{"id":34540,"date":"2021-10-06T13:00:21","date_gmt":"2021-10-06T13:00:21","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=34540"},"modified":"2025-09-08T05:49:42","modified_gmt":"2025-09-08T05:49:42","slug":"refractio-in-interfaciebus-sphaericis","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/la\/refractio-in-interfaciebus-sphaericis\/","title":{"rendered":"Refractio in Interfaciebus Sphaericis"},"content":{"rendered":"<p><center><\/p>\n<h1>Refractio in Interfaciebus Sphaericis<\/h1>\n<p><em><strong>Summarium:<\/strong><br \/>\nIn hac lectione perpendemus Refrationem in Interfaciebus Sphaericis, ostendentes quomodo lux se habeat transiens per superficies sphaericas et quomodo imagines formantur. Exhibentur aequationes praecipuae ad locum et magnitudinem imaginum computanda. Item explorantur casus practici, ut lentes et aestimatio profunditatum apparentium.<\/em><\/p>\n<p><strong>Proposita Discendi:<\/strong><br \/>\nPost hanc lectionem discipulus poterit<\/p>\n<ol style=\"text-align:left;\">\n<li><strong>Intellegere<\/strong> refractionem lucis transeuntis per interfacies sphaericas.<\/li>\n<li><strong>Deducere<\/strong> et adhibere relationem obiecti-imaginis pro interfaciebus sphaericis.<\/li>\n<li><strong>Applicare<\/strong> legem Snell in contextu interfaciis sphaericis.<\/li>\n<li><strong>Determinare<\/strong> situm imaginis a superficie sphaerica formatae.<\/li>\n<li><strong>Computare<\/strong> magnificationem imaginis per refractionem in superficiebus sphaericis.<\/li>\n<li><strong>Intellegere<\/strong> conventionem signorum pro positione et magnitudine obiectuum atque imaginum.<\/li>\n<li><strong>Referre<\/strong> interfacies sphaericas ad interfacies planas tamquam casum limitis.<\/li>\n<li><strong>Analyzare<\/strong> formationem imaginum extensarum per interfacies sphaericas.<\/li>\n<\/ol>\n<p><strong>INDEX CONTENTORUM<\/strong><br \/>\n<a href=\"#1\"><strong>Introductio<\/strong><\/a><br \/>\n<a href=\"#2\"><strong>Relatio obiecti-imaginis pro refractione in interfaciebus sphaericis<\/strong><\/a><br \/>\n<a href=\"#3\">Excerpere relationes inter angulos<\/a><br \/>\n<a href=\"#4\">Introducere legem Snell<\/a><br \/>\n<a href=\"#5\"><strong>Formatio imaginum extensarum per refractionem ad alteram partem interfaciis sphaericis<\/strong><\/a><br \/>\n<a href=\"#6\">Synopsis<\/a><br \/>\n<a href=\"#7\"><strong>Interfacies planae ut casus limes sphaericarum<\/strong><\/a><br \/>\n<a href=\"#8\"><strong>Exercitia<\/strong><\/a><\/p>\n<p><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/13kntUA9n-I\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\n<\/center><br \/>\n<a name=\"1\"><\/a><\/p>\n<h2>Introductio<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=13kntUA9n-I&amp;t=146s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Iam studuimus quomodo refractionis operetur;<\/span><\/strong><\/a> id est, quid accidat cum lux ex uno medio in aliud transeat. Sed hoc totum egimus in casu quo interfacies quae media separat est superficies plana. Attamen, tam in natura quam in applicationibus practicis, non difficile est invenire processus refractionis in interfaciebus sphaericis. Exempla horum sunt oculus humanus (atque fere cuiuslibet animalis revera) et pleraque instrumenta optica adhibita sive in vita cotidiana sive in usibus industrialibus.<\/p>\n<p style=\"text-align: justify;\">In sequenti figura monstratur quomodo lens ex duabus superficiebus sphaericis construatur.<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-xuCmdVLtNcc\/YVuCvXoOfZI\/AAAAAAAAFmc\/mCuYiHaNfxM4I6RcxEJcLln6uLC5MMIKQCLcBGAsYHQ\/s0\/lente%2Bde%2Bvidrio.PNG\" width=\"661\" height=\"378\" alt=\"Lente vitrea ex duabus superficiebus sphaericis formata\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-xuCmdVLtNcc\/YVuCvXoOfZI\/AAAAAAAAFmc\/mCuYiHaNfxM4I6RcxEJcLln6uLC5MMIKQCLcBGAsYHQ\/s0\/lente%2Bde%2Bvidrio.PNG\" width=\"661\" height=\"378\" alt=\"Lente vitrea ex duabus superficiebus sphaericis formata\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align: justify;\">Ad studium accuratum huius generis instrumentorum necesse est investigare quomodo lux se habeat cum ex uno medio in aliud per interfaciem sphaericam transeat.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Relatio obiecti-imaginis pro refractione in interfaciebus sphaericis<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=13kntUA9n-I&amp;t=235s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Initium studii nostri faciemus investigando<\/span><\/strong><\/a> quomodo lux se habeat transeundo ex uno medio in aliud per interfaciem sphaericam. Ad hoc considerabimus sphaeram radii <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">R<\/span><\/span> factam ex materia cuius index refractionis est <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_b<\/span><\/span> immersam in medio cuius index refractionis est <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_a.<\/span><\/span><\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-x0g44iZ6LDo\/YVuDO2nkD3I\/AAAAAAAAFmk\/8gQHFBnQxvEG-5B6XUAgTTZHtxntx6YogCLcBGAsYHQ\/s0\/inteface%2Besf%25C3%25A9rica.PNG\" width=\"987\" height=\"371\" alt=\"Interfacies sphaerica quae duo media separat\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-x0g44iZ6LDo\/YVuDO2nkD3I\/AAAAAAAAFmk\/8gQHFBnQxvEG-5B6XUAgTTZHtxntx6YogCLcBGAsYHQ\/s0\/inteface%2Besf%25C3%25A9rica.PNG\" width=\"987\" height=\"371\" alt=\"Interfacies sphaerica quae duo media separat\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><br \/>\n<a name=\"3\"><\/a><\/p>\n<h3>Excerpere relationes inter angulos<\/h3>\n<p style=\"text-align: justify;\">Si angulos in hac figura implicatos perpendamus animadvertimus:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n{(1)}&amp; \\theta_a &amp; =\\alpha + \\phi \\\\ \\\\\n\n{(2)}&amp; \\phi &amp; =\\beta + \\theta_b\n\n\\end{array}<\/span><\/span><\/p>\n<h4>Demonstratio<\/h4>\n<p style=\"text-align: justify;\">Prima aequatio obtinetur ex eo quod summa angulorum internorum trianguli aequatur duobus angulis rectis:<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-0ILYFKDJRo0\/YVvJaKwoZxI\/AAAAAAAAFms\/arUYkIwSz18wvNntSE_WHuknjoYolOXSQCLcBGAsYHQ\/s0\/triangulorectangulo-optica.PNG\" width=\"346\" height=\"148\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-0ILYFKDJRo0\/YVvJaKwoZxI\/AAAAAAAAFms\/arUYkIwSz18wvNntSE_WHuknjoYolOXSQCLcBGAsYHQ\/s0\/triangulorectangulo-optica.PNG\" width=\"346\" height=\"148\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n&amp; \\alpha + \\phi + (\\pi - \\theta_a) = \\pi\\\\ \\\\\n\n\\equiv &amp; \\alpha + \\phi - \\theta_a = 0 \\\\ \\\\\n\n\\equiv &amp; \\color{blue}{\\theta_a = \\alpha + \\phi}\n\n\\end{array}<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Secunda analogice obtinetur:<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-R5siuLq4gJs\/YVvJ22k7m6I\/AAAAAAAAFm0\/D6gmIAHjXFY4cNmUwrnq4yvxQ_2PcBSjgCLcBGAsYHQ\/s0\/triangulorectangulo-optica2.PNG\" width=\"329\" height=\"216\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-R5siuLq4gJs\/YVvJ22k7m6I\/AAAAAAAAFm0\/D6gmIAHjXFY4cNmUwrnq4yvxQ_2PcBSjgCLcBGAsYHQ\/s0\/triangulorectangulo-optica2.PNG\" width=\"329\" height=\"216\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n&amp; \\beta + \\theta_b + (\\pi - \\phi) = \\pi\\\\ \\\\\n\n\\equiv &amp;  \\beta + \\theta_b - \\phi = 0\\\\ \\\\\n\n\\equiv &amp; \\color{blue}{\\phi = \\beta + \\theta_b }\n\n\\end{array}<\/span><\/span><\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h3>Introducens legem Snell<\/h3>\n<p style=\"text-align: justify;\">Ex figura etiam habentur sequentia enuntiata:<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-x0g44iZ6LDo\/YVuDO2nkD3I\/AAAAAAAAFmk\/8gQHFBnQxvEG-5B6XUAgTTZHtxntx6YogCLcBGAsYHQ\/s0\/inteface%2Besf%25C3%25A9rica.PNG\" width=\"987\" height=\"371\" alt=\"Interfacies sphaerica quae duo media separat\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-x0g44iZ6LDo\/YVuDO2nkD3I\/AAAAAAAAFmk\/8gQHFBnQxvEG-5B6XUAgTTZHtxntx6YogCLcBGAsYHQ\/s0\/inteface%2Besf%25C3%25A9rica.PNG\" width=\"987\" height=\"371\" alt=\"Interfacies sphaerica quae duo media separat\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n{(3)}&amp;\\tan(\\alpha) &amp;=\\displaystyle \\frac{h}{s+\\delta}\\\\ \\\\\n\n{(4)}&amp;\\tan(\\beta) &amp;=\\displaystyle \\frac{h}{s^\\prime - \\delta}\\\\ \\\\\n\n{(5)}&amp;\\tan(\\phi) &amp;=\\displaystyle \\frac{h}{R - \\delta}\n\n\\end{array}<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Et ex lege Snell habemus<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n{(6)} &amp;  n_a\\sin(\\theta_a) = n_b \\sin(\\theta_b)\\end{array}<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Nunc, si accipimus approximationem qua <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta_a<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta_b<\/span><\/span> sunt parvi, tum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha, \\beta<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi<\/span><\/span> quoque parvi erunt atque accidet ut:<\/p>\n<p style=\"text-align: justify;\">Ex figura etiam habentur sequentia enuntiata:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n\\sin(\\theta_a) &amp;\\approx \\theta_a \\\\ \\\\\n\n\\sin(\\theta_b) &amp;\\approx \\theta_b \\\\ \\\\\n\n\\delta &amp;\\approx 0 \\\\ \\\\\n\n\\tan(\\alpha) &amp;\\approx \\alpha \\\\ \\\\\n\n\\tan(\\beta) &amp;\\approx \\beta \\\\ \\\\\n\n\\tan(\\phi) &amp;\\approx \\phi\n\n\\end{array}<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Deinde, ex hoc et lege Snell habetur:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n{(7)} &amp; n_a \\theta_a \\approx n_b \\theta_b \\\\ \\\\\n\n\\equiv &amp;  \\theta_b \\approx \\displaystyle \\frac{n_a}{n_b} \\theta_a\n\n\\end{array}<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Nunc, ex (7), (1) et (2) habetur<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n{(8)} &amp; \\phi - \\beta \\approx \\displaystyle \\frac{n_a}{n_b}(\\alpha + \\phi) \\\\ \\\\\n\n\\equiv &amp; \\phi \\approx \\beta + \\displaystyle  \\frac{n_a}{n_b}(\\alpha + \\phi) \\\\ \\\\\n\n{}\\equiv &amp;  n_b\\phi \\approx n_b\\beta + n_a \\alpha + n_a\\phi \\\\ \\\\\n\n\\equiv &amp; \\color{blue}{n_a \\alpha + n_b\\beta \\approx (n_b - n_a) \\phi }\n\n\\end{array}<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Denique, ex (8), approximationibus et aequationibus (3), (4) et (5), pervenitur ad:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n{(9)} &amp; \\displaystyle n_a \\left( \\frac{\\color{red}{h}}{S + \\underbrace{\\delta}_{\\to 0}} \\right) + n_b \\left(\\frac{\\color{red}{h}}{S^\\prime - \\underbrace{\\delta}_{\\to 0} } \\right) \\approx (n_b - n_a) \\left(\\frac{\\color{red}{h}}{R-\\underbrace{\\delta}_{\\to 0}}\\right) \\\\ \\\\\n\n\\equiv &amp; \\displaystyle \\color{blue}{\\frac{n_a}{S } + \\frac{ n_b}{S^\\prime } \\approx \\frac{n_b - n_a}{R} }\n\n\\end{array}<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Hoc ultimum appellamus <strong>Relationem Obiecti-Imaginis pro refractione in Interfaciebus Sphaericis.<\/strong><\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Formatio imaginum extensarum per refractionem ad alteram partem interfaciis sphaericis<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=13kntUA9n-I&amp;t=1211s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Nunc videamus quid fiat cum mutamus<\/span><\/strong><\/a> fontem lucis punctalem pro obiecto extenso. Hoc illustratur in sequenti figura:<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-Nnk1qsJET0k\/YVvLLFbEWPI\/AAAAAAAAFm8\/zWNnZMpqHSMT_38PJgKqIhqzLO53CCMSwCLcBGAsYHQ\/s0\/objeto-extendido-frente-interfaz-esferica.PNG\" width=\"1005\" height=\"409\" alt=\"obiectum extensum ante interfaciem sphaericam\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-Nnk1qsJET0k\/YVvLLFbEWPI\/AAAAAAAAFm8\/zWNnZMpqHSMT_38PJgKqIhqzLO53CCMSwCLcBGAsYHQ\/s0\/objeto-extendido-frente-interfaz-esferica.PNG\" width=\"1005\" height=\"409\" alt=\"obiectum extensum ante interfaciem sphaericam\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align: justify;\">Analysis superior iam indicat relationem inter <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">S<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime,<\/span><\/span> nunc tantum restat invenire relationem inter magnitudines obiecti et imaginis.<\/p>\n<p style=\"text-align: justify;\">Ex figura habemus:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n\\tan(\\theta_a) &amp; =\\displaystyle \\frac{y}{S} \\\\ \\\\\n\n\\tan(\\theta_b) &amp; =\\displaystyle - \\frac{y^\\prime}{S^\\prime}\n\n\\end{array}\n\n<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Hoc iungemus cum lege Snell<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_a\\sin(\\theta_a) = n_b\\sin(\\theta_b). <\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Et ad hoc nitimur in eo quod pro angulis parvis valet approximatio<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n\\sin(\\theta_a) &amp; \\approx \\tan(\\theta_a) \\\\ \\\\\n\n\\sin(\\theta_b) &amp; \\approx \\tan(\\theta_b)\n\n\\end{array}\n\n<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Ita scribere possumus<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n&amp;\\displaystyle n_a \\frac{y}{S} \\approx- n_b \\dfrac{y^\\prime}{S^\\prime} \\\\ \\\\\n\n\\equiv &amp; \\displaystyle \\dfrac{y^\\prime}{y} \\approx - \\dfrac{n_a S^\\prime}{n_b S} \\\\ \\\\\n\n\\end{array}\n\n<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Nunc, reminiscendo quod de speculis sphaericis vidimus, habemus aliquid simile. Hoc loco possumus (iterum) definire factor magnificationis <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">m<\/span><\/span> per:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\nm=\\displaystyle \\frac{y^\\prime}{y}\n\n<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">ita ut:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\displaystyle \\color{blue}{m\\approx -\\frac{n_a S^\\prime}{n_b S}}\n\n<\/span><\/span><\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h3>Synthesis<\/h3>\n<p style=\"text-align: justify;\">Summatim, adhuc duo eventus excerpsimus qui nobis sinunt intellegere formationem imaginum cum lux ab obiecto emissa per interfaciem sphaericam transit. Hae sunt sequentes aequationes:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\displaystyle \\dfrac{n_a}{S} + \\dfrac{n_b}{S^\\prime} &amp; \\approx \\dfrac{n_b - n_a}{R} \\\\ \\\\\n\nm &amp; \\displaystyle \\approx - \\dfrac{n_a S^\\prime}{n_b S}\n\n\\end{array}\n\n<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">His duabus aequationibus potes tam positionem imaginis quam orientationem et magnitudinem imaginis computare, et illae valent sive superficies interfaciei concava sive convexa est. Hoc autem loco necesse est de conventione signorum exponere.<\/p>\n<h4>Conventio signorum<\/h4>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=13kntUA9n-I&amp;t=1682s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">His duabus aequationibus potes calcula<\/span><\/strong><\/a>re tam positionem imaginis quam orientationem et magnitudinem imaginis, et illae valent sive superficies interfaciei concava sive convexa est. Hoc autem loco necesse est de conventione signorum exponere.<\/p>\n<p style=\"text-align: justify;\">Interfacies spatium in duas regiones separat, unam ubi inveniri potest obiectum et alteram ubi imago reperitur. Ex hoc sequitur quod:<\/p>\n<ul style=\"text-align: justify;\">\n<li><strong>Positio obiecti <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">S<\/span><\/span>:<\/strong> Positiva si in parte obiecti est, negativa si in parte imaginis est.<\/li>\n<li><strong>Positio imaginis <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span><\/span> et radius curvaturae <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">R<\/span><\/span>:<\/strong> Positiva si in parte imaginis est, negativa si in parte obiecti est.<\/li>\n<li><strong>Magnitudo obiecti et imaginis, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y^\\prime<\/span><\/span>:<\/strong> Positiva si supra axem opticam est, negativa si infra axem opticam est.<\/li>\n<\/ul>\n<p><a name=\"7\"><\/a><\/p>\n<h2>Interfacies planae ut casus limes sphaericarum<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=13kntUA9n-I&amp;t=1897s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Omnia quae de interfaciebus<\/span><\/strong><\/a> sphaericis explicavimus valent etiam ad intellegendum melius interfacies planas. Revera, interfaciem planam possumus intellegere ut partem interfaciei sphaericae cum radio curvaturae maximo; etenim, si limites sumimus in relatione obiecti-imaginis pro interfaciebus sphaericis cum radius ad infinitum tendit, habemus:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{n_a}{S } + \\frac{ n_b}{S^\\prime} = \\lim_{R\\to \\infty} \\frac{n_a}{S } + \\frac{ n_b}{S^\\prime } \\approx \\lim_{R\\to \\infty} \\frac{n_b - n_a}{R} = 0\n\n<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Et si ex hoc computamus factor magnificationis, obtinemus:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">m=1\n\n<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Id est, imago suam magnitudinem et orientationem servat, quod vero variatur est eius positio observata.<\/p>\n<p><a name=\"8\"><\/a><\/p>\n<h2>Exercitia<\/h2>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/bwGJYhN3Gv8\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/p>\n<ol style=\"text-align: justify;\">\n<li>Coram virga vitrea cylindrica particula ponitur ut infra monstratur<center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-ayoBx1gP-cM\/YVvMd0Dv6rI\/AAAAAAAAFnE\/TBz3MtCOMvw5SxoNk0XlR70M11UCopfbACLcBGAsYHQ\/s0\/varilla-de-vidrio1.PNG\" width=\"945\" height=\"327\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-ayoBx1gP-cM\/YVvMd0Dv6rI\/AAAAAAAAFnE\/TBz3MtCOMvw5SxoNk0XlR70M11UCopfbACLcBGAsYHQ\/s0\/varilla-de-vidrio1.PNG\" width=\"945\" height=\"327\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center>Si particula ad 30[cm] a virga posita est et eius extremitas est fere sphaerica cum radio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">R=1,5[cm],<\/span><\/span> calcula positionem imaginis intra virgam generatae.<\/li>\n<li>Consideremus eandem virgam ex exercitio priore, sed nunc sub aqua est. Si ante eam acus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1[cm]<\/span><\/span> altitudinis ad eandem distantiam <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">30[cm],<\/span><\/span> ponitur, calcula locum et altitudinem imaginis.<center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-2Zr1OfLvpAU\/YVvM1MQr1UI\/AAAAAAAAFnM\/9HCljhcB8Ss97L3tLzyXM2E3JzqioJNHgCLcBGAsYHQ\/s0\/varilla-de-vidrio2.PNG\" width=\"948\" height=\"446\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-2Zr1OfLvpAU\/YVvM1MQr1UI\/AAAAAAAAFnM\/9HCljhcB8Ss97L3tLzyXM2E3JzqioJNHgCLcBGAsYHQ\/s0\/varilla-de-vidrio2.PNG\" width=\"948\" height=\"446\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/li>\n<li>Homo in piscinam spectat ut eius profunditatem aestimet. Ut ducem utitur sagitta in fundo picta. Quae relatio inter profunditatem veram et apparentem exsistit?<center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-G8Lr7rWgyCw\/YVvNQqC6rLI\/AAAAAAAAFnU\/mF2xOr5TPPgG62N8FZqQYijhYQg8_co0ACLcBGAsYHQ\/s0\/profundidad-real-y-aparente.PNG\" width=\"957\" height=\"658\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-G8Lr7rWgyCw\/YVvNQqC6rLI\/AAAAAAAAFnU\/mF2xOr5TPPgG62N8FZqQYijhYQg8_co0ACLcBGAsYHQ\/s0\/profundidad-real-y-aparente.PNG\" width=\"957\" height=\"658\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Refractio in Interfaciebus Sphaericis Summarium: In hac lectione perpendemus Refrationem in Interfaciebus Sphaericis, ostendentes quomodo lux se habeat transiens per superficies sphaericas et quomodo imagines formantur. Exhibentur aequationes praecipuae ad locum et magnitudinem imaginum computanda. Item explorantur casus practici, ut lentes et aestimatio profunditatum apparentium. Proposita Discendi: Post hanc lectionem discipulus poterit Intellegere refractionem lucis [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":27723,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":2,"footnotes":""},"categories":[1280,1250],"tags":[],"class_list":["post-34540","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-optica-geometrica-la","category-physica"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Refractio in Interfaciebus Sphaericis - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Disce de refractione in interfaciebus sphaericis, quomodo imagines formantur et aequationes praecipuae ad earum positionem et magnitudinem computandam.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"http:\/\/toposuranos.com\/material\/la\/refractio-in-interfaciebus-sphaericis\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Refractio in Interfaciebus Sphaericis\" \/>\n<meta property=\"og:description\" content=\"Disce de refractione in interfaciebus sphaericis, quomodo imagines formantur et aequationes praecipuae ad earum positionem et magnitudinem computandam.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/la\/refractio-in-interfaciebus-sphaericis\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2021-10-06T13:00:21+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-09-08T05:49:42+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/10\/interfaces-esfericas.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Refractio in Interfaciebus Sphaericis\" \/>\n<meta name=\"twitter:description\" content=\"Disce de refractione in interfaciebus sphaericis, quomodo imagines formantur et aequationes praecipuae ad earum positionem et magnitudinem computandam.\" \/>\n<meta name=\"twitter:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/10\/interfaces-esfericas.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minuto\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/refractio-in-interfaciebus-sphaericis\\\/#article\",\"isPartOf\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/refractio-in-interfaciebus-sphaericis\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Refractio in Interfaciebus Sphaericis\",\"datePublished\":\"2021-10-06T13:00:21+00:00\",\"dateModified\":\"2025-09-08T05:49:42+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/refractio-in-interfaciebus-sphaericis\\\/\"},\"wordCount\":1365,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/refractio-in-interfaciebus-sphaericis\\\/#primaryimage\"},\"thumbnailUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2021\\\/10\\\/interfaces-esfericas.jpg\",\"articleSection\":[\"Optica Geometrica\",\"Physica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/refractio-in-interfaciebus-sphaericis\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/refractio-in-interfaciebus-sphaericis\\\/\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/refractio-in-interfaciebus-sphaericis\\\/\",\"name\":\"Refractio in Interfaciebus Sphaericis - 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