{"id":33722,"date":"2021-04-28T13:00:53","date_gmt":"2021-04-28T13:00:53","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=33722"},"modified":"2025-07-30T20:54:36","modified_gmt":"2025-07-30T20:54:36","slug":"aequatio-ellipsium-et-circumferentiarum","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/la\/aequatio-ellipsium-et-circumferentiarum\/","title":{"rendered":"\u00c6quatio Ellipsium et Circumferentiarum"},"content":{"rendered":"<p><center><\/p>\n<h1>\u00c6quatio Ellipsium et Circumferentiarum<\/h1>\n<p><em><strong>Summarium:<\/strong><br \/>\n   In hac lectione exponitur quomodo \u00e6quatio ellipsium ex definitione geometrica obtineatur, qu\u00e6 statuit summam distantiarum cuiuslibet puncti elliptici ad duo puncta fixa, qu\u00e6 foci appellantur, esse constantem. Per explicatum processum algebraicum, deducitur \u00e6quatio generalis ellipsium eiusque forma canonica, necnon nexus inter ellipses et circumferentias, demonstrando circumferentiam esse casum particularem ellipsis cum semiaxes sunt \u00e6quales.<br \/>\n   <\/em><\/p>\n<p>   <strong>Propositi Discendi:<\/strong><br \/>\n   Hac lectione confecta, discipulus poterit<\/p>\n<ol style=\"text-align:left;\">\n<li><strong>Deducere<\/strong> \u00e6quationem ellipsium ex eorum definitione geometrica.<\/li>\n<li><strong>Agnoscere<\/strong> formam generalem et formam canonicam \u00e6quationis ellipsium.<\/li>\n<\/ol>\n<p>   INDEX RERUM<br \/>\n   <a href=\"#1\">Formatio geometrica<\/a><br \/>\n   <a href=\"#2\">\u00c6quationis ellipsium derivatio<\/a><br \/>\n   <a href=\"#3\">\u00c6quatio generalis ellipsium<\/a><br \/>\n   <a href=\"#4\">\u00c6quatio canonica ellipsium<\/a><br \/>\n   <a href=\"#5\">Reductio ad \u00c6quationem Circumferentiarum<\/a>\n   <\/p>\n<p>   <\/center><\/p>\n<p>   <center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/HHiC0bp-Vyc\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><br \/>\n   <a name=\"1\"><\/a><\/p>\n<h2>Formatio geometrica<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=HHiC0bp-Vyc&amp;t=133s\" target=\"_blank\" rel=\"noopener\"><strong>Ut \u00e6quationem ellipsium exprimamus,<\/strong> <\/a>ratiocinari debemus, ut in parabolis, de earum significatione geometrica. Ellipsis est collectio omnium punctorum in plano, talium ut summa distantiarum inter illa et duo alia puncta, qu\u00e6 foci nominantur, sit semper eadem.<\/p>\n<p>   <img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-rHroj77w4-o\/YIhoGfTvE_I\/AAAAAAAAFAw\/2Yoa3Q2yrmknQMPObDz8wuyDoOehCug5QCLcBGAsYHQ\/s0\/elipse.PNG\" alt=\"Ellipsis\" class=\" aligncenter lazyload\" width=\"338\" height=\"241\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-rHroj77w4-o\/YIhoGfTvE_I\/AAAAAAAAFAw\/2Yoa3Q2yrmknQMPObDz8wuyDoOehCug5QCLcBGAsYHQ\/s0\/elipse.PNG\" alt=\"Ellipsis\" class=\" aligncenter lazyload\" width=\"338\" height=\"241\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">Id est, verificabitur quod:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">d(f_1,p) + d(f_2,p) = constans<\/span>\n<p>   <a name=\"2\"><\/a><\/p>\n<h2>\u00c6quationis ellipsium derivatio<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=HHiC0bp-Vyc&amp;t=311s\" target=\"_blank\" rel=\"noopener\"><strong>Ex definitione geometrica<\/strong><\/a> ellipsium exprimi potest \u00e6quatio algebraica qu\u00e6 eam describit. Hoc autem facilius perfici potest per quasdam simplificationes. Sine detrimento generalitatis, supponamus foci sitos esse in <span class=\"katex-eq\" data-katex-display=\"false\">f_1 =(-c,0)<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">f_2 =(c,0),<\/span> ita ut, si punctum aliquod <span class=\"katex-eq\" data-katex-display=\"false\">p=(x,y)<\/span> ad ellipsim pertinet, tunc verificabitur quod<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sqrt{(x+c)^2 + y^2} + \\sqrt{(x-c)^2 + y^2} = 2a<\/span>\n<p>   <img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-LtAamnh5D78\/YIiBshjM70I\/AAAAAAAAFA4\/hGiHx6jf_nMOOUHfH-Ywj34TyDJDGEv-wCLcBGAsYHQ\/s0\/ecuacion%2Bde%2Blas%2Belipses.PNG\" alt=\"\u00c6quatio Ellipsium\" class=\" aligncenter lazyload\" width=\"412\" height=\"333\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-LtAamnh5D78\/YIiBshjM70I\/AAAAAAAAFA4\/hGiHx6jf_nMOOUHfH-Ywj34TyDJDGEv-wCLcBGAsYHQ\/s0\/ecuacion%2Bde%2Blas%2Belipses.PNG\" alt=\"\u00c6quatio Ellipsium\" class=\" aligncenter lazyload\" width=\"412\" height=\"333\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">Ubi <span class=\"katex-eq\" data-katex-display=\"false\">a\\in\\mathbb{R}<\/span> est constantia fixa. Ex hoc ratiocinatio sequens constitui potest<\/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\">\\sqrt{(x+c)^2 + y^2} + \\sqrt{(x-c)^2 + y^2} = 2a<\/span><\/td>\n<td>; Definitio geometrica ellipsis<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\sqrt{(x-c)^2 + y^2} = 2a - \\sqrt{(x+c)^2 + y^2}<\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(x-c)^2 + \\cancel{y^2} = 4a^2 -4a \\sqrt{(x+c)^2 + y^2} + (x+c)^2 + \\cancel{y^2}<\/span><\/td>\n<td>; Elevando ad quadratum (1)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(x-c)^2 = 4a^2 -4a \\sqrt{(x+c)^2 + y^2} + (x+c)^2 <\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\cancel{x^2} -2xc + \\cancel{c^2} = 4a^2 -4a \\sqrt{(x+c)^2 + y^2} + \\cancel{x^2} +2xc + \\cancel{c^2} <\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">-2xc = 4a^2 -4a \\sqrt{(x+c)^2 + y^2} +2xc <\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">4a \\sqrt{(x+c)^2 + y^2} = 4a^2 +4xc = 4(a^2 + xc) <\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">a \\sqrt{(x+c)^2 + y^2} = a^2 + xc <\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">a^2 [(x+c)^2 + y^2] = (a^2 + xc)^2 <\/span><\/td>\n<td>; Elevando ad quadratum (2)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">a^2 [x^2 + 2xc + c2 + y^2] = a^4 +2a^2xc + x^2c^2 <\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> a^2 x^2 + \\cancel{2xca^2} + a^2 c2 + a^2 y^2 = a^4 + \\cancel{2a^2xc} + x^2c^2 <\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> a^2 x^2 + a^2 c2 + a^2 y^2 = a^4 + x^2c^2 <\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> x^2 (a^2 - c^2) + a^2 y^2 = a^4 - a^2 c^2 =a^2(a^2-c^2) <\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> \\dfrac{x^2}{a^2} +\\dfrac{ y^2}{a^2-c^2} = 1 <\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">0\\lt a^2 - c^2 =: b^2 <\/span><\/td>\n<td>; Numerus a <span class=\"katex-eq\" data-katex-display=\"false\">b^2<\/span> repraesentatus est positivus, ut ex imagine apparet.<\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">{\\dfrac{x^2}{a^2} +\\dfrac{ y^2}{b^2} = 1}<\/span><\/td>\n<td>; Ex (3) et (4)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\left(\\dfrac{x}{a}\\right)^2 + \\left(\\dfrac{y}{b}\\right)^2 = 1}<\/span><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center;\">Hoc ultimum est quod \u00ab\u00e6quatio ellipsium\u00bb appellatur.<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>\u00c6quatio generalis ellipsium<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=HHiC0bp-Vyc&amp;t=706s\" target=\"_blank\" rel=\"noopener\"><strong>\u00c6quatio quam modo obtinuimus<\/strong><\/a> ad formam generalem reduci potest per transformationes translationis, faciendo substitutiones <span class=\"katex-eq\" data-katex-display=\"false\">x\\longmapsto (x-h)<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">y\\longmapsto (y-k).<\/span> Hoc modo pervenimus ad formam generalem \u00e6quationis ellipsium<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\left(\\dfrac{x-h}{a}\\right)^2 + \\left(\\dfrac{y-k}{b}\\right)^2 = 1}<\/span>\n<p style=\"text-align: justify;\">H\u00e6c est ellipsis cui centrum est in puncto <span class=\"katex-eq\" data-katex-display=\"false\">(h,k)<\/span>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-lkxt91FvMTs\/YIiQaL9wpII\/AAAAAAAAFBA\/sUxc6ajd6tcPymC8g4oh3M0l2CTI-xOvgCLcBGAsYHQ\/s0\/elipsegeneral.PNG\" alt=\"elipsis generalis\" class=\" aligncenter lazyload\" width=\"469\" height=\"373\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-lkxt91FvMTs\/YIiQaL9wpII\/AAAAAAAAFBA\/sUxc6ajd6tcPymC8g4oh3M0l2CTI-xOvgCLcBGAsYHQ\/s0\/elipsegeneral.PNG\" alt=\"elipsis generalis\" class=\" aligncenter lazyload\" width=\"469\" height=\"373\" \/><\/noscript><br \/>\n<a name=\"4\"><\/a><\/p>\n<h2>\u00c6quatio canonica ellipsium<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=HHiC0bp-Vyc&amp;t=761s\" target=\"_blank\" rel=\"noopener\"><strong>Per operationes algebraicas<\/strong><\/a> hinc pervenitur ad \u00e6quationem canonicam ellipsium:<\/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\">\\left(\\dfrac{x-h}{a}\\right)^2 + \\left(\\dfrac{y-k}{b}\\right)^2 = 1<\/span><\/td>\n<td>; \u00e6quatio generalis ellipsium<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">b^2 (x-h)^2 + a^2(y-k)^2 = a^2 b^2<\/span><\/td>\n<td>; Multiplicatio totius per <span class=\"katex-eq\" data-katex-display=\"false\">a^2b^2<\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">b^2 [x^2-2xh+h^2] + a^2[y^2-2yk + k^2] = a^2 b^2<\/span><\/td>\n<td>; evolutio quadratorum<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> b^2 x^2-2hb^2 x + h^2b^2 + a^2 y^2-2ka^2y + k^2a^2 = a^2 b^2<\/span><\/td>\n<td>; explicatio parenthesium<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> b^2 x^2- 2hb^2 x + a^2 y^2-2ka^2y +(h^2b^2 + k^2a^2 - a^2 b^2) = 0 <\/span><\/td>\n<td>; aggregatio terminorum constantium<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">In hac ultima expressione fieri possunt substitutiones <span class=\"katex-eq\" data-katex-display=\"false\">A:=b^2,<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">B:=-2hb^2,<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">C:=a^2,<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">D:=-2ka^2<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">E:=h^2b^2 + k^2a^2 - a^2 b^2.<\/span> Ita videre possumus ellipses describi per \u00e6quationes huiusmodi<\/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;\">Hoc est quod \u00ab\u00e6quatio canonica ellipsium\u00bb appellatur.<\/p>\n<p style=\"text-align: justify;\">Ex his deductionibus extrahi possunt qu\u00e6dam restrictiones circa constantes \u00e6quationis canonic\u00e6. Maxima momenti est quod <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">B<\/span> eandem significationem habere debent; si aliter, non iam de ellipsi, sed de hyperbola agitur. Sunt etiam ali\u00e6 conditiones in constantibus huius repraesentationis canonic\u00e6, sed nunc eas tractare non est opportunum: eas exacte tractabimus cum ad ellipsium et hyperbolarum characterizationem pervenerimus.<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Reductio ad \u00c6quationem Circumferentiarum<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=HHiC0bp-Vyc&amp;t=948s\" target=\"_blank\" rel=\"noopener\"><strong>Aliquid quod tractabimus cum disseremus<\/strong><\/a> de characterizatione ellipsium est quod constantes <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> in \u00e6quatione generali ellipsis correspondent semiaxibus eius. Si utrumque semiaxem \u00e6quamus, scilicet <span class=\"katex-eq\" data-katex-display=\"false\">a=b=r,<\/span> tunc ellipsis in circumferentiam radii <span class=\"katex-eq\" data-katex-display=\"false\">r<\/span> convertetur.<\/p>\n<h3>\u00c6quatio generalis circumferentiarum<\/h3>\n<p style=\"text-align: justify;\">Hoc modo obtinetur \u00e6quatio generalis circumferentiarum, nempe:<\/p>\n<p style=\"text-align: justify;\"><span class=\"katex-eq\" data-katex-display=\"false\">(x-h)^2 + (y-k)^2 = r^2<\/span>\n<h3>\u00c6quatio canonica circumferentiarum<\/h3>\n<p style=\"text-align: justify;\">Similiter, obtinetur \u00e6quatio canonica circumferentiarum<\/p>\n<p style=\"text-align: justify;\"><span class=\"katex-eq\" data-katex-display=\"false\">Ax^2 + Bx + Cy^2 + Dy + E = 0<\/span>\n<p style=\"text-align: justify;\">In sua forma canonica cum ellipsibus congruit, quia, ut iam vidimus, circumferenti\u00e6 sunt casus particularis ellipsis.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u00c6quatio Ellipsium et Circumferentiarum Summarium: In hac lectione exponitur quomodo \u00e6quatio ellipsium ex definitione geometrica obtineatur, qu\u00e6 statuit summam distantiarum cuiuslibet puncti elliptici ad duo puncta fixa, qu\u00e6 foci appellantur, esse constantem. Per explicatum processum algebraicum, deducitur \u00e6quatio generalis ellipsium eiusque forma canonica, necnon nexus inter ellipses et circumferentias, demonstrando circumferentiam esse casum particularem ellipsis [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":28959,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":1,"footnotes":""},"categories":[1304,1298],"tags":[],"class_list":["post-33722","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>\u00c6quatio Ellipsium et Circumferentiarum - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Disce quomodo \u00e6quationem ellipsium deducere possis eiusque nexum cum circumferentiis intelligere. 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