{"id":34498,"date":"2021-08-26T13:00:30","date_gmt":"2021-08-26T13:00:30","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=34498"},"modified":"2025-09-08T04:59:02","modified_gmt":"2025-09-08T04:59:02","slug":"problemata-soluta-speculorum-planorum","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/la\/problemata-soluta-speculorum-planorum\/","title":{"rendered":"Problemata Soluta Speculorum Planorum"},"content":{"rendered":"<p><center><\/p>\n<h1>Specula plana, problemata soluta<\/h1>\n<p><em><strong>Summarium:<\/strong><br \/>\nIn hac lectione recognoscemus quaedam problemata soluta speculorum planorum. Determinatur angulus reflexionis <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span><\/span> in functione anguli <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta<\/span><\/span> inter duo specula plana cardine iuncta et exempla specifica computantur. Examinauntur valores critici <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> ut radius semel in quodque speculum reflectatur et formula pro <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span><\/span> confirmatur. Item identificantur anguli incidentiae qui faciunt ut radius in se redeat, computando seriem angulorum reditus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha_n = n\\theta<\/span><\/span>.<br \/>\n<\/em><br \/>\n<u><strong>Proposita Discendi<\/strong><\/u><br \/>\nHoc cursu confecto, discipulus poterit:<\/p>\n<ol style=\"text-align:left;\">\n<li><strong>Intellegere<\/strong> formulas fundamentales opticae speculorum planorum.<\/li>\n<li><strong>Applicare<\/strong> legem reflexionis in problematibus cum speculis planis.<\/li>\n<li><strong>Determinare<\/strong> angulum reflexionis <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span><\/span> in functione anguli <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta<\/span><\/span> inter duo specula plana.<\/li>\n<li><strong>Analyzare<\/strong> limites formularum speculorum et condiciones earum validitatis.<\/li>\n<\/ol>\n<p><strong>INDEX CONTENTORUM<\/strong><br \/>\n<a href=\"#1\"><strong>Introductio<\/strong><\/a><br \/>\n<a href=\"#2\"><strong>Specula cardine iuncta<\/strong><\/a><br \/>\n<a href=\"#3\">Examinatio limitum ratiocinii<\/a><br \/>\n<a href=\"#4\">Anguli reditus<\/a><\/p>\n<p><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/sX--0tertWI\" 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>Introductio<\/h2>\n<p style=\"text-align: justify;\">In <a href=\"http:\/\/toposuranos.com\/material\/es\/reflexion-en-espejos-planos-y-esfericos\/\" rel=\"noopener\" target=\"_blank\">lectione praeterita<\/a> recognovimus plerasque formulas ad opticam speculorum planorum et sphaericorum pertinentes; tamen, ut melior intellectus horum argumentorum obtineatur, necesse est inspicere modum quo hae in resolutione problematum ad haec pertinentium apparent. Quam ob rem hanc partem solummodo destinabimus ad solutionem nonnullorum problematum recognoscendam. Hoc tempore nos tantum in speculis planis concentramur.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Specula cardine iuncta<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=sX--0tertWI&amp;t=64s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Duo specula plana cardine iuncta<\/span><\/strong><\/a> extremo inter se angulum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta.<\/span><\/span> sustinent. Si radius lucis in unum ex speculis cum angulo <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> respectu ad normalem incidit ita ut lux semel tantum in quodque speculum reflectatur et secum ipsam intersecaretur angulum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma:<\/span><\/span> formans:<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-CA7kLZjaxyo\/YSMSe1ea97I\/AAAAAAAAFc8\/i5snILhLzT8XlVM88Hs8JDZiJvum5zmlgCLcBGAsYHQ\/s0\/ESPEJOPLANO1.PNG\" width=\"400\" height=\"100\" alt=\"anguli in speculis planis\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-CA7kLZjaxyo\/YSMSe1ea97I\/AAAAAAAAFc8\/i5snILhLzT8XlVM88Hs8JDZiJvum5zmlgCLcBGAsYHQ\/s0\/ESPEJOPLANO1.PNG\" width=\"400\" height=\"100\" alt=\"anguli in speculis planis\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<ol style=\"text-align: justify;\">\n<li type=\"a\">Inveni formulam quae sinat determinare angulum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span><\/span> ex ceteris datis.<\/li>\n<li type=\"a\">Si radius lucis in primum speculum cum angulo <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha=30^o<\/span><\/span> incidit et angulus inter specula est <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta=50^o<\/span><\/span>, quis erit angulus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span><\/span>?<\/li>\n<\/ol>\n<span class=\"collapseomatic \" id=\"id6abf267c509e5\"  tabindex=\"0\" title=\"SOLUTIO\"    >SOLUTIO<\/span><div id=\"target-id6abf267c509e5\" class=\"collapseomatic_content \">\n<ol style=\"text-align: justify;\">\n<li type=\"a\">Definiendo angulum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span><\/span> inter normalem secundi speculi et radium lucis a primo speculo reflexum, et utens lege reflexionis in speculis planis figuram complere possumus hoc modo:<br \/>\n<center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-zheSpWUbupU\/YSMU24o_WkI\/AAAAAAAAFdE\/iOtRNY_vBWMWaY24ycU8llcenhsBRVpDQCLcBGAsYHQ\/s0\/espejoplano2.PNG\" width=\"400\" height=\"100\" alt=\"angulus inter specula plana\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-zheSpWUbupU\/YSMU24o_WkI\/AAAAAAAAFdE\/iOtRNY_vBWMWaY24ycU8llcenhsBRVpDQCLcBGAsYHQ\/s0\/espejoplano2.PNG\" width=\"400\" height=\"100\" alt=\"angulus inter specula plana\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><br \/>\nHis consideratis, fieri potest sequens ratiocinatio:<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"width: 50px;\"><span dir=\"ltr\"> [\/latex]<\/span><\/td>\n<td style=\"width: 350px;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(90^o - \\alpha) + (90^o - \\beta) + \\theta = 180^o<\/span><\/span><\/td>\n<td>; Quia summa angulorum interiorum trianguli est <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">180^o<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\equiv<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\alpha + \\beta = \\theta <\/span><\/span><\/td>\n<td> <\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"> [\/latex]<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> 2\\alpha +2\\beta + \\gamma = 180 <\/span><\/span><\/td>\n<td>; Quia summa angulorum interiorum trianguli est <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">180^o<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\equiv<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\gamma = 180 - 2(\\alpha + \\beta)<\/span><\/span><\/td>\n<td> <\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"> [\/latex]<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\color{blue}{\\gamma = 180 - 2\\theta}<\/span><\/span><\/td>\n<td>; Ex (1,2)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Ergo colligitur angulum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span><\/span> solum functionem esse anguli <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta<\/span><\/span> quem specula formant et eius formula erit <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma(\\theta) = 180^0 - 2\\theta<\/span><\/span><\/li>\n<li type=\"a\">Ex ratione in priori inciso habetur <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma = 180^o - 2\\cdot 50^o = 80^o <\/span><\/span><\/li>\n<\/ol>\n<\/div>\n<a name=\"3\"><\/a><\/p>\n<h3>Examinatio limitum ratiocinii<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=sX--0tertWI&amp;t=464s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Exercitium prius habet subtilem<\/span><\/strong><\/a> quaestionem. Si enuntiatum inspicis videbis requiri ut radius lucis semel tantum in quodque speculum reflectatur; tamen, ut id fiat, non quilibet valor <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> sufficit. Inveni valores <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> qui talem condicionem satisfaciant et qui igitur sinant formulam in exercitio priore obtentam validam esse.<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-FIE6EKNxu1Q\/YSM34HUa-NI\/AAAAAAAAFdM\/DVM38fUBrxggrGsXoJcAI4SgDC2u5gm1gCLcBGAsYHQ\/s0\/espejosplanos3.PNG\" width=\"1021\" height=\"485\" alt=\"Radii lucis in speculis planis reflectentes\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-FIE6EKNxu1Q\/YSM34HUa-NI\/AAAAAAAAFdM\/DVM38fUBrxggrGsXoJcAI4SgDC2u5gm1gCLcBGAsYHQ\/s0\/espejosplanos3.PNG\" width=\"1021\" height=\"485\" alt=\"Radii lucis in speculis planis reflectentes\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><span class=\"collapseomatic \" id=\"id6abf267c50bd7\"  tabindex=\"0\" title=\"SOLUTIO\"    >SOLUTIO<\/span><div id=\"target-id6abf267c50bd7\" class=\"collapseomatic_content \">\n<p style=\"text-align: justify;\">Habemus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> valorem \u00abcriticum\u00bb attingere cum efficit <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta=0^o;<\/span><\/span> et cum hoc fit, possumus sumere angulum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> qui sequens ratiocinium sinit:<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-b_rcYFH-moo\/YSM--pA_6sI\/AAAAAAAAFdU\/VQRBmlh7I-oAhjuWi83GOphA1JKQHrrbACLcBGAsYHQ\/s0\/espejoplano4.PNG\" width=\"288\" height=\"319\" alt=\"radius reflectens in speculis planis cum angulo critico\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-b_rcYFH-moo\/YSM--pA_6sI\/AAAAAAAAFdU\/VQRBmlh7I-oAhjuWi83GOphA1JKQHrrbACLcBGAsYHQ\/s0\/espejoplano4.PNG\" width=\"288\" height=\"319\" alt=\"radius reflectens in speculis planis cum angulo critico\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align: justify;\">Accidere debet quod hae duae aequationes indicant:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha + x = 90^o<\/span><\/span><\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta + x = 90^o<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Et hoc fieri potest solum si:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha = \\theta<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Id est: valor <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha=\\theta<\/span><\/span> est valor anguli incidentiae criticus talis ut, si superetur, tunc radius plus quam bis in speculo reflectetur et consequenter formulam in exercitio priore obtentam invalidaret. Ex his resultatis possumus corrigere exitum exercitii prioris scribentes:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma(\\theta, \\alpha) = 180^0 - 2\\theta \\;\\;\\;\\; ; \\;\\;\\;\\; \\alpha \\in ]0,\\theta[ <\/span><\/span><\/p>\n<\/div>\n<a name=\"4\"><\/a><\/p>\n<h3>Anguli reditus<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=sX--0tertWI&amp;t=809s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Ex his resultatis videre possumus quod,<\/span><\/strong><\/a> pro certis angulis incidentiae, radius lucis in se ipsum revertitur. Hoc fit cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha = 0^o<\/span><\/span> vel cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha = \\theta,<\/span><\/span> ubi <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta<\/span><\/span> est angulus inter duo specula plana formatus. Existentne plures anguli reditus?; et si sunt, quomodo computari possint?<\/p>\n<span class=\"collapseomatic \" id=\"id6abf267c50d3f\"  tabindex=\"0\" title=\"SOLUTIO\"    >SOLUTIO<\/span><div id=\"target-id6abf267c50d3f\" class=\"collapseomatic_content \">\n<p style=\"text-align: justify;\">Ad hunc problematem solvendi debemus imaginari condicionem quae eveniet cum radius lucis in primum speculum cum angulo respectu ad normalem <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha\\in ]\\theta, 180^o[<\/span><\/span> incidit. Cum hoc fit, habemus condicionem sicut in figura sequenti monstratur:<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-T3bmPeT9SHU\/YSNnIfiDnzI\/AAAAAAAAFdc\/68ireIOvkuYdVvLw0dZD4oqeP2YL8Ml3QCLcBGAsYHQ\/s0\/espejo5.PNG\" width=\"650\" height=\"294\" alt=\"radius contra specula plana\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-T3bmPeT9SHU\/YSNnIfiDnzI\/AAAAAAAAFdc\/68ireIOvkuYdVvLw0dZD4oqeP2YL8Ml3QCLcBGAsYHQ\/s0\/espejo5.PNG\" width=\"650\" height=\"294\" alt=\"radius contra specula plana\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align: justify;\">Cum summa angulorum interiorum trianguli sit <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">180^o<\/span><\/span>:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(90^o - \\alpha) + (90^o + \\beta) + \\theta = 180<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Simplificando hanc relationem possumus obtinere angulum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span><\/span> in terminis <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta.<\/span><\/span><\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta=\\alpha - \\theta<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Haec expressio magni momenti est quia si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta=\\theta,<\/span><\/span> tunc secundum rationem exercitii prioris radius debet in se ipsum redire in proxima reflexione. Itaque <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha=2\\theta.<\/span><\/span> Ergo hoc ratiocinium inductive extendi potest per:<\/p>\n<ul style=\"text-align: justify;\">\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha_0 = 0^o<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha_1 = \\theta<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha_{n-1} = \\alpha_n - \\theta<\/span><\/span><\/li>\n<\/ul>\n<p style=\"text-align: justify;\">Et ex hoc, habemus seriem angulorum reditus:<\/p>\n<ul style=\"text-align: justify;\">\n<li style=\"list-style-type: none;\">\n<ul style=\"text-align: justify;\">\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha_0 = 0^o<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha_1 = \\theta<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha_{2} = 2\\theta<\/span><\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdots<\/span><\/span><\/p>\n<ul style=\"text-align: justify;\">\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha_{n} = n\\theta<\/span><\/span><\/li>\n<\/ul>\n<p style=\"text-align: justify;\">Praeterea notandum est tam angulum inter specula plana quam singulos angulos incidentiae acutos esse debere.<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Specula plana, problemata soluta Summarium: In hac lectione recognoscemus quaedam problemata soluta speculorum planorum. Determinatur angulus reflexionis in functione anguli inter duo specula plana cardine iuncta et exempla specifica computantur. Examinauntur valores critici ut radius semel in quodque speculum reflectatur et formula pro confirmatur. Item identificantur anguli incidentiae qui faciunt ut radius in se redeat, [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":27598,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":2,"footnotes":""},"categories":[1280,1250],"tags":[],"citadela-post-location":[],"class_list":["post-34498","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 v28.0 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Problemata Soluta Speculorum Planorum - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Problemata soluta speculorum planorum: angulos reflexionis determinat, valores criticos cum exemplis practicis examinat.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/toposuranos.com\/material\/la\/problemata-soluta-speculorum-planorum\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Problemata Soluta Speculorum Planorum\" \/>\n<meta property=\"og:description\" content=\"Problemata soluta speculorum planorum: angulos reflexionis determinat, valores criticos cum exemplis practicis examinat.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/toposuranos.com\/material\/la\/problemata-soluta-speculorum-planorum\/\" \/>\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=\"2021-08-26T13:00:30+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-09-08T04:59:02+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/08\/espejosplanos-1024x342.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Problemata Soluta Speculorum Planorum\" \/>\n<meta name=\"twitter:description\" content=\"Problemata soluta speculorum planorum: angulos reflexionis determinat, valores criticos cum exemplis practicis examinat.\" \/>\n<meta name=\"twitter:image\" content=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/08\/espejosplanos.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\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/problemata-soluta-speculorum-planorum\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/problemata-soluta-speculorum-planorum\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Problemata Soluta Speculorum Planorum\",\"datePublished\":\"2021-08-26T13:00:30+00:00\",\"dateModified\":\"2025-09-08T04:59:02+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/problemata-soluta-speculorum-planorum\\\/\"},\"wordCount\":812,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/problemata-soluta-speculorum-planorum\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2021\\\/08\\\/espejosplanos.jpg\",\"articleSection\":[\"Optica Geometrica\",\"Physica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/problemata-soluta-speculorum-planorum\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/problemata-soluta-speculorum-planorum\\\/\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/problemata-soluta-speculorum-planorum\\\/\",\"name\":\"Problemata Soluta Speculorum Planorum - 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