• Title/Summary/Keyword: tangent line

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Isotomic and Isogonal Conjugates Tangent Lines of Lines at Vertices of Triangle

  • Seo, Min Young;Ahn, Young Joon
    • Journal of Integrative Natural Science
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    • v.10 no.1
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    • pp.27-32
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    • 2017
  • In this paper we consider the two tangent lines of isogonal and isotomic conjugates of the line at both vertices of a given triangle. We find the necessary and sufficient condition for the two tangent lines of isogonal or isotomic conjugates of the line at both vertices and the median line to be concurrent. We also prove that every line whose isogonal conjugate tangent lines at both vertices are concurrent with the median line intersects at a unique point. Moreover, we show that the three intersection points correspond to the vertices of triangle are collinear.

On effective way of teaching concept of tangent line using Lakatos theory and GSP (Lakatos 이론과 GSP를 활용한 접선지도연구)

  • An, Byung-Kuk;Kim, Byung-Hak;Park, Youn-Keun
    • Communications of Mathematical Education
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    • v.24 no.3
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    • pp.627-658
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    • 2010
  • The introduce and accurate understanding of the concept of tangent line is very important in Mathematics education and its applications. In this study, we investigated the introduction method in the textbook for tangent line and its concepts. And we studied the effective teaching methods for the accurate understanding using Lakatos learning theory, GSP and related precedence studies. Finally we suggest the teaching plan for the equation of for the tangent line in the high school and apply to the High school students.

Students' Reinvention of Derivative Concept through Construction of Tangent Lines in the Context of Mathematical Modeling (수학적 모델링 과정에서 접선 개념의 재구성을 통한 미분계수의 재발명과 수학적 개념 변화)

  • Kang, Hyang Im
    • School Mathematics
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    • v.14 no.4
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    • pp.409-429
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    • 2012
  • This paper reports the process two 11th grade students went through in reinventing derivatives on their own via a context problem involving the concept of velocity. In the reinvention process, one of the students conceived a tangent line as the limit of a secant line, and then the other student explained to a peer that the slope of a tangent line was the geometric mean of derivative. The students also used technology to concentrate on essential thinking to search for mathematical concepts and help visually understand them. The purpose of this study was to provide meaningful implications to school practices by describing students' process of reinvention of derivatives. This study revealed certain characteristics of the students' reinvention process of derivatives and changes in the students' thinking process.

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The maxillary incisor labial face tangent: clinical evaluation of maxillary incisor inclination in profile smiling view and idealized aesthetics

  • Naini, Farhad B.;Manouchehri, Shaadi;Al-Bitar, Zaid B.;Gill, Daljit S.;Garagiola, Umberto;Wertheim, David
    • Maxillofacial Plastic and Reconstructive Surgery
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    • v.41
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    • pp.31.1-31.7
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    • 2019
  • Background: To test the hypothesis that in profile smiling view, for ideal aesthetics, a tangent to the labial face of the maxillary central incisor crowns should be approximately parallel to the true vertical line and thereby perpendicular to the true horizontal line. Methods: An idealized female image was created with computer software and manipulated using the same software to construct an "ideal" female profile image with proportions, and linear and angular soft tissue measurements, based on currently accepted criteria for idealized Caucasian profiles. The maxillary incisor labial face tangent was altered in 5° increments from 70 to 120°, creating a range of images, shown in random order to 70 observers (56 lay people and 14 clinicians), who ranked the images from the most to the least attractive. The main outcome was the preference ranks of image attractiveness given by the observers. Results: The most attractive inclination of a tangent to the labial face of the maxillary incisor crowns in profile view in relation to the true horizontal line was 85°, i.e. 5° retroclined from a perpendicular 90° inclination. The most attractive range appears to be between 80 and 90°. Excessive proclination appeared to be less desirable than retroclination. Beyond 105° most observers recommend treatment. Conclusion: In natural head position, the ideal inclination of the maxillary incisor crown labial face tangent in profile view will be approximately parallel to the true vertical line and thereby approximately perpendicular to the true horizontal line.

A STUDY ON THE FACIAL ESTHETIC PREFERENCES AMONG KOREAN YOUTHS: ASSESSMENT OF PROFILE PREFERENCES (한국 젊은이의 안면미 선호경향에 관한 연구 : 얼굴의 측모평가를 중심으로)

  • Song, Sejin;Choi, Ik-chan
    • The korean journal of orthodontics
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    • v.22 no.4 s.39
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    • pp.881-920
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    • 1992
  • This study was designed to assess profile preferences among Korean youths in the year 1992. Facial esthetics was evaluated by means of silhouette profiles, eliminating the influence of a number of aspects that may affect judgment when normal lateral photographs are used. The main points of preference to be clarified here are as follows. First, on facial convexity, Second, on nasion depth, Third, on mentolabial sulcus depth, Fourth, on the position of upper and lower lips, Fifth, on facial type according to Angle's classification of malocclusion, Sixth, on Song's tangents. The 54 subjects printed in questionnaire as black and white silhouettes were selected from 300 tracings from cephalometric radiographs of people whose age ranging from 11 to 20 years. Photographs of six female subjects were retouched by computer graphic software and printed in color and black/white photographs which were used for adaptation of eyes of participants in selecting profiles in silhouette. They constitute 2 questions. The 54 subjects were grouped as 22 questions, each of them composed of 6 subjects, according to the aspects to be clarified. Twenty four questions in total were asked to assess profile preferences. For the assessment, the profile line, the facial esthetic triangle, Song's tangents, and Angle's classification of malocclusion were introduced. The profile line is composed of 11 component points which are Trichion, Glabella, Nasion, Pronasale, Subnasale, Labrale superius, Stomion, Labrale inferius, Supramentale, Pogonion, and Gnathion. The facial esthetic triangle is composed of 3 tangents: A-tangent which is the tangent of dorsum of nose, B-tangent which is the line passing through Sn and Ls, and C-tangent which is drawn on the turning point of the curve which lies between mentolabial sulcus (Sm) and pogonion (Pg). Angle's classification has 3 types of malocclusion which are Class I, Class II, and Class III. Class II malocclusion is subdivided into Division 1 and Division 2. The participants of the survey were composed of 861 college students (448 male students, 413 female students) whose majors grouped as Fine Arts. Liberal Arts, and Natural Sciences, and whose mean age 21.8 years. The statistics program SPSS/PC + of SPSS Inc. was used to analyze answers of participants. Crosstabulation, Chi-square test, and Kendall test were done. The conclusions are as follows: First, Korean youths have a tendency to prefer the slightly convex face to the flat or concave face. Second, they prefer a moderately deep nasion. Third, they prefer a moderately deep mentolabial sulcus. Fourth, they prefer the position of lips which are near to Ricketts' E-line. The position of the upper lip which is slightly posterior to E-line is preferred. The upper lip which lies too far anterior or posterior to the lower lip is not perferred. Fifth, they prefer most, according to Angle's Classification of Malocclusion, Class I facial profile which has a slight inclination to Class II division 2. The order of preference is Class I, Class II division 2, Class III, and Class II division 1. Sixth, they prefer the type 2 and 3 of Song's tangents. The facial profile within which A-and B-tangent meet is preferred. The facial profile which has Cotangent that .meets with A-tangent slightly posterior to the crossing point of A-and B-tangent or that parallels with B-tangent is preferred.

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Influence of Tangent Line Angle on Surface Roughness at Fused Deposition (FDM에서 곡면부의 접선기울기가 제품의 표면에 미치는 영향)

  • Ha, Man-Kyung;Jun, Jae-Uhk
    • Journal of the Korean Society of Manufacturing Process Engineers
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    • v.1 no.1
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    • pp.23-28
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    • 2002
  • Fused deposition modelling(FDM) is a rapid prototyping(RP) process that fabricates part layer by layer by deposition of molte thermoplastic material extrude from a nozzle. RP system has benefits. Benefit would be the ability to experiment with physical objects of any complexity m a relatively short period of time. But it has a matter of surface roughness and geometric accuracy. We study on Influence of tangent line angle on surface roughness at fused deposition.

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Measurements of Dielectric constant and Loss tangent of Microwave Substrates (초고주파 대역에서의 기판의 유전상수 및 손실계수 측정)

  • 이선하;한상민;김영식
    • Proceedings of the Korea Electromagnetic Engineering Society Conference
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    • 2000.11a
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    • pp.238-242
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    • 2000
  • In this paper, the method for dielectric constant and less tangent measurements of microwave substrates using a microstrip line is presented. The two samples of an open-ended line and a short-ended tine are made and complex propagation constant is evaluated from the one-port scattering parameter measurements. This method has been automated fully using a personal computer and a vector network analyzer.

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A study on the pedagogical consideration of the related knowledge for teaching 'Approximation' conception (근사개념 지도를 위한 관련 지식의 교수학적 고찰)

  • Chung, Young-Woo;Lee, Mok-Hwa;Kim, Boo-Yoon
    • Communications of Mathematical Education
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    • v.26 no.1
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    • pp.137-154
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    • 2012
  • Approximation' is one of central conceptions in calculus. A basic conception for explaining 'approximation' is 'tangent', and 'tangent' is a 'line' with special condition. In this study, we will study pedagogically these mathematical knowledge on the ground of a viewpoint on the teaching of secondary geometry, and in connection with these we will suggest the teaching program and the chief end for the probable teaching. For this, we will examine point, line, circle, straight line, tangent line, approximation, and drive meaningfully mathematical knowledge for algebraic operation through the process translating from the above into analytic geometry. And we will construct the stream line of mathematical knowledge for approximation from a view of modern mathematics. This study help mathematics teachers to promote the pedagogical content knowledge, and to provide the basis for development of teaching model guiding the mathematical knowledge. Moreover, this study help students to recognize that mathematics is a systematic discipline and school mathematics are activities constructed under a fixed purpose.