{"id":33746,"date":"2021-05-05T13:00:31","date_gmt":"2021-05-05T13:00:31","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=33746"},"modified":"2025-07-30T22:39:27","modified_gmt":"2025-07-30T22:39:27","slug":"sectiones-conicae-characterizatio-et-repraesentatio-graphica-parabolarum-ellipseon-et-hyperbolarum","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/la\/sectiones-conicae-characterizatio-et-repraesentatio-graphica-parabolarum-ellipseon-et-hyperbolarum\/","title":{"rendered":"Sectiones Conicae: Characterizatio et Repraesentatio Graphica Parabolarum, Ellipseon et Hyperbolarum"},"content":{"rendered":"<p><center><\/p>\n<h1>Sectiones Conicae: Characterizatio et Graphice Parabolarum, Ellipseon et Hyperbolarum<\/h1>\n<p><em><strong>Summarium:<\/strong><br \/>\nIn hac lectione sectiones conicas (parabolas, ellipses et hyperbolas) recognoscemus, ab eorum aequationibus canonicis et generalibus incipientes. Explicatur quomodo unaquaeque curva cognosci et characterizari possit, cum attentione speciali ad elementa clavis sicut vertex, focus et axis symmetriae in parabolis, atque ad distinctionem inter ellipses et hyperbolas secundum signa coefficientium suorum.<br \/>\n<\/em><br \/>\n<strong>Proposita Discendi:<\/strong><br \/>\nHoc capite perfecto discipulus poterit<\/p>\n<ol style=\"text-align:left;\">\n<li><strong>Agoscere<\/strong> aequationes canonicas sectionum conicarum (parabolarum, ellipseon, hyperbolarum)<\/li>\n<li><strong>Computare<\/strong> unamquamque proprietatem sectionum conicarum: longitudinem semiaxium, distantiam focalem, directricem, etc.<\/li>\n<\/ol>\n<p><strong>INDEX CONTENTORUM<\/strong><br \/>\n<a href=\"#1\">Sectiones conicae<\/a><br \/>\n<a href=\"#2\">Recognitio Parabolarum<\/a><br \/>\n<a href=\"#3\">Recognitio Ellipseon et Hyperbolarum<\/a><br \/>\n<a href=\"#4\">Characterizatio Ellipseos<\/a><br \/>\n<a href=\"#5\">Characterizatio Hyperbolae<\/a><br \/>\n<a href=\"#6\">Exercitia Soluta<\/a>\n<\/p>\n<p><\/center><\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/d21_9EHUv_M\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><br \/>\n<a name=\"1\"><\/a><\/p>\n<h2>Sectiones conicae<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=d21_9EHUv_M&amp;t=126s\" target=\"_blank\" rel=\"noopener\"><strong>Sectiones conicae appellantur omnes<\/strong><\/a> curvae quae oriuntur ex sectione superficiei coni plano facto. Familia sectionum conicarum constat ex circulis et ellipsibus, et hyperbolis: omnes sunt curvae quas iam investigavimus.<\/p>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-5eckvpNxzlg\/YJJimcxkMYI\/AAAAAAAAFEA\/dfGTvXblcD4dZXSjpWvonYFN8O0EMNqtwCLcBGAsYHQ\/s0\/curvas-conicas-secciones-cono.png\" alt=\"Sectiones Conicae\" class=\" aligncenter lazyload\" width=\"531\" height=\"272\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-5eckvpNxzlg\/YJJimcxkMYI\/AAAAAAAAFEA\/dfGTvXblcD4dZXSjpWvonYFN8O0EMNqtwCLcBGAsYHQ\/s0\/curvas-conicas-secciones-cono.png\" alt=\"Sectiones Conicae\" class=\" aligncenter lazyload\" width=\"531\" height=\"272\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">Nunc recensionem faciemus de technicis quibus unaquaeque harum curvarum agnosci et characterizari potest. In formis canonicis specialiter incumbemus, quia hae formae saepissime occurrunt atque minime informationis explicite exhibent. Aequationes generales, contra, fere totam characterisationem geometricam manifestant.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Recognitio Parabolarum<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=d21_9EHUv_M&amp;t=160s\" target=\"_blank\" rel=\"noopener\"><strong>Omnis parabola repraesentatur<\/strong><\/a> per aequationem huius formae<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">y=ax^2 + bx + c,<\/span> cum <span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span>\n<p style=\"text-align: justify;\">Ex hac forma obtinuimus<\/p>\n<ul style=\"text-align: justify;\">\n<li>Coordinatae Verticis: <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle (x_0, y_0)=\\left( -\\dfrac{b}{2a}, c - \\dfrac{b^2}{4a} \\right)<\/span><\/li>\n<li>Positio focalis: <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle f=\\dfrac{1}{4a}<\/span><\/li>\n<li>Coordinatae foci: <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle focus=\\left( -\\dfrac{b}{2a}, c - \\dfrac{b^2}{4a} + f \\right) =\\left( -\\dfrac{b}{2a}, c + \\dfrac{1- b^2}{4a} \\right)<\/span><\/li>\n<li>Aequatio directricis: <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle y= c - \\dfrac{b^2}{4a} - f = c - \\dfrac{1+b^2}{4a}<\/span><\/li>\n<li>Aequatio axis symmetriae: <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle x= -\\dfrac{b}{2a} <\/span><\/li>\n<li>Intersections cum axe x (si existunt): <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle x_{1,2}= \\dfrac{-b \\pm \\sqrt{b^2-4ac}}{2a} <\/span><\/li>\n<\/ul>\n<p style=\"text-align: justify;\">Et his iam habemus omnem informationem necessariam ad graphice repraesentandam quamlibet parabolam.<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Recognitio Ellipseon et Hyperbolarum<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=d21_9EHUv_M&amp;t=449s\" target=\"_blank\" rel=\"noopener\"><strong>Ellipses et hyperbolae, ut vidimus,<\/strong><\/a> habent expressionem canonicam huiusmodi formae.<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">Ax^2 + Bx + Cy^2 + Dy + E = 0<\/span>\n<p style=\"text-align: justify;\">Ubi <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">C<\/span> sunt constantes a zero distinctae, et ex iis quae studuimus, habetur quod:<\/p>\n<ul style=\"text-align: justify;\">\n<li>Si <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">C<\/span> idem signum habent, tunc est ellipsis.<\/li>\n<li>Si <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">C<\/span> signa contraria habent, tunc est hyperbola.<\/li>\n<\/ul>\n<p style=\"text-align: justify;\">Ad casus utrumque clare separandos, scribemus:<\/p>\n<ul style=\"text-align: justify;\">\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha x^2+ \\beta x + \\gamma y^2 + \\delta y + \\epsilon = 0<\/span> est ellipsis.<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha x^2+ \\beta x - \\gamma y^2 + \\delta y + \\epsilon = 0<\/span> est hyperbola.<\/li>\n<\/ul>\n<p style=\"text-align: justify;\">Ubi <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha, \\beta, \\gamma, \\delta<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon<\/span> sunt numeri reales quicumque, et <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span> semper positi sunt. Sic scribere nobis permittit casus ambos clare distinguere. Ex hoc possumus sequentia concludere:<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Characterizatio Ellipseos<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=d21_9EHUv_M&amp;t=552s\" target=\"_blank\" rel=\"noopener\"><strong>Partiendo ab aequatione canonica<\/strong><\/a> habemus deductionem sequentem:<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td width=\"50\">(1)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha x^2+ \\beta x + \\gamma y^2 + \\delta y + \\epsilon = 0<\/span><\/td>\n<td>; aequatio canonica ellipsium.<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\alpha \\left( x^2+ \\dfrac{\\beta}{\\alpha }x\\right) + \\gamma \\left(y^2 + \\dfrac{\\delta}{\\gamma }y\\right) =- \\epsilon<\/span><\/td>\n<td>; factorando et terminos regruppando<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\alpha \\left( x + \\dfrac{\\beta}{2 \\alpha }\\right)^2 + \\gamma \\left(y + \\dfrac{\\delta}{2 \\gamma } \\right)^2 =\\dfrac{\\beta^2}{4\\alpha } + \\dfrac{\\delta^2}{4\\gamma } - \\epsilon<\/span><\/td>\n<td>; quadrata complendo et terminos regruppando<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\alpha \\dfrac{\\left( x + \\dfrac{\\beta}{2 \\alpha }\\right)^2}{\\left(\\dfrac{\\beta^2}{4\\alpha } + \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)} + \\gamma \\dfrac{\\left(y + \\dfrac{\\delta}{2 \\gamma } \\right)^2}{\\left(\\dfrac{\\beta^2}{4\\alpha } + \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)} = 1<\/span><\/td>\n<td>; dividendo omnia per <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\dfrac{\\beta^2}{4\\alpha } + \\dfrac{\\delta^2}{4\\gamma } - \\epsilon<\/span><\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\dfrac{\\left( x + \\dfrac{\\beta}{2 \\alpha }\\right)^2}{\\dfrac{1}{\\alpha }\\left(\\dfrac{\\beta^2}{4\\alpha } + \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)} + \\dfrac{\\left(y + \\dfrac{\\delta}{2 \\gamma } \\right)^2}{\\dfrac{1}{ \\gamma}\\left(\\dfrac{\\beta^2}{4\\alpha } + \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)} = 1<\/span><\/td>\n<td>; reordinando <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span><\/td>\n<\/tr>\n<tr>\n<td>(6)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left( \\dfrac{ x - \\left(-\\dfrac{\\beta}{2 \\alpha }\\right)}{\\sqrt{\\dfrac{1}{\\alpha}\\left(\\dfrac{\\beta^2}{4\\alpha } + \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)}}\\right)^2 + \\left( \\dfrac{y - \\left(-\\dfrac{\\delta}{2 \\gamma } \\right)}{\\sqrt{\\dfrac{1}{\\gamma}\\left(\\dfrac{\\beta^2}{4\\alpha } + \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)}}\\right)^2 = 1<\/span><\/td>\n<td>; restruendo per radices<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">In hac deductione gradus (3) est praesertim delicatus, quia si coefficientis <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\dfrac{\\beta^2}{4\\alpha } + \\dfrac{\\delta^2}{4\\gamma } - \\epsilon<\/span> valor est negativus, tunc ellipsis exsistere non potest.<\/p>\n<p style=\"text-align: justify;\">Recordemur aequationem generalem ellipsium huius formae esse:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left( \\dfrac{x-h}{a} \\right)^2 + \\left(\\dfrac{y-k}{b} \\right)^2 = 1<\/span>\n<p style=\"text-align: justify;\">Hoc ultimo resultato habemus nunc relationem directam inter parametra formulae generalis quae permittit nobis totam informationem contentam in expressione canonica manifestare:<\/p>\n<ul style=\"text-align: justify;\">\n<li>Coordinatae centri: <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle (h,k) = \\left( -\\dfrac{\\beta}{2\\alpha}, -\\dfrac{\\delta}{2\\gamma}\\right)<\/span><\/li>\n<li>Longitudo semiaxis horizontalis: <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle a = \\sqrt{\\dfrac{1}{\\alpha}\\left(\\dfrac{\\beta^2}{4\\alpha } + \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)}<\/span><\/li>\n<li>Longitudo semiaxis verticalis: <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle b = \\sqrt{\\dfrac{1}{\\gamma}\\left(\\dfrac{\\beta^2}{4\\alpha } + \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)}<\/span><\/li>\n<\/ul>\n<p style=\"text-align: justify;\">His iam possibile est ellipsin agnoscere et graphice repraesentare directe ex forma canonica. Figura eius talis videbitur:<\/p>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-aVD7YQ7DfL0\/YJKBv9QXeTI\/AAAAAAAAFEQ\/urCuFtrn-YYBQ_fVSGXsmhMqExFumag-ACLcBGAsYHQ\/s0\/elipse.PNG\" class=\" aligncenter lazyload\" width=\"414\" height=\"291\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-aVD7YQ7DfL0\/YJKBv9QXeTI\/AAAAAAAAFEQ\/urCuFtrn-YYBQ_fVSGXsmhMqExFumag-ACLcBGAsYHQ\/s0\/elipse.PNG\" class=\" aligncenter lazyload\" width=\"414\" height=\"291\" \/><\/noscript><\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Characterizatio Hyperbolae<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=d21_9EHUv_M&amp;t=911s\" target=\"_blank\" rel=\"noopener\"><strong>Ratione omnino simili utens potes,<\/strong><\/a> ab aequatione canonica, characterisationem completam hyperbolarum efficere. Re vera, analysis tam similis est ut exemplum ellipseos hic applicetur, cum paucis mutationibus.<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td width=\"50\">(1)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha x^2+ \\beta x - \\gamma y^2 + \\delta y + \\epsilon = 0<\/span><\/td>\n<td>; aequatio canonica hyperbolarum.<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\alpha \\left( x^2+ \\dfrac{\\beta}{\\alpha }x\\right) - \\gamma \\left(y^2 - \\dfrac{\\delta}{\\gamma }y\\right) =- \\epsilon<\/span><\/td>\n<td>; factorando et terminos regruppando<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\alpha \\left( x + \\dfrac{\\beta}{2 \\alpha }\\right)^2 - \\gamma \\left(y - \\dfrac{\\delta}{2 \\gamma } \\right)^2 =\\dfrac{\\beta^2}{4\\alpha } - \\dfrac{\\delta^2}{4\\gamma } - \\epsilon<\/span><\/td>\n<td>; quadrata complendo et terminos regruppando<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\alpha \\dfrac{\\left( x + \\dfrac{\\beta}{2 \\alpha }\\right)^2}{\\left(\\dfrac{\\beta^2}{4\\alpha } - \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)} - \\gamma \\dfrac{\\left(y - \\dfrac{\\delta}{2 \\gamma } \\right)^2}{\\left(\\dfrac{\\beta^2}{4\\alpha } - \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)} = 1<\/span><\/td>\n<td>; dividendo omnia per <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\dfrac{\\beta^2}{4\\alpha } - \\dfrac{\\delta^2}{4\\gamma } - \\epsilon<\/span><\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\dfrac{\\left( x + \\dfrac{\\beta}{2 \\alpha }\\right)^2}{\\dfrac{1}{\\alpha}\\left(\\dfrac{\\beta^2}{4\\alpha } - \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)} - \\dfrac{\\left(y - \\dfrac{\\delta}{2 \\gamma } \\right)^2}{\\dfrac{1}{\\gamma}\\left(\\dfrac{\\beta^2}{4\\alpha } - \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)} = 1<\/span><\/td>\n<td>; reordinando terminos <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span><\/td>\n<\/tr>\n<tr>\n<td>(6)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left( \\dfrac{ x - \\left(-\\dfrac{\\beta}{2 \\alpha }\\right)}{\\sqrt{\\dfrac{1}{\\alpha}\\left(\\dfrac{\\beta^2}{4\\alpha } - \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)}}\\right)^2 - \\left( \\dfrac{y - \\left(\\dfrac{\\delta}{2 \\gamma } \\right)}{\\sqrt{\\dfrac{1}{\\gamma}\\left(\\dfrac{\\beta^2}{4\\alpha } - \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)}}\\right)^2 = 1<\/span><\/td>\n<td>; restruendo per radices<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Ex hoc nunc habemus relationem directam inter aequationem canonicam et aequationem hyperbolarum, quae nobis permittit celeriter figuram earum depingere.<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left(\\dfrac{x-h}{a} \\right)^2 - \\left(\\dfrac{y-k}{b} \\right)^2 =1 <\/span>\n<p style=\"text-align: justify;\">Nunc autem, dissimiliter ab ellipsibus, hic rectius est loqui de \u00abarca generatrice\u00bb, ut in sequenti figura demonstrabitur:<\/p>\n<ul style=\"text-align: justify;\">\n<li>Coordinatae centri: <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle (h,k) = \\left( -\\dfrac{\\beta}{2\\alpha}, \\dfrac{\\delta}{2\\gamma}\\right)<\/span><\/li>\n<li>Longitudo semiaxis horizontalis: <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle a = \\sqrt{\\dfrac{1}{\\alpha}\\left(\\dfrac{\\beta^2}{4\\alpha } - \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)}<\/span><\/li>\n<li>Longitudo semiaxis verticalis: <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle b = \\sqrt{\\dfrac{1}{\\gamma}\\left(\\dfrac{\\beta^2}{4\\alpha } - \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)}<\/span><\/li>\n<\/ul>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-bd0n_BuEFiE\/YJKJ1fPDhMI\/AAAAAAAAFEY\/-QjR2QbycSkKJihjHnwmdIDESYgNDyuBgCLcBGAsYHQ\/s0\/hiperbola.PNG\" alt=\"Hyperbola\" class=\" aligncenter lazyload\" width=\"428\" height=\"305\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-bd0n_BuEFiE\/YJKJ1fPDhMI\/AAAAAAAAFEY\/-QjR2QbycSkKJihjHnwmdIDESYgNDyuBgCLcBGAsYHQ\/s0\/hiperbola.PNG\" alt=\"Hyperbola\" class=\" aligncenter lazyload\" width=\"428\" height=\"305\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">Ex his analysibus iam possumus quemlibet membrum familiae sectionum conicarum sine ulla difficultate graphice repraesentare.<\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h2>Exercitia Soluta<\/h2>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/PKjQrcC0HG4\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Sectiones Conicae: Characterizatio et Graphice Parabolarum, Ellipseon et Hyperbolarum Summarium: In hac lectione sectiones conicas (parabolas, ellipses et hyperbolas) recognoscemus, ab eorum aequationibus canonicis et generalibus incipientes. Explicatur quomodo unaquaeque curva cognosci et characterizari possit, cum attentione speciali ad elementa clavis sicut vertex, focus et axis symmetriae in parabolis, atque ad distinctionem inter ellipses et [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":29020,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":13,"footnotes":""},"categories":[1304,1298],"tags":[],"class_list":["post-33746","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-algebra-et-geometria","category-mathematica"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Sectiones Conicae: Characterizatio et Repraesentatio Graphica Parabolarum, Ellipseon et Hyperbolarum - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Sectiones conicae sunt curvae quae ex intersectione coni cum plano oriuntur; 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