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New horizons in geometry

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The collaborative work of Tom Apostol and Mamikon Mnatsakanian has been lauded for its clarity and originality. in this volume the authors present an impressive collection of geometric results that reveal surprising connections between lengths, areas and volumes in various shapes, and allow one to compute difficult integrals, all using intuitive visual calculations. One noteworthy idea that the reader will encounter is Mamikon's Sweeping Tangent Theorem from which the authors obtain a visual derivation of the property that the length of an arc of a catenary is proportional to the area under the arc. This is one of many 'proofs without words' contained within. in addition, a variety of results are derived visually for cycloids, conic sections, and many more geometric objects. As befits a book that emphasises visual thinking, the text is beautifully illustrated. This is a book that will inspire students and enrich any geometry or calculus course.. © 2012 by The Mathematical Association of America (Incorporated) and Mamikon A. Mnatsakanian.

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... Closely-connected approaches are employed by [9][10][11]. 20 Ref. [9] uses the normal curvature in transformed coordinates to set the zero tangential stress boundary condition for a streamfunction-vorticity numerical formulation of axisymmetric flow of a Newtonian fluid. Herrera, in [10] and [12] develop and utilize a relationship between 25 vorticity components and the curvatures and torsions of a streamline. ...
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... The eccentricity of these conics are given by e = tan (θ) [20], and the radius of curvature 1/κ n,s at the vertex of any of these conics is r tan θ. These Euclidean geometry findings are readily obtainable with the Dandelin (focal) sphere construction [20] (see supplementary material for a relevant visualization ,which corresponds to θ < π/4). Using this radius for 1/κ ns in Equation (5) ...
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... Este trabalho tem como base o livro New Horizons in Geometry [1], de Tom Apostol e Mamikon Mnatsakanian, que traz uma abordagem visual e inovadora, com métodos geométricos que requerem pouco ou nenhuma fórmula, para resolver muitos problemas clássicos do Cálculo. Como tal, grande parte dos resultados e aplicações aqui presentes são uma releitura e um detalhamento daquilo ali apresentado. ...
... Artists [33], architects [41], film makers, engineers and designers draw inspiration from visual mathematics. Well illustrated books like [21,49,85,48,34,49,10,3] advertise mathematics with figures and illustrations. Such publications help to counterbalance the impression that mathematics is difficult to communicate to non-mathematicians. ...
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