{"id":33491,"date":"2021-08-16T13:00:37","date_gmt":"2021-08-16T13:00:37","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=33491"},"modified":"2025-07-26T07:10:59","modified_gmt":"2025-07-26T07:10:59","slug":"reflexio-in-speculis-planis-et-sphaericis","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/la\/reflexio-in-speculis-planis-et-sphaericis\/","title":{"rendered":"Reflexio in speculis planis et sphaericis"},"content":{"rendered":"<p><center><\/p>\n<h1>Reflexio in speculis planis et sphaericis<\/h1>\n<p><em><strong>Summarium:<\/strong><br \/>\nIn hac lectione recensibimus principia fundamentalia optic\u00e6 geometric\u00e6, quae in reflexione in speculis planis et sphaericis versantur. Definit terminos claves sicut radius lucis, obiectum punctiforme et imago punctiformis. Praeterea tractat regulam signorum pro speculis et relationem Cartesii ad locum imaginum computandum. Explorantur etiam proprietates speculorum concavorum et convexorum, et quomodo formationem imaginum realium et virtualium afficiant. Denique introducitur coefficientis amplificationis ad describendum mutationem in magnitudine et orientatione imaginis respectu obiecti originalis.<\/em>\n<\/p>\n<p><strong>Propositi Discendi<\/strong><br \/>\nPost finem lectionis, discipulus capax erit<\/p>\n<ol style=\"text-align:left;\">\n<li><strong>Opticam geometricam intellegere<\/strong> ut simplificationem opticae electromagneticae quae comprehensionem formationis imaginum per usum geometriae et calculi faciliorem reddit.<\/li>\n<li><strong>Leges reflectionis et refractionis intelligere<\/strong> atque earum applicationem in formatione imaginum cum speculis et lentibus.<\/li>\n<li><strong>Conceptus claves intellegere et distinguere<\/strong> ut radius lucis, radius proiectus, obiectum punctiforme et imago punctiformis.<\/li>\n<li><strong>Regulam signorum speculorum applicare<\/strong> ad positionem obiectorum et imaginum determinandam.<\/li>\n<li><strong>Formationem imaginum in speculis planis analyzare<\/strong>, symmetriam et naturam virtualem imaginum extollens.<\/li>\n<\/ol>\n<p><u><strong>Index Contentorum<\/strong><\/u><br \/>\n<a href=\"#1\"><strong>Notiones fundamentales in Optica Geometrica<\/strong><\/a><br \/>\n<a href=\"#2\">Definitiones<\/a><br \/>\n<a href=\"#3\">Regula signorum pro speculis<\/a><br \/>\n<a href=\"#4\"><strong>Specula plana et reflexio specularis<\/strong><\/a><br \/>\n<a href=\"#5\">Obiectum punctiforme coram speculo plano<\/a><br \/>\n<a href=\"#6\">Obiectum extensum coram speculo plano<\/a><br \/>\n<a href=\"#7\"><strong>Reflexio in Speculis sphaericis<\/strong><\/a><br \/>\n<a href=\"#8\">Relatio inter positionem obiecti et imaginis in speculo sphaerico<\/a><br \/>\n<a href=\"#9\">Casus limes cum <span class=\"katex-eq\" data-katex-display=\"false\">s\\to +\\infty<\/span><\/a><br \/>\n<a href=\"#10\">Reflexio obiectorum extensarum in speculis sphaericis<\/a><br \/>\n<a href=\"#11\">Specula concava et convexa<\/a><br \/>\n<a href=\"#12\">Coefficientes amplificationis et eius interpretatio<\/a>\n<\/p>\n<p><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/Ze0lpO0gDys\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Notiones fundamentales in Optica Geometrica<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=Ze0lpO0gDys&amp;t=179s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>Optica geometrica est simplificatio<\/strong><\/span><\/a> opticae electromagneticae quae permittit facile intelligere formationem imaginum et earum proprietates. Per Geometriam et Calculum possibile est leges refractionis et reflectionis inferre quae permittunt <strong>formationem imaginum cum speculis et lentibus intellegere.<\/strong> In hac prima parte studebimus conceptus fundamentales <strong>opticae geometricae<\/strong> et <strong>reflexionem in speculis planis et sphaericis.<\/strong><\/p>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=Ze0lpO0gDys&amp;t=315s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Ut haec consilia aggrediamur<\/span><\/strong><\/a> et deductiones faciamus, aliquos conceptus claves definiremus:<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h3>Definitiones<\/h3>\n<table style=\"color: #000000;\">\n<tbody>\n<tr>\n<td><strong>Radius Lucis<\/strong><\/td>\n<td>Linea imaginaria est quae repraesentat trajectoriam propagationis lucis. Si fons est un <strong>obiectum punctiforme,<\/strong> lux ex eo emergit in forma undarum (electromagneticarum) sphaericarum; radii lucis consequenter habent directionem fluxus energiae vel, si mavis, directionem vectoris Poynting.<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 200px;\" ><strong>Radius proiectus<\/strong><\/td>\n<td>Linea imaginaria quae repraesentat extensionem unius <strong>radii lucis.<\/strong><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><strong>Obiectum punctiforme vel Fons punctiformis<\/strong><\/td>\n<td>Punctum spatii unde procedunt <strong>radii lucis,<\/strong> sive proprii sive reflexi. Obiectum potest esse punctiforme vel extensum; si punctiforme est, formam non habet, sed solum positionem; si extensum est, volumen finitum non nullum et superficiem circa se habet.<\/td>\n<\/tr>\n<tr>\n<td><strong>Imago punctiformis<\/strong><\/td>\n<td>Locus spatii ubi conveniunt <strong>radii lucis<\/strong> vel <strong>radii proiecti.<\/strong><\/td>\n<\/tr>\n<tr>\n<td><strong>Reflexio<\/strong><\/td>\n<td>Processus quo <strong>radii lucis<\/strong> directionem mutant incidendo super <strong>superficiem reflectentem.<\/strong><\/td>\n<\/tr>\n<tr>\n<td><strong>Refractio<\/strong><\/td>\n<td>Processus quo <strong>radii lucis<\/strong> directionem et velocitatem mutant transeundo de uno medio in aliud.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"3\"><\/a><\/p>\n<h3>Regula signorum pro speculis<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=Ze0lpO0gDys&amp;t=596s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Conceptus utilis ad systematizandum<\/span><\/strong><\/a> opticam geometricam est regula signorum quae infra introducitur:<\/p>\n<ul>\n<li><strong>Positio obiecti:<\/strong> Si obiectum in latere est quo lux ad superficiem reflectentem pervenit, tunc magnitudo positioni eius associata <span class=\"katex-eq\" data-katex-display=\"false\">s<\/span> est numerus positivus, aliter negativa.<\/li>\n<li><strong>Positio imaginis:<\/strong> Si imago eodem latere est quo lux superficiem reflectentem relinquit, magnitudo positioni eius associata <span class=\"katex-eq\" data-katex-display=\"false\">s^\\prime<\/span> erit positiva, aliter negativa.<\/li>\n<\/ul>\n<p style=\"text-align: justify;\">In speculo plano semper satisfit aequatio <span class=\"katex-eq\" data-katex-display=\"false\">s=-s^\\prime.<\/span>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Specula plana et reflexio specularis<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=Ze0lpO0gDys&amp;t=795s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Genus simplicissimum superficiei reflectentis<\/span><\/strong><\/a> est speculum planum. In his observatur omnem radium incidente angulo <span class=\"katex-eq\" data-katex-display=\"false\">\\theta<\/span> respectu normalis speculi referri angulo <span class=\"katex-eq\" data-katex-display=\"false\">\\theta^\\prime =\\theta.<\/span> Propter hoc, observator radium reflexum conspiciens videbit quasi obiectum reflexum post speculum situm sit.<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h3>Obiectum punctiforme coram speculo plano<\/h3>\n<p style=\"text-align: justify;\">Imago formata in speculo plano est symmetrica et virtualis. Symmetrica significat distantiam inter obiectum et speculum eamdem esse ac inter imaginem et speculum, et virtualis significat imaginem \u00abpost speculum\u00bb esse.<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-mMoRSOOSEvk\/YRXkH9OiYDI\/AAAAAAAAFZg\/_dJmGpPbn30eWs4eZGlpOAjlTg1ZAd9FACLcBGAsYHQ\/s0\/espejo-plano.PNG\" width=\"618\" height=\"264\" alt=\"Obiectum et imago reflexa in speculo plano\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-mMoRSOOSEvk\/YRXkH9OiYDI\/AAAAAAAAFZg\/_dJmGpPbn30eWs4eZGlpOAjlTg1ZAd9FACLcBGAsYHQ\/s0\/espejo-plano.PNG\" width=\"618\" height=\"264\" alt=\"Obiectum et imago reflexa in speculo plano\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h3>Obiectum extensum coram speculo plano<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=Ze0lpO0gDys&amp;t=948s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Si observator ignoraret<\/span><\/strong><\/a> existentiam obiecti extensi et speculi, radiis reflexis acceptis eos interpretaretur quasi ex imagine emergentes, quasi imago esset obiectum reale.<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-H1UGdQb5sgY\/YRbmbS-_H1I\/AAAAAAAAFZo\/x1cCRQAdXIYV6uMlWyAhQHbLa8_S13mkQCLcBGAsYHQ\/s0\/objeto-extendido-espejo%2Bplano.PNG\" width=\"618\" height=\"264\" alt=\"obiectum extensum et imago reflexa coram speculo plano\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-H1UGdQb5sgY\/YRbmbS-_H1I\/AAAAAAAAFZo\/x1cCRQAdXIYV6uMlWyAhQHbLa8_S13mkQCLcBGAsYHQ\/s0\/objeto-extendido-espejo%2Bplano.PNG\" width=\"618\" height=\"264\" alt=\"obiectum extensum et imago reflexa coram speculo plano\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><br \/>\n<a name=\"7\"><\/a><\/p>\n<h2>Reflexio in Speculis sphaericis<\/h2>\n<p><a name=\"8\"><\/a><\/p>\n<h3>Relatio inter positionem obiecti et imaginis in speculo sphaerico<\/h3>\n<p><center><\/center><\/p>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=Ze0lpO0gDys&amp;t=1092s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Consideremus speculum sphaericum<\/span><\/strong><\/a> cum radio curvature <span class=\"katex-eq\" data-katex-display=\"false\">r.<\/span> Si obiectum ad distantiam <span class=\"katex-eq\" data-katex-display=\"false\">s<\/span> a vertice ponimus, imago apparebit in puncto <span class=\"katex-eq\" data-katex-display=\"false\">s^\\prime,<\/span> ut in figura monstratur:<\/p>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-XVOtDyTIR_Q\/YRcog7ZzhuI\/AAAAAAAAFaI\/l20l-kAWQjUqthdJjouuFR6xuWYtKKjqgCLcBGAsYHQ\/s0\/objeto-espejo-esferico.PNG\" width=\"618\" height=\"264\" alt=\"obiectum punctiforme reflexum coram speculo sphaerico\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-XVOtDyTIR_Q\/YRcog7ZzhuI\/AAAAAAAAFaI\/l20l-kAWQjUqthdJjouuFR6xuWYtKKjqgCLcBGAsYHQ\/s0\/objeto-espejo-esferico.PNG\" width=\"618\" height=\"264\" alt=\"obiectum punctiforme reflexum coram speculo sphaerico\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">Quoniam summa angulorum interiorum trianguli est <span class=\"katex-eq\" data-katex-display=\"false\">\\pi[rad],<\/span>, habetur:<\/p>\n<p style=\"text-align: justify;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{lr}\n\n\\phi + \\theta + \\pi - \\beta =\\pi\\; &amp;\\Longrightarrow {\\beta = \\phi + \\theta}\\\\ \\\\\n\n\\alpha + \\theta + \\pi - \\phi =\\pi\\; &amp;\\Longrightarrow {\\theta = \\phi - \\alpha}\n\n\\end{array}\n\n<\/span>\n<p style=\"text-align: justify;\">Ex hoc inferimus <span class=\"katex-eq\" data-katex-display=\"false\">\\beta = 2\\phi - \\alpha <\/span> et ideo<\/p>\n<p style=\"text-align: center; background-color: #88ff88;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\color{blue}{\\alpha + \\beta = 2\\phi}.<\/span>\n<p style=\"text-align: justify;\">His notitiis possibile est relationem inter positiones <span class=\"katex-eq\" data-katex-display=\"false\">s<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">s^\\prime<\/span> obiecti et imaginis inferre. Ad hoc observamus quod:<\/p>\n<p style=\"text-align: justify;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n\\tan(\\alpha) &amp;\\displaystyle = \\frac{h}{s - \\delta} \\\\ \\\\\n\n\\tan(\\beta) &amp;\\displaystyle = \\frac{h}{s^\\prime - \\delta} \\\\ \\\\\n\n\\tan(\\phi) &amp;\\displaystyle = \\frac{h}{s - \\delta}\n\n\\end{array}\n\n<\/span>\n<p style=\"text-align: justify;\">Nunc, si obiectum satis longe a speculo est vel radius curvature satis magnus est, licet assumere angulos <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha, \\beta<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\phi<\/span> proximos esse ad zerum et in hoc contextu approximationes sequentes valent:<\/p>\n<p style=\"text-align: justify;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n\\delta &amp; \\approx 0 \\\\ \\\\\n\n\\alpha &amp;\\displaystyle \\approx \\tan(\\alpha) \\approx \\frac{h}{s} \\\\ \\\\\n\n\\beta &amp;\\displaystyle \\approx \\tan(\\beta) \\approx \\frac{h}{s^\\prime} \\\\ \\\\\n\n\\phi &amp;\\displaystyle \\approx \\tan(\\phi) \\approx \\frac{h}{r}\n\n\\end{array}<\/span>\n<p style=\"text-align: justify;\">His approximationibus in aequationem viridem insertis obtinemus:<\/p>\n<p style=\"text-align: justify;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\displaystyle \\frac{h}{s}+\\frac{h}{s^\\prime}\\approx\\frac{2h}{r}\n\n<\/span>\n<p style=\"text-align: justify;\">Denique, simplificatis <span class=\"katex-eq\" data-katex-display=\"false\">h<\/span> et substituendo <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle f = \\frac{r}{2}<\/span> habemus quod<\/p>\n<p style=\"text-align: justify;\"><strong><span class=\"katex-eq\" data-katex-display=\"false\">\n\\displaystyle\\color{blue}{\\frac{1}{s}+\\frac{1}{s^\\prime}\\approx\\frac{1}{f}}\n\n<\/span><\/strong><\/p>\n<p style=\"text-align: justify;\">Hoc est quod appellatur \u00abrelatio Cartesiana\u00bb pro speculis sphaericis parvae aperturae, ubi valor <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> foco lensis respondet.<\/p>\n<p><a name=\"9\"><\/a><\/p>\n<h4>Casus limes cum <span class=\"katex-eq\" data-katex-display=\"false\">s\\to+\\infty<\/span><\/h4>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=Ze0lpO0gDys&amp;t=1748s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Si valorem<\/span><\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">s^\\prime<\/span> computamus et limitem cum <span class=\"katex-eq\" data-katex-display=\"false\">s\\to+\\infty,<\/span> calculamus, tunc habebimus:<\/p>\n<p style=\"text-align: justify;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle s^\\prime = \\frac{1}{\\frac{1}{f}-\\frac{1}{s}} =\\frac{sf}{s-f}<\/span>\n<p style=\"text-align: justify;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{s\\to +\\infty}s^\\prime = \\lim_{s\\to +\\infty}\\frac{sf}{s-f}=f<\/span>\n<p style=\"text-align: justify;\">Aliter, si fontem longe ponimus, tunc radius ex ea egrediens et ad speculum perveniens fere horizontem sequetur et, in speculo reflectens, per focum transibit ut in figura ostenditur:<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-DuLxeEzpreA\/YRiDferq6uI\/AAAAAAAAFag\/nJs3uKnA5cAqQ4xhFxGJVB715kRJ4NHFgCLcBGAsYHQ\/s0\/espejo-esferico-casolimite.PNG\" width=\"618\" height=\"264\" alt=\"radius qui ab infinito venit in speculum sphaericum reflectitur\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-DuLxeEzpreA\/YRiDferq6uI\/AAAAAAAAFag\/nJs3uKnA5cAqQ4xhFxGJVB715kRJ4NHFgCLcBGAsYHQ\/s0\/espejo-esferico-casolimite.PNG\" width=\"618\" height=\"264\" alt=\"radius qui ab infinito venit in speculum sphaericum reflectitur\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><br \/>\n<a name=\"10\"><\/a><\/p>\n<h3>Reflexio obiectorum extensarum in speculis sphaericis<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=Ze0lpO0gDys&amp;t=1878s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Resultata quae hactenus recensuimus<\/span><\/strong><\/a> sinent nos locum geometrice invenire ubi imago obiecti formabitur cum lux quam emittit vel reflectit in speculo sphaerico reflectitur. Ad hoc sufficit notare omnes radios horizontales reflexos per focum transire, omnes radios per focum transeuntes horizontaliter reflexos esse, et quod localiter (in puncto ubi radius speculum sphaericum attingit) speculum se gerit ut speculum planum, unde angulus incidentiae aequatur angulo reflexo.<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-dBrwLgxx-Hw\/YRif1YsG-JI\/AAAAAAAAFaw\/UURe6b9HaXoAlb-cKOM33EwZ0aFxAigMwCLcBGAsYHQ\/s0\/objeto-extendido-espejo-esferico.PNG\" width=\"720\" height=\"594\" alt=\"formatio imaginum obiectorum extensarum supra specula sphaerica\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-dBrwLgxx-Hw\/YRif1YsG-JI\/AAAAAAAAFaw\/UURe6b9HaXoAlb-cKOM33EwZ0aFxAigMwCLcBGAsYHQ\/s0\/objeto-extendido-espejo-esferico.PNG\" width=\"720\" height=\"594\" alt=\"formatio imaginum obiectorum extensarum supra specula sphaerica\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align: justify;\">Quisque punctus obiecti extensi radios lucis emittit qui, postquam a speculo reflectuntur, in puncto respondente imaginis intersecantur.<\/p>\n<p><a name=\"11\"><\/a><\/p>\n<h3>Specula concava et convexa<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=Ze0lpO0gDys&amp;t=2227s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Specula sphaerica quae hactenus<\/span><\/strong><\/a> recensuimus omnia exempla speculorum concavorum sunt. Haec sunt illa in quibus curvatura ex parte est unde radii lucis veniunt. Cum curvatura ad partem oppositam dirigitur, speculum convexus dicitur. Cum formatio imaginum in huiusmodi speculis geometrice analysatur, primum quod animadvertitur est radios reflexos, pro conspiratione in punctum, dispergi; ad locum invenire ubi imago formatur necesse est, consequenter, radios reflexos projicere ita ut imago virtualis obtineatur.<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-cfrp8zRyqOk\/YRinI5L1BkI\/AAAAAAAAFa4\/Uj1etPhThK8_Zr1N3vU7Sw2ua0U3svpgwCLcBGAsYHQ\/s0\/reflejo%2Ben%2Bespejo%2Bconvexo.PNG\" width=\"869\" height=\"503\" alt=\"Imago virtualis in speculo convexo\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-cfrp8zRyqOk\/YRinI5L1BkI\/AAAAAAAAFa4\/Uj1etPhThK8_Zr1N3vU7Sw2ua0U3svpgwCLcBGAsYHQ\/s0\/reflejo%2Ben%2Bespejo%2Bconvexo.PNG\" width=\"869\" height=\"503\" alt=\"Imago virtualis in speculo convexo\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align: justify;\">Hoc loco sequentia vocabula consideranda sunt:<\/p>\n<ul style=\"text-align: justify;\">\n<li><strong>Imago realis:<\/strong> est quando imago a radiis reflexis formatur et ideo ante speculum est.<\/li>\n<li><strong>Imago virtualis:<\/strong> est quando imago a radiis proiectis formatur et idcirco \u00abpost speculum\u00bb est.<\/li>\n<\/ul>\n<p><a name=\"12\"><\/a><\/p>\n<h3>Coefficientes amplificationis et eius interpretatio<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=Ze0lpO0gDys&amp;t=2393s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Ut in figuris superioribus videre potuimus<\/span><\/strong><\/a>, cum reflexio fit in speculis sphaericis, concavis vel convexis, imago magnitudinem vel orientationem respectu obiecti originalis mutare potest. Tunc quaestio oritur: num existat modus ad hoc incrementum vel decrementum et mutationem orientationis imaginis modellandum? Responsum est affirmativum et deducitur ex relationibus similitudinis triangulorum in quacumque figura quam iam recensuimus. Infra ostensum erit analysis pro speculo concavo; pro speculis convexis ratio similis est. Ad singulos gradus rite sequendos, memineris <strong>regulas signorum pro speculis<\/strong> quas in principio vidimus.<\/p>\n<p><center><br \/>\n<img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-cRVOSOADzXI\/YRix4uOudBI\/AAAAAAAAFbA\/vgqeiZEXhGATgnNu3f2zSvL-H8Rp8VVRACLcBGAsYHQ\/s0\/determinaci%25C3%25B3n%2Bdel%2Bcoeficiente%2Bde%2Bmagnificacion.PNG\" width=\"716\" height=\"488\" alt=\"similitudo triangulorum inter radios incidentes et reflexos\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-cRVOSOADzXI\/YRix4uOudBI\/AAAAAAAAFbA\/vgqeiZEXhGATgnNu3f2zSvL-H8Rp8VVRACLcBGAsYHQ\/s0\/determinaci%25C3%25B3n%2Bdel%2Bcoeficiente%2Bde%2Bmagnificacion.PNG\" width=\"716\" height=\"488\" alt=\"similitudo triangulorum inter radios incidentes et reflexos\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align: justify;\">Cum triangula caeruleum et viride similia sint, tunc habetur coefficientem amplificationis <span class=\"katex-eq\" data-katex-display=\"false\">m=y^\\prime\/y<\/span> qui indicat quantum imago reflexa relativa ad magnitudinem obiecti originalis augeatur, per relationem sequentem computari posse:<\/p>\n<p style=\"text-align: justify;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{y}{s} = \\frac{-y^\\prime}{s^\\prime}<\/span>\n<p style=\"text-align: justify;\">Hic <span class=\"katex-eq\" data-katex-display=\"false\">y^\\prime<\/span> signo negativo comitatur quia imago deorsum orientatur (invertitur), et secundum regulam signorum speculorum, <span class=\"katex-eq\" data-katex-display=\"false\">s<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">s^\\prime<\/span> sunt ambo positi. Consequenter habebitur:<\/p>\n<p style=\"text-align: justify;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\color{blue}{m=\\frac{y^\\prime}{y} = - \\frac{s^\\prime}{s}}<\/span>\n<p style=\"text-align: justify;\">Id est, cognitis positionibus obiecti et imaginis possibile est coefficientem amplificationis speculi computare.<\/p>\n<p style=\"text-align: justify;\">Haec formula componi potest cum relatione Cartesiana ad coefficientem amplificationis computandum ex foco et positione obiecti. Sufficit meminisse quod<\/p>\n<p style=\"text-align: justify;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle s^\\prime=\\frac{sf}{s-f}.<\/span>\n<p style=\"text-align: justify;\">et habebitur:<\/p>\n<p style=\"text-align: justify;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\color{blue}{m= - \\frac{1}{s}\\frac{sf}{s-f} = \\frac{f}{f-s}}<\/span>\n<p style=\"text-align: justify;\">Ex hoc sequitur quod:<\/p>\n<ul style=\"text-align: justify;\">\n<li>Si <span class=\"katex-eq\" data-katex-display=\"false\">|m|\\lt 1<\/span>, imago contrahitur; cum <span class=\"katex-eq\" data-katex-display=\"false\">|m|\\gt 1<\/span>, imago expanditur; et cum <span class=\"katex-eq\" data-katex-display=\"false\">|m|=1,<\/span>, magnitudinem suam servat.<\/li>\n<li>Si <span class=\"katex-eq\" data-katex-display=\"false\">m\\gt 0<\/span>, imago orientationem obiecti originalis retinet; et cum <span class=\"katex-eq\" data-katex-display=\"false\">m\\lt 0<\/span>, imago respectu obiecti originalis invertitur.<\/li>\n<li>Imago ad punctum redigitur cum <span class=\"katex-eq\" data-katex-display=\"false\">m=0.<\/span><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Reflexio in speculis planis et sphaericis Summarium: In hac lectione recensibimus principia fundamentalia optic\u00e6 geometric\u00e6, quae in reflexione in speculis planis et sphaericis versantur. Definit terminos claves sicut radius lucis, obiectum punctiforme et imago punctiformis. Praeterea tractat regulam signorum pro speculis et relationem Cartesii ad locum imaginum computandum. Explorantur etiam proprietates speculorum concavorum et convexorum, [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":27537,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":7,"footnotes":""},"categories":[1280,1250],"tags":[],"class_list":["post-33491","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>Reflexio in speculis planis et sphaericis - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Explora principia opticae geometricae in reflexione speculorum planorum et sphaericorum. 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