{"id":553,"date":"2012-07-30T16:02:04","date_gmt":"2012-07-30T04:02:04","guid":{"rendered":"https:\/\/blog.waikato.ac.nz\/physicsstop\/2012\/07\/30\/the_shortest_distance_between\/"},"modified":"2012-07-30T16:02:04","modified_gmt":"2012-07-30T04:02:04","slug":"the-shortest-distance-between","status":"publish","type":"post","link":"https:\/\/blog.waikato.ac.nz\/physicsstop\/2012\/07\/30\/the-shortest-distance-between\/","title":{"rendered":"The shortest distance between two points"},"content":{"rendered":"<p>I remember as a student being presented with the proof that the shortest distance between two points is a straight line (at least, on a 2 dimensional flat surface). Although it&#8217;s almost blatantly obvious, it can be formally proved through <a href=\"http:\/\/mathworld.wolfram.com\/CalculusofVariations.html\" target=\"_blank\">Calculus of Variations<\/a>.<\/p>\n<p>However, the quickest route between two points is not necessarily a straight line. Take the example of my walk from my office, on the 3rd floor of EF block at Waikato, to the lecture room G.3.33 on the third floor of G block. Normally it is a short (1 minute) stroll down the third floor corridor in a straight line &#8211; the walkway goes from EF building into F building into G building and G.3.33 is straight in front of me. (Those &#8216;in-the-know&#8217; will realise I&#8217;ve over-simplified the situation &#8211; there are a couple of 22.5 degree bends in the corridor and a few steps to negotiate, but basically it is a straight line between the two.) But not now. There is building work going on in FG link block. This means that the straight-line route is unavailable. Instead, I have to go down three floors, out the door, walk about 100 metres through the rain around the back of F and G buildings to the G front door, then up three flights of stairs, and walk back through the corridor to G.3.33. Total time more like 6 minutes for a distance of about 50 metres.<\/p>\n<p>Here&#8217;s another example:&nbsp; You are a lifeguard on a beach, say 50 metres from the water, and  you see someone in trouble in the sea, say 25 metres off shore and 50 metres along the shore from where you are.&nbsp; Which direction do you run in? You don&#8217;t run straight towards them, because you know you can run faster than you can swim. Covering the shortest distance in total would mean that you are swimming a longer distance than necessary. It is better to run more along the shore, and enter the water at a point closer to the man in trouble. You cover a longer distance in total, but less of that distance is spent in the water where you are going slowly.<\/p>\n<p>The problem of refraction of light is analogous to the lifesaver description above. The light travels the path that gives the  shortest time between two points. Suppose a light ray starts at point A,  in air, and ends at point B, in a block of glass. Now, we know that  glass has a refractive index. This means that the speed of light in the  glass is lower than that in air.&nbsp; The light doesn&#8217;t travel in a straight line between A and B,  but rather in two straight line segments &#8211; in a straight line&nbsp; from  point A to point C on the surface of the glass, then in a straight line  from point C to point B.&nbsp; Where is point C? It turns out that it&#8217;s such that the total time the ray takes from A to B is minimized. Clever or what? How does it &#8216;know&#8217; that this is the shortest path?<\/p>\n<p>Many problems in physics are similar minimization problems.<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>I remember as a student being presented with the proof that the shortest distance between two points is a straight line (at least, on a 2 dimensional flat surface). Although it&#8217;s almost blatantly obvious, it can be formally proved through Calculus of Variations. However, the quickest route between two points is not necessarily a straight [&hellip;]<\/p>\n","protected":false},"author":20,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[16,19,6],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v22.3 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>The shortest distance between two points : Physics Stop<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/blog.waikato.ac.nz\/physicsstop\/2012\/07\/30\/the-shortest-distance-between\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"The shortest distance between two points : Physics Stop\" \/>\n<meta property=\"og:description\" content=\"I remember as a student being presented with the proof that the shortest distance between two points is a straight line (at least, on a 2 dimensional flat surface). Although it&#8217;s almost blatantly obvious, it can be formally proved through Calculus of Variations. However, the quickest route between two points is not necessarily a straight [&hellip;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/blog.waikato.ac.nz\/physicsstop\/2012\/07\/30\/the-shortest-distance-between\/\" \/>\n<meta property=\"og:site_name\" content=\"Physics Stop\" \/>\n<meta property=\"article:published_time\" content=\"2012-07-30T04:02:04+00:00\" \/>\n<meta name=\"author\" content=\"Marcus\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Marcus\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"3 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/blog.waikato.ac.nz\/physicsstop\/2012\/07\/30\/the-shortest-distance-between\/\",\"url\":\"https:\/\/blog.waikato.ac.nz\/physicsstop\/2012\/07\/30\/the-shortest-distance-between\/\",\"name\":\"The shortest distance between two points : Physics Stop\",\"isPartOf\":{\"@id\":\"https:\/\/blog.waikato.ac.nz\/physicsstop\/#website\"},\"datePublished\":\"2012-07-30T04:02:04+00:00\",\"dateModified\":\"2012-07-30T04:02:04+00:00\",\"author\":{\"@id\":\"https:\/\/blog.waikato.ac.nz\/physicsstop\/#\/schema\/person\/34d359280e44f926f2df12eb3e50aa22\"},\"breadcrumb\":{\"@id\":\"https:\/\/blog.waikato.ac.nz\/physicsstop\/2012\/07\/30\/the-shortest-distance-between\/#breadcrumb\"},\"inLanguage\":\"en-NZ\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/blog.waikato.ac.nz\/physicsstop\/2012\/07\/30\/the-shortest-distance-between\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/blog.waikato.ac.nz\/physicsstop\/2012\/07\/30\/the-shortest-distance-between\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/blog.waikato.ac.nz\/physicsstop\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"The shortest distance between two points\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/blog.waikato.ac.nz\/physicsstop\/#website\",\"url\":\"https:\/\/blog.waikato.ac.nz\/physicsstop\/\",\"name\":\"Physics Stop\",\"description\":\"A light-hearted look at everyday physics going on in the world, written by a physicist but intended for a general audience.\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/blog.waikato.ac.nz\/physicsstop\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"en-NZ\"},{\"@type\":\"Person\",\"@id\":\"https:\/\/blog.waikato.ac.nz\/physicsstop\/#\/schema\/person\/34d359280e44f926f2df12eb3e50aa22\",\"name\":\"Marcus\",\"url\":\"https:\/\/blog.waikato.ac.nz\/physicsstop\/author\/mtwilson\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"The shortest distance between two points : Physics Stop","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/blog.waikato.ac.nz\/physicsstop\/2012\/07\/30\/the-shortest-distance-between\/","og_locale":"en_US","og_type":"article","og_title":"The shortest distance between two points : Physics Stop","og_description":"I remember as a student being presented with the proof that the shortest distance between two points is a straight line (at least, on a 2 dimensional flat surface). Although it&#8217;s almost blatantly obvious, it can be formally proved through Calculus of Variations. However, the quickest route between two points is not necessarily a straight [&hellip;]","og_url":"https:\/\/blog.waikato.ac.nz\/physicsstop\/2012\/07\/30\/the-shortest-distance-between\/","og_site_name":"Physics Stop","article_published_time":"2012-07-30T04:02:04+00:00","author":"Marcus","twitter_card":"summary_large_image","twitter_misc":{"Written by":"Marcus","Est. reading time":"3 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/blog.waikato.ac.nz\/physicsstop\/2012\/07\/30\/the-shortest-distance-between\/","url":"https:\/\/blog.waikato.ac.nz\/physicsstop\/2012\/07\/30\/the-shortest-distance-between\/","name":"The shortest distance between two points : Physics Stop","isPartOf":{"@id":"https:\/\/blog.waikato.ac.nz\/physicsstop\/#website"},"datePublished":"2012-07-30T04:02:04+00:00","dateModified":"2012-07-30T04:02:04+00:00","author":{"@id":"https:\/\/blog.waikato.ac.nz\/physicsstop\/#\/schema\/person\/34d359280e44f926f2df12eb3e50aa22"},"breadcrumb":{"@id":"https:\/\/blog.waikato.ac.nz\/physicsstop\/2012\/07\/30\/the-shortest-distance-between\/#breadcrumb"},"inLanguage":"en-NZ","potentialAction":[{"@type":"ReadAction","target":["https:\/\/blog.waikato.ac.nz\/physicsstop\/2012\/07\/30\/the-shortest-distance-between\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/blog.waikato.ac.nz\/physicsstop\/2012\/07\/30\/the-shortest-distance-between\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/blog.waikato.ac.nz\/physicsstop\/"},{"@type":"ListItem","position":2,"name":"The shortest distance between two points"}]},{"@type":"WebSite","@id":"https:\/\/blog.waikato.ac.nz\/physicsstop\/#website","url":"https:\/\/blog.waikato.ac.nz\/physicsstop\/","name":"Physics Stop","description":"A light-hearted look at everyday physics going on in the world, written by a physicist but intended for a general audience.","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/blog.waikato.ac.nz\/physicsstop\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-NZ"},{"@type":"Person","@id":"https:\/\/blog.waikato.ac.nz\/physicsstop\/#\/schema\/person\/34d359280e44f926f2df12eb3e50aa22","name":"Marcus","url":"https:\/\/blog.waikato.ac.nz\/physicsstop\/author\/mtwilson\/"}]}},"_links":{"self":[{"href":"https:\/\/blog.waikato.ac.nz\/physicsstop\/wp-json\/wp\/v2\/posts\/553"}],"collection":[{"href":"https:\/\/blog.waikato.ac.nz\/physicsstop\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.waikato.ac.nz\/physicsstop\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.waikato.ac.nz\/physicsstop\/wp-json\/wp\/v2\/users\/20"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.waikato.ac.nz\/physicsstop\/wp-json\/wp\/v2\/comments?post=553"}],"version-history":[{"count":0,"href":"https:\/\/blog.waikato.ac.nz\/physicsstop\/wp-json\/wp\/v2\/posts\/553\/revisions"}],"wp:attachment":[{"href":"https:\/\/blog.waikato.ac.nz\/physicsstop\/wp-json\/wp\/v2\/media?parent=553"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.waikato.ac.nz\/physicsstop\/wp-json\/wp\/v2\/categories?post=553"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.waikato.ac.nz\/physicsstop\/wp-json\/wp\/v2\/tags?post=553"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}