted fever is caused by Rickettsia japonica. Lethal cases are reported yearly and AKI has occurred in the context of multiple organ failure.
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Book chapters on the topic "Similar Triangles"

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van Gulik-Gulikers, Iris, Jenneke Krüger, and Jan van Maanen. "Eighteenth Century Land Surveying as a Context for Learning Similar Triangles and Measurement." In ICME-13 Monographs, 235–61. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-33824-4_13.

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Joseph, Anthony. "Proper Self-Similar Triangle Tiling and Representing Weight Diagrams in the Plane." In Representations and Nilpotent Orbits of Lie Algebraic Systems, 357–409. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-23531-4_9.

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Drilling, Matthias, Hannah Grove, Byron Ioannou, and Thibauld Moulaert. "Towards a Structural Embeddedness of Space in the Framework of the Social Exclusion of Older People." In International Perspectives on Aging, 193–207. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-51406-8_15.

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AbstractCombating social exclusion of older adults is a key objective of European social policy. But from the beginning of the debate, social exclusion has mainly been interpreted as arising from economic circumstances. Doubts have been raised as to whether this narrow focus contributes to solving the problems identified. In particular, spatial aspects come to the fore, highlighting the fact that exclusion always happens in a specific place. However, spatial exclusion is often reduced to a territorial concept of ‘where’ the exclusion takes place. – but it is simplistic to state that everything happens in a space. The aim of this chapter is twofold. First, it presents a theoretical model, developed during and through the ROSEnet COST Action. The model integrates dimensions of age, space and exclusion in one perspective – the ASE Triangle. Second, this chapter explores the potential of the ASE Triangle to enhance our understanding of two specific European case studies of older people’s exclusion, in Greater Dublin – Ireland, and Nicosia – Cyprus. By ‘upgrading space’ as a theory-led idea we contribute to and challenge existing (human) gerontological theory of space. Similar to the exploration of a relational vision of space, our model does not only consider “experience” of space exclusion, but offers the possibility to simultaneously encompass it in societal processes.
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"Similar triangles." In Plain Plane Geometry, 81–103. WORLD SCIENTIFIC, 2015. http://dx.doi.org/10.1142/9789814740456_0004.

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"Similar Triangles." In Solving Problems in Geometry, 35–85. WORLD SCIENTIFIC, 2017. http://dx.doi.org/10.1142/9789814583756_0002.

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"Tiling by Similar Triangles." In Algebra and Tiling, 135–53. Providence, Rhode Island: American Mathematical Society, 1994. http://dx.doi.org/10.5948/upo9781614440246.007.

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"Chapter 13. Square Roots, Pythagoras, and Similar Triangles." In Experiencing Geometry, 169–83. Ithaca, NY: Daina Taimina, 2020. http://dx.doi.org/10.3792/euclid/9781429799850-17.

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Wilson, Robin. "5. More triangles and squares." In Number Theory: A Very Short Introduction, 79–96. Oxford University Press, 2020. http://dx.doi.org/10.1093/actrade/9780198798095.003.0005.

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‘More triangles and squares’ explores Diophantine equations, named after the mathematician Diophantus of Alexandria. These are equations requiring whole number solutions. Which numbers can be written as the sum of two perfect squares? Joseph-Louis Lagrange’s theorem guarantees that every number can be written as the sum of four squares, and Edward Waring correctly suggested that there are similar results for higher powers. In 1637, Fermat conjectured that no three positive integers, a, b, and c, can satisfy the equation an+bn=cn, if n is greater than 2. Known as ‘Fermat’s last theorem’, this conjecture was eventually proved by Andrew Wiles in 1995.
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"Anthropometric Algorithm Used for Automatic Body Dimensions and Skin Color Detection Aimed for Homeland Security Systems." In Advances in Multimedia and Interactive Technologies, 141–56. IGI Global, 2014. http://dx.doi.org/10.4018/978-1-4666-4896-8.ch011.

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Automatic body detection and identification of a person is one of the most recent research topics that has gained a lot of attention from researchers. Automated systems that will store human biometrics along with the personal information can be of significant assistance in investigations and security issues. Biometrics represents unique aspects of the body that are measurable, robust, distinctive, physical characteristic, or personal traits of an individual by which a person can be identified. Biometric surveillance systems measure and analyze human physical and behavioral characteristics for identification purposes. A method of body measurement can be used for human identification, with the means of using a static camera. Body measurement calculation based on similar triangles is proposed. The focal length of the camera is a very important aspect of the method. This process can provide the means for obtained image segmentation, measurement of the body parts of the subject, and finally, these measurements can be used for identification of the person.
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Levitin, Anany, and Maria Levitin. "Hints." In Algorithmic Puzzles. Oxford University Press, 2011. http://dx.doi.org/10.1093/oso/9780199740444.003.0008.

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1. A Wolf, a Goat, and a Cabbage With one insignificant exception, the puzzle can be solved by making a sequence of the only moves available in each situation. 2. Glove Selection Imagine a malevolent adversary who wants you to draw as many gloves as possible before getting what you need. Note that gloves are not socks: they can be right-handed and left-handed. 3. Rectangle Dissection Triangles in question need not be of the same size. 4. Ferrying Soldiers Solve the problem of ferrying one soldier first. 5. Row and Column Exchanges The answer is “no”; determine why. 6. Predicting a Finger Count Reenact the girl’s count long enough to see a pattern that makes the answer obvious. 7. Bridge Crossing at Night The answer is “yes,” and the solution does not involve any tricks. 8. Jigsaw Puzzle Assembly A similar problem is discussed in the book’s tutorial on algorithm analysis techniques. 9. Mental Arithmetic There are at least two different ways to compute this sum. Both use the methods discussed in the tutorial on algorithm analysis techniques. 10. A Fake Among Eight Coins “Three” is not the correct answer to the puzzle. 11. A Stack of Fake Coins The answer is “one.” Take advantage of the fact that the scale gives the exact weight. 12. Questionable Tiling The answer is “no.” 13. Blocked Paths Use dynamic programming as explained in the tutorial on algorithm design strategies. 14. Chessboard Reassembly What parts of the board do you have to cut to solve the puzzle? 15. Tromino Tilings Only one of the three questions has a “yes” answer. 16. Making Pancakes What is the fastest way to make three pancakes? Also note that n = 1 is, in fact, a special case here. 17. A King’s Reach The puzzle statement does not forbid the king to visit the same square more than once. Also, make sure that your answer is correct for every value of n ≥ 1. 18. A Corner-to-Corner Journey Observe the colors of the squares the knight jumps through.
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Conference papers on the topic "Similar Triangles"

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Pham, Tuan Minh. "Similar triangles and orientation in plane elementary geometry for Coq-based proofs." In the 2010 ACM Symposium. New York, New York, USA: ACM Press, 2010. http://dx.doi.org/10.1145/1774088.1774358.

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Jian, Ming, Cui-Fang Zhang, Fei Yan, and Mo-Zhen Tang. "A global line extraction algorithm for indoor robot mapping based on noise eliminating via similar triangles rule." In 2016 35th Chinese Control Conference (CCC). IEEE, 2016. http://dx.doi.org/10.1109/chicc.2016.7554319.

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Kearney, Daniel, Jeff Punch, and Ronan Grimes. "An Analysis of the Flow Fields Within Geometrically-Similar Miniature Scale Centrifugal Pumps." In ASME 2008 Heat Transfer Summer Conference collocated with the Fluids Engineering, Energy Sustainability, and 3rd Energy Nanotechnology Conferences. ASMEDC, 2008. http://dx.doi.org/10.1115/ht2008-56195.

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Thermal management has become a key constraint in the development of contemporary electronics systems. It is evident that heat fluxes are currently approaching the limits of conventional forced air cooling, and that liquid cooling technologies are now under consideration. As the space available to incorporate a pump is often limited, miniature-scale pumps are required. Because such pumps operate at low Reynolds numbers, their operation may deviate from that predicted from the conventional pump scaling laws, and their efficiencies reduced. This paper investigates such deviations, and reduced efficiency, through experimental measurements of the performance of two geometrically-similar pumps of – a fully scaled pump of diameter 34.3mm, and a half scale version of the same construction. A facility for the measurement of bulk pressure-flow performance characteristics is described. Particle Image Velocimetry (PIV) measured velocity profiles were extracted at varying radii in the blade passages, and at varying angular positions in the volutes. The absolute, relative, radial and whirl velocity vectors were evaluated for each flow field at three operating points and compared with conventional pump theory. The data was plotted non-dimensionally to investigate points of similitude. Fluidic phenomena occurring in the impeller passage at both pressure and suction sides of the impeller blades are addressed. The theoretical velocity triangles occurring at the impeller tip are compared with the experimental data. The blade angle at inlet and discharge are found to have a large bearing on the poor pumping performance. The quantitative velocity characteristics are discussed in the context of efficiency degradation at decreasing Reynolds numbers.
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Helmers, Lennard, and Jens Klingmann. "Unshrouded Rotor Tip Clearance Effects in Expander Cycle Turbines." In ASME Turbo Expo 2002: Power for Land, Sea, and Air. ASMEDC, 2002. http://dx.doi.org/10.1115/gt2002-30338.

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Steady flow in axial one-stage turbines is assessed numerically and experimentally. The simulations are performed on coarse meshes using a standard numerical approach (3D, steady state, kε-turbulence model, wall function at solid boundaries). In order to allow for conclusions drawn from these rapid numerical studies, the approach was compared with an explicit LDA (Laser Doppler anemometry) mapping of the velocity field downstream the rotor on a representative turbine stage. A two-component LDA system allowed for measurements of axial and tangential velocity components at varying depth (radius) in the flow channel, Measurements thus correspond to a full plane at constant axial position in the rotating frame of reference of the rotor. Comparison between LDA velocity mapping and CFD results shows good agreement. A series of subsequent simulations is thus used to judge the impact of varied blade/stage design parameters. Two turbine layouts are defined for identical operating conditions and shaft power. The flow in the unshrouded rotor blade row is analyzed for the influence of varying tip clearance size and the dependency on stage velocity triangles. – Known correlations for tip clearance losses (typically used in mean line predictions) are used, though the blade row geometry considered is beyond the limits the correlations are intended for. The absolute loss level found in CFD simulations differs significantly from what is expected when using loss correlations. Still the variation with tip gap size is predicted well by some of the investigated models. As dependency of tip clearance losses on stage velocity triangles is considered, none of the tested correlations gives results consistent with the numerical simulations. The use of standard correlations ‘beyond the limits’ is thus considered to introduce high uncertainty. Due to the good consistency between LDA and numerical results, the conclusions are considered to be valid for stage designs similar to the ones analyzed.
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Shibukawa, Naoki, Tomohiko Tsukuda, Tadayuki Hashidate, Hiroyuki Kawagishi, Tatsuro Uchida, and Koichi Goto. "An Experimental Flow Investigation of Low Pressure Turbine Stages With Various Wet Conditions." In ASME Turbo Expo 2012: Turbine Technical Conference and Exposition. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/gt2012-69613.

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Detail flow characteristics of an actual size low pressure steam turbine stages under real operating conditions were examined in this paper. The main purpose of the experimental work was to obtain the radial distribution of the velocity triangles in the wet flow of the large size turbine, so that a series of tests were carried out with various wet conditions. Some particular changes of the flow pattern were observed at the exit of both the stator and the blade rows which would not predicted by steam turbine design tools. A kind of flow coefficient was defined and investigated as well as the steam velocities. With an assumption of the steam wetness distribution along the blade span, a tendency of the flow coefficient was appeared which was similar to previous work [7]. The wetness assumption was qualitatively verified by bore-scope observation of the water concentration on the stator surface and the fog condition of the steam path.
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6

Sangan, Carl M., Kunyuan Zhou, J. Michael Owen, Oliver J. Pountney, Mike Wilson, and Gary D. Lock. "Experimental Measurements of Ingestion Through Turbine Rim Seals: Part 1—Externally-Induced Ingress." In ASME 2011 Turbo Expo: Turbine Technical Conference and Exposition. ASMEDC, 2011. http://dx.doi.org/10.1115/gt2011-45310.

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This paper describes a new research facility which experimentally models hot gas ingestion into the wheel-space of an axial turbine stage. Measurements of CO2 gas concentration in the rim-seal region and inside the cavity are used to assess the performance of two generic (though engine-representative) rim-seal geometries in terms of the variation of concentration effectiveness with sealing flow rate. The variation of pressure in the turbine annulus, which governs this externally-induced (EI) ingestion, was obtained from steady pressure measurements downstream of the vanes and near the rim seal upstream of the rotating blades. Although the ingestion through the rim seal is a consequence of an unsteady, three-dimensional flow field and the cause-effect relationship between pressure and the sealing effectiveness is complex, the experimental data is shown to be successfully calculated by simple effectiveness equations developed from a previously published orifice model. The data illustrate that, for similar turbine-stage velocity triangles, the effectiveness can be correlated using a non-dimensional sealing parameter, Φo. In principle, and within the limits of dimensional similitude, these correlations should apply to a geometrically-similar engine at the same operating conditions. Part 2 of this paper describes an experimental investigation of rotationally-induced (RI) ingress, where there is no mainsteam flow and consequently no circumferential variation of external pressure.
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Willinger, Reinhard, and Michael Köhler. "Influence of Blade Loading Criteria and Design Limits on the Cordier-Line for Axial Flow Fans." In ASME Turbo Expo 2014: Turbine Technical Conference and Exposition. American Society of Mechanical Engineers, 2014. http://dx.doi.org/10.1115/gt2014-25140.

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Volume flow rate, specific isentropic enthalpy difference, rotor outer diameter and rotational speed of a fan can be transformed to speed number and diameter number. These two non-dimensional numbers are related together in the so-called Cordier-diagram. For axial, radial and mixed flow fans, there is a single empirical relationship between both quantities and it is well accepted that this line represents “optimum” fan designs with high efficiency. Based on velocity triangles, a relationship between flow coefficient and pressure coefficient exists. This so-called performance curve captures off-design operating points as well as the design point of a fan. Therefore, the performance curve can be transformed to the Cordier-diagram to predict the relationship between speed number and diameter number. It is shown that the Cordier-line depends mainly on velocity triangles and the common argument of high efficiency, claimed in the majority of the literature, plays only a secondary role. Nevertheless, the requirement of high efficiency influences the fan design for a certain design point. This paper focuses mainly on axial flow fans. It gives a theoretical interpretation of the influence of blade loading criteria and design limits on the Cordier-line: (1) De Haller number, (2) cascade loading parameter, (3) Lieblein diffusion factor, (4) Strscheletzky swirl number. Criterion (1) reflects the minimum velocity ratio to avoid endwall separation in a linear compressor cascade. Criterion (2) is a combination of lift coefficient and cascade solidity. It reflects the aerodynamic loading of the suction side blade boundary layer. Criteria (1) and (2) are included in criterion (3). Finally, criterion (4) indicates the risk of hub separation due to strong swirl flow. The investigation shows that the transformation of these criteria to the Cordier-diagram gives very similar results. Furthermore, it is shown that the axial fan design limits in the Cordier-diagram are represented by certain hub-to-tip radius ratios.
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8

Sangan, Carl M., James A. Scobie, J. Michael Owen, Oliver J. Pountney, Mike Wilson, and Gary D. Lock. "Experimental Measurements of Ingestion Through Turbine Rim Seals: Part 3—Single and Double Seals." In ASME Turbo Expo 2012: Turbine Technical Conference and Exposition. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/gt2012-68493.

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This paper describes experimental results from a research facility which experimentally models hot gas ingress into the wheel-space of an axial turbine stage. Measurements of CO2 gas concentration in the rim-seal region and inside the wheel-space are used to assess the performance of generic (though engine-representative) single and double seals in terms of the variation of concentration effectiveness with sealing flow rate. The variation of pressure in the turbine annulus, which governs externally-induced ingress, was obtained from steady pressure measurements downstream of the vanes. The benefit of using double seals is demonstrated: the ingested gas is shown to be predominately confined to the outer wheel-space radially outward of the inner seal; in the inner wheel-space, radially inward of the inner seal, the effectiveness is shown to be significantly higher. Criteria for ranking the performance of single and double seals are proposed, and the performance limit for any double seal is shown to be one in which the inner seal is exposed to rotationally-induced ingress. Although the ingress is a consequence of an unsteady, three-dimensional flow field and the cause-effect relationship between pressure and the sealing effectiveness is complex, the experimental data is shown to be successfully calculated by simple effectiveness equations developed from a theoretical model. The data illustrate that, for similar turbine-stage velocity triangles, the effectiveness can be correlated using two empirical parameters. In principle, these correlations could be extrapolated to a geometrically-similar turbine operating at engine-representative conditions.
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9

Jordal, Kristin, Olav Bolland, and A˚ke Klang. "Aspects of Cooled Gas Turbine Modelling for the Semi-Closed O2/CO2 Cycle With CO2 Capture." In ASME Turbo Expo 2003, collocated with the 2003 International Joint Power Generation Conference. ASMEDC, 2003. http://dx.doi.org/10.1115/gt2003-38067.

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In order to capture the behaviour of the oxyfuel cycle operating with high combustor-outlet temperature, the impact of blade and vane cooling on cycle performance must be included in the thermodynamic model. As a basis for a future transient model, three thermodynamic models for the cooled gas turbine are described and compared. The first model, known previously from the literature, models expansion as a continuous process with simultaneous heat and work extraction. The second model is a simple stage-by-stage model and the third is a more detailed stage-by-stage model that includes velocity triangles and enables the use of advanced loss correlations. An airbreathing aeroderivative gas turbine is modelled, and the same gas turbine operating in an oxyfuel cycle is studied. The two simple models show very similar performance trends in terms of variation of pressure ratio and turbine inlet temperature in both cases. With the more detailed model, it was found that, without any change of geometry, the turbine rotational speed increases significantly and performance drops for the maintained geometry and pressure ratio. A tentative increase of blade angles or compressor pressure ratio is found to increase turbine performance and decrease rotational speed. This indicates that a turbine will require re-design for operation in the oxyfuel cycle.
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10

Suzuki, Hirotaka. "Skew quadrilateral membrane folding for lampshade design." In The 13th International Conference on Engineering and Computer Graphics BALTGRAF-13. Vilnius Gediminas Technical University, 2015. http://dx.doi.org/10.3846/baltgraf.2015.016.

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Historically, Japanese traditional lampstand 'Andon', was manufactured from paper. And paper folding method was adopted into some Andon or western lampshade design. For example, Yoshimura/Diamond pattern, known as a structure of crashed cylinder, or a structure of building roof, that has been of beneficial use in commercial lampshade products. Yoshimura pattern structure includes a set of skew quadrilaterals which are not on a plane surface and each quadrilateral is constructed with 2 planar triangles. Development of Yoshimura pattern is constructed by one set of horizontal parallel lines at even intervals for valley fold and two sets of oblique parallel lines at even intervals for mountain fold. The author found that similar shape can be constructed from development which includes only mountain fold lines of Yoshimura pattern and this method has various applications. With proposed paper folding method, each skew quadrilateral is constructed by single curved surface. In this paper, the author first defined the principle of the proposed paper folding method, second, explained the. features of the shape made by the proposed method and the luminance distribution on the shape, third, indicated examples of applications of the proposed method. Finally, examples of application of the shape made by SQMF in the field of education are explained.
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Reports on the topic "Similar Triangles"

1

Shaw, Dennis J., Ralph S. Meldahl, John S. Kush, and Greg L. Somers. A Tree Taper Model Based on Similar Triangles and Use of Crown Ratio as a Measure of Form in Taper Equations for Longleaf Pine. Asheville, NC: U.S. Department of Agriculture, Forest Service, Southern Research Station, 2003. http://dx.doi.org/10.2737/srs-gtr-66.

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Shaw, Dennis J., Ralph S. Meldahl, John S. Kush, and Greg L. Somers. A Tree Taper Model Based on Similar Triangles and Use of Crown Ratio as a Measure of Form in Taper Equations for Longleaf Pine. Asheville, NC: U.S. Department of Agriculture, Forest Service, Southern Research Station, 2003. http://dx.doi.org/10.2737/srs-gtr-66.

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You might also be interested in the extended bibliographies on the topic 'Similar Triangles' for particular source types:
Journal articles
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