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According to this technique, the needle trajectory from the skin to the vessel corresponds to the hypotenuse of the triangle. [...]while controlling the long-axis plane by the previously described technique, the use of the short axis view during the vessel puncture allows checking structures lateral to the vessel, in particular arteries and nerves in the case of vein puncture. [...]the basis of Euclidean geometry might be an educational tool for the critical care physician teaching fellows for their first venous catheterizations, giving them a simple step by step procedure. 1.

This study furthers the theory of conceptual blending as a useful tool for revealing the structure and process of student reasoning in relation to the Sierpinski triangle (ST). We use conceptual blending to investigate students' reasoning, revealing how students engage with the ST and coordinate their understandings of its area and perimeter. Our analysis of ten individual interviews with mathematics education masters' student documents diverse ways in which students reason about this situation through the constituent processes of blending: composition, completion, and elaboration. This reveals that students who share basic understandings of the area and perimeter of the ST recruit idiosyncratic ideas to engage with and resolve the paradox of a figure with infinite perimeter and zero area.

Grunert’s system of equations is commonly used as a basis for mathematical investigations into the Perspective 3-Point Pose Problem, for camera resectioning and tracking. This consists of three quadratic equations involving three unknown distances. The discriminant of this system helps to determine the number of real-valued solutions, in terms of the system’s parameters. In its raw form, this is a very complicated and seemingly unintelligible polynomial. However, through a series of algebraic manipulations, this article manages to bring this polynomial into a far more sensible form. In addition, by making substitutions suggested by the system of equations, the discriminant is realized as a rational function of the Cartesian coordinates of the ambient space containing the control points. Moving perpendicular to the plane containing the control points, and moving away from this plane, cross sections of the surface on which this rational function vanishes approach a deltoid curve, together with the deltoid’s inscribed circle (a cross section of the danger cylinder). As long as such a cross section of the surface is not too close to the control points plane, it is homeomorphic to a union of the deltoid and its inscribed circle. The orthogonal projection of the deltoid onto the control points plane contains in its interior, the control points triangle’s orthocenter.

Recently, the triangle features have been applied in digit recognition by adopting the angle as a part of the features. Most of the studies in digit recognition area which applied these features have given impressive result. However, the issue of big gap values that occurred between angle, ratio and gradient has given big impact to the accuracy of result. Therefore, we introduce our proposed method which is data normalization that has adopted the nature of triangle geometry in order to resolve the issue. Besides, we have applied other techniques such as Z-score, Minimax and LibSVM function in the experiment. There are four digit datasets used which are HODA, MNIST, IFCHDB and BANGLA. The result of classification have shown our proposed method have given better result compared to other technique as aforementioned.

In this paper I examine three versions of a case added to Euclid's Elements, Book I, proposition 1, which are found in three late medieval Hebrew texts based on the Elements. The added case explains how to construct an isosceles triangle. Following a brief discussion of this problem and its solutions in the Greek, Arabic, and Latin traditions, I examine the constructions in the Hebrew texts and their peculiarities. In an Appendix I present a partial English translation of the version of this added case found in Ibn al-Haytham's On the Resolution of Doubts on Euclid's Elements.

Features are normally modelled based on color, texture and shape. However, some features may have different constraints based on types, styles and pattern of an image. The Arabic subword handwriting, for example, cannot be recognized by color and not suitable to be characterized based on texture. Therefore, features based on shape are suitable to be used for recognizing Arabic subword handwriting since each of the character has various characteristics such as diacritics, thinning and strokes. These characteristics can contribute to particular a shape that is unique and can represent Arabic subword handwriting. Currently, geometry shape such as triangle has been adopted to extract useful features based on triangle properties without implicating any triangle form. In order to increase classification accuracy, these properties have been categorized into several zones where the number of features produced is directly proportional to the number of zones. Nevertheless, shape representation does not implicate any triangle properties such as ratio of side, angle and gradient. By using shape representation, it helps in reducing the number of features. Thus, this paper presents feature based on triangle shape that can represent the identity of Arabic subword handwriting. The method based on triangle shape identifies three main coordinates of triangle formed based on selected centroids. The AHDB dataset is used as a testing data. The Support Vector Machine (SVM) and Random Forest (RF), respectively were used to measure the accuracy of the proposed method using triangle shape as a feature. The accuracy results have shown better outcome with 77.65% (SVM) and 76.43% (RF), which prove the feature based on triangle shape is applicable for Arabic subword handwriting recognition.

This study explored whether people create Euclidean representations of 2-dimensional right triangles from touch and use them to make spatial inferences in accord with Euclidean distance axioms. Blindfolded participants who were instructed to form visual images of triangles felt the vertical and horizontal sides of right triangles, then estimated the lengths (but not the angles) of the 3 triangle sides. In these 3 experiments, length estimates conformed closely to the Euclidean metric when evaluated on application of the Pythagorean theorem. Participants who used a visual imaging strategy were accurate more often than those who used visual imagery less often. In Experiments 2 and 3, a hypotenuse inference was as accurate as a direct haptic judgment of the hypotenuse. These results demonstrated similar accuracy of the hypotenuse judgments when participants made verbal rather than haptic estimates. The findings indicate that participants can form Euclidean representations under certain conditions from felt 2-dimensional right triangles based on visual images.