Elementary Geometrical Research Articles

Miquel's Theorem and its Elementary Geometric Relatives

Elementarnogeometrijski Miquelov teorem odnosi se na trokut ABC i točke R, S, T na njegovim stranicama, i kaže da kružnice k(ART), k(BRS), k(CST) imaju jednu zajedničku točku M, Miquelovu točku zadane figure. Posebnim odabirom točaka R, S, T dobivaju se takozvani beermat teorem, Brocardovi teoremi i Steinerov Simson-Wallaceov teorem kao specijalni slučajevi Miquelovog teorema. Stoga, činjenice vezane za Brocardov teorem slijede iz svojstava Miquelovog teorema. Ako npr. R, S, T zadavoljavaju Cevin uvjet, Miquelova konstrucija inducira preslikavanje Cevine točke u Miquelovu točku. Proučavamo ovo i druga preslikavanja koja su direktna posljedica Miquelovog teorem. Nadalje, kreću li se točke R, S, T stranicama trokuta tako da su npr. afini omjeri ar(ARB), ar(BSC), ar(CTA) jednaki, Miquelove točke M opisuju kružnicu opisanu trokutu čiji su vrhovi dvije Brocardove točke i središte opisane kružnice trokuta. Osim spomenutih triju posebnih točaka trokuta, na ovoj kružnici leži još nekoliko osobitih točaka trokuta. Iako je većina predstavljenih tema dobro poznata, čini se da njihove međusobne veze još nisu razmatrane u standardnoj literaturi iz geometrije trokuta i stoga bi mogle opravdati dodatnu obradu.

On the Mastery of Elementary Geometric Concepts¤.

We publish here the report by N.F. Talyzina to a Conference on Psychology in 1955 and released two years later in the proceedings of that conference (N.F. Talyzina [1957]. In B.G. Anan′yeva et al. [Eds.], Materialy soveshchaniia po psikhologii. July 1–6, 1955. Moscow: Publishing House of the Academy of Pedagogical Sciences of the RSFSR.) This report is of particular interest, since it describes some of the first formative experiments based on the theory of planned stage-by-stage formation of mental actions; the text does not yet discuss the formation of actions, but only the application of the attributes of concepts to solving problems. The text presents the results of a study of the formation of elementary geometric concepts according to the theory of planned stage-by-stage formation of mental actions and concepts. Twenty-two subjects from grades 6–9 formed geometric concepts such as “line,” “angle,” “angle bisector,” “perpendicular,” “adjacent angles,” and “supplementary angles.” These attributes were formed in the process of their application to solving different types of problems. Observational and then formative (training) experiments were performed individually. The article contains excerpts from the subjects’ protocols as they solved the problems. It was shown that stage-by-stage development (identifying attributes from a definition, saying them aloud, and then applying them to solving problems) leads to mastery not only of the geometric concepts themselves, but also of the method of action with definitions in general, allowing students to transfer the method they have learned to concepts from another field of knowledge.

Model-Based Roentgen Stereophotogrammetric Analysis Using Elementary Geometrical Shape Models: Reliability of Migration Measurements for an Anatomically Shaped Femoral Stem Component

Elementary Geometrical Shape (EGS) models present an alternative approach to detect in vivo migration of total hip arthroplasty using model-based Roentgen Stereophotogrammetric Analysis (mbRSA). However, its applicability for an irregular-shaped femoral stem and the reliability of this mbRSA approach has not been proven so far. The aim of this study is to assess the effect of multi-rater and an anatomically shaped femoral stem design onto resulting implant to bone migration results. The retrospective analysis included 18 clinical cases of anatomically shaped stem with 10-year RSA follow-ups. Three raters repeatedly measured all RSA follow-ups for evaluating the rater equivalence and intra-rater reliability. The results proved the equivalence between different raters for mbRSA using EGS models (mbRSA-EGS), hence it simplified the investigation of rater reliability to intra-rater reliability. In all in-plane migration measurements, mbRSA-EGS shows good intra-rater reliability and small intra-rater variability (translation: <0.15 mm; rotation: <0.18 deg). However, the reliability is worse in the out-of-plane measurements, especially the cranial-caudal rotation (intra-rater variability: 0.99–1.81 deg). Overall, mbRSA-EGS can be an alternative approach next to surface models while the in-plane migration of femoral stem (e.g., the implant subsidence for loosening prediction) have more research interested than other directions.

Complex Parts Partitioning by Elementary Geometric Objects Clustering in CAD/CAM Process

Complex shaped parts are omnipresent in the industry: aeronautics, molds, etc. CAD/CAM software is used in their production process. To simulate this process, three geometric representation models can be considered: workpiece model, tool model and swept volume model which can be discrete or continuous. The use of discrete models is simple to implement compared to continuous models but requires a tight discretization to represent faithfully theoretical shapes. This is necessary for a high accuracy but increases processing time. To overcome this computing constrains, the main contribution of this work consists in partitioning the part space into regions by pooling the geometric elements (triangles, Dexels) into sets while introducing for the first time in this context two methods knows as Fast Global K-means (FGK) and Incremental K-means (IK), that arise from Basic K-means (BK) method presented in previous works. Another key point is developed; it consists on establishing a pretreatment on the Triple Dexels model in order to be able to use BK, IK and FGK methods in the material removal simulation process. A comparative study has been established to evaluate the performances of these methods. FGK demonstrated better performances in terms of computational time, cluster uniformity and reduction of regions overlapping areas, compared to BK and IK methods.

A Didactic Sequence of Elementary Geometric Optics Informed by History and Philosophy of Science

The concepts and instruments required for the teaching and learning of geometric optics are introduced in the didactic process without a proper didactic transposition. This claim is secured by the ample evidence of both wide- and deep-rooted alternative concepts on the topic. Didactic transposition is a theory that comes from a reflection on the teaching and learning process in mathematics but has been used in other disciplinary fields. It will be used in this work in order to clear up the main obstacles in the teaching-learning process of geometric optics. We proceed to argue that since Newton’s approach to optics, in his Book I of Opticks, is independent of the corpuscular or undulatory nature of light, it is the most suitable for a constructivist learning environment. However, Newton’s theory must be subject to a proper didactic transposition to help overcome the referred alternative concepts. Then is described our didactic transposition in order to create knowledge to be taught using a dialogical process between students’ previous knowledge, history of optics and the desired outcomes on geometrical optics in an elementary pre-service teacher training course. Finally, we use the scheme-facet structure of knowledge both to analyse and discuss our results as well as to illuminate shortcomings that must be addressed in our next stage of the inquiry.

Molecular Jigsaw: Pattern Diversity Encoded by Elementary Geometrical Features

Scanning tunneling microscopy (STM) images of self-organized monolayers of Frechet dendrons display a variety of two-dimensional ordering motifs, which are influenced by engineering the molecular interactions. An interaction-site model condenses the essential molecular properties determined by molecular mechanics modeling, which in a Monte Carlo approach successfully predicts the various ordering motifs. This confirms that geometry as well as a few salient weak interaction sites encode these structural motifs.

The Elementary Geometric Structure of Compact Lie Groups

We give geometric proofs of some of the basic structure theorems for compact Lie groups. The goal is to take a fresh look at these theorems, prove some that are difficult to find in the literature, and illustrate an approach to the theorems that can be imitated in the homotopy theoretic setting of p-compact groups. 1991 Mathematics Subject Classification 22E15, 55P35.

Pattern Change and Distance Variation in the Square

Geographical AnalysisVolume 14, Issue 1 p. 72-78 Free Access Pattern Change and Distance Variation in the Square Arthur Getis, Arthur Getis Arthur Getis is professor of geography, University of Illinois at Urbana-Champaign. The author has benefited from comments made by Tames O. Huff and computer programming by Marc Armstrong, both of the Department of Geography, University of Illinois, and Ralph Lenz of Wittenberg University.Search for more papers by this author Arthur Getis, Arthur Getis Arthur Getis is professor of geography, University of Illinois at Urbana-Champaign. The author has benefited from comments made by Tames O. Huff and computer programming by Marc Armstrong, both of the Department of Geography, University of Illinois, and Ralph Lenz of Wittenberg University.Search for more papers by this author First published: January 1982 https://doi.org/10.1111/j.1538-4632.1982.tb00056.xCitations: 4 Arthur Getis is professor of geography, University of Illinois at Urbana-Champaign. The author has benefited from comments made by Tames O. Huff and computer programming by Marc Armstrong, both of the Department of Geography, University of Illinois, and Ralph Lenz of Wittenberg University. AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat LITERATURE CITED Dacey, M. F. (1965). “Numerical Measures of Random Sets.” Research Report,16. Evanston., Ill: Department of Geography, Northwestern University. Dacey, M. F. (1975). “Evaluation of the Poisson Approximation to Measures of the Random Pattern in the Square.” Geographical Analysis, 7: 351– 67. Gaile, G. L., and J. E. Burt (1980). “Directional Statistics.” Concepts and Techniques in Modem Geography (CATMOG), No. 25. London: Institute of British Geographers. Getis, A., and B. Boots (1978). Models of Spatial Processes: An Approach to the Study of Point, Line and Area Patterns. Cambridge: Cambridge University Press. Ghosh, B. (1951). “Random Distances within a Rectangle and between Two Rectangles.” Bulletin of the Calcutta Mathematical Society. 43: 17– 24. Horowitz, M. (1965). “Probability of Random Paths across Elementary Geometrical Shapes.” Journal of Applied Probability, 2: 169– 77. Taylor, P. J. (1971). “Distances within Shapes: An Introduction to a Family of Finite Frequency Distributions.” Geografiska Annaler, 53: 40– 53. Tobler, W. (1977). “Bidimensional Regression.” Santa Barbara, Calif.: Department of Geography, University of California. Tobler, W. (1978). “Comparison of Plane Forms.” Geographical Analysis, 10: 154– 62. Citing Literature Volume14, Issue1January 1982Pages 72-78 ReferencesRelatedInformation

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Note on Certain Elementary Geometrical Notions and Determinations

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