{"id":845,"date":"2022-03-27T11:26:16","date_gmt":"2022-03-27T11:26:16","guid":{"rendered":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/?page_id=845"},"modified":"2022-07-04T11:06:58","modified_gmt":"2022-07-04T11:06:58","slug":"on-constructing-angles-with-prescribed-vertex-and-measure-in-the-upper-half-plane-modelof-hyperbolic-geometry","status":"publish","type":"page","link":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/on-constructing-angles-with-prescribed-vertex-and-measure-in-the-upper-half-plane-modelof-hyperbolic-geometry\/","title":{"rendered":"On constructing angles with prescribed vertex and measure in the upper half plane model of hyperbolic geometry"},"content":{"rendered":"\n
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Moroccan Journal of Algebra and Geometry with Applications<\/strong><\/p>\n\n\n\n

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On constructing angles with prescribed vertex and measure
<\/em><\/strong><\/em><\/strong>in the upper half-plane model of hyperbolic geometry <\/em><\/strong><\/p>\n<\/div>\n<\/div>\n\n\n\n

David E. Dobbs <\/strong><\/i>
Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1320, USA<\/p>\n\n\n\n

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Pages 18-48 | Received 11 April 2021, Accepted 30 May 2021, Published 01 June<\/span> 2022 <\/span><\/p>\n\n\n\n

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Abstract<\/span><\/strong><\/p>\n\n\n\n

It is proved that if k,m \\in \\mathbb{R} <\/span> and P_0 \\, (x_0,y_0) <\/span> is a point in the Euclidean plane \\mathbb{R}^2 <\/span> with y_0 \\neq k <\/span> and with \\mathcal{X} <\/span> denoting the (horizontal) line with Cartesian equation y=k <\/span>, then there exists a unique circle, say \\mathcal{K}, <\/span> such that the center of \\mathcal{K} <\/span> is on \\mathcal{X}, \\ P_0 <\/span> lies on \\mathcal{K}<\/span>, and the tangent to \\mathcal{K} <\/span> at P_0 <\/span> has slope m<\/span>. An ensuing multi-step algorithmic result that is proved here for the Euclidean upper half-plane determines the angular measure of any directed angle that is formed by counterclockwise rotation from a designated initial side to a designated terminal side, in case each of those “sides” is a hyperbolic line segment (that is, either a vertical (Euclidean) line segment or an arc of a (Euclidean) circle centered on the x <\/span>-axis). One consequence (for the Euclidean upper half-plane) is the construction of (the unique hyperbolic line segment playing the role of terminal (resp., initial) side of) a unique directed angle having a prescribed vertex, a prescribed measure between 0 and \\pi <\/span>, and a prescribed hyperbolic line segment as initial (resp., terminal) side. As the only prerequisites assumed here are related topics in analytic geometry and trigonometry that can be covered in a precalculus course, this paper could be used as enrichment material for a precalculus course, a calculus course, or a course on the classical geometries that features the upper half-plane model of hyperbolic plane geometry.<\/p>\n\n\n\n

Keywords<\/strong>:<\/span>  Hyperbolic plane geometry, upper half-plane model, directed angle, vertex, Euclidean geometry, slope, tangent line, inverse tangent function, angle of inclination, inverse cosine function, bowedgeodesic, straight geodesic. <\/p>\n\n\n\n

MSC numbers<\/strong>:<\/span> Primary 51-02; Secondary 33B10, 51N20, 51M04.<\/p>\n\n\n\n

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Moroccan Journal of Algebra and Geometry with Applications Latest articles On constructing angles with prescribed vertex and measure in the upper half-plane model of hyperbolic geometry David E. Dobbs  Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1320, USA Pages 18-48 | Received 11 April 2021, Accepted 30 May 2021, Published 01 June 2022 Abstract Read More<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"template-page-builder.php","meta":{"footnotes":""},"class_list":["post-845","page","type-page","status-publish","hentry","entry"],"_links":{"self":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/845","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/comments?post=845"}],"version-history":[{"count":25,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/845\/revisions"}],"predecessor-version":[{"id":1686,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/845\/revisions\/1686"}],"wp:attachment":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/media?parent=845"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}