CINXE.COM

<!DOCTYPE html><html><head><title>Geometry and the Imagination - David Hilbert, Stephan Cohn-Vossen - Google Books</title><link rel="stylesheet" href="/books/css/_ff77d0f0508c7e0bad470e7d6b7f28fa/kl_viewport_kennedy_full_bundle.css" type="text/css" /><link rel="stylesheet"href="https://fonts.googleapis.com/css2?family=Product+Sans:wght@400"><script src="/books/javascript/v2_ff77d0f0508c7e0bad470e7d6b7f28fa__en.js"></script><script>_OC_Hooks = ["_OC_Page", "_OC_SearchReload", "_OC_TocReload", "_OC_EmptyFunc", "_OC_SearchPage", "_OC_QuotePage" ];for (var _OC_i = 0; _OC_i < _OC_Hooks.length; _OC_i++) {eval("var " + _OC_Hooks[_OC_i] + ";");}function _OC_InitHooks () {for (var i = 0; i < _OC_Hooks.length; i++) {var func = arguments[i];eval( _OC_Hooks[i] + " = func;");}}</script><link rel="canonical" href="https://books.google.com/books/about/Geometry_and_the_Imagination.html?id=7WY5AAAAQBAJ"/><meta property="og:url" content="https://books.google.com/books/about/Geometry_and_the_Imagination.html?id=7WY5AAAAQBAJ"/><meta name="title" content="Geometry and the Imagination"/><meta name="description" content="This remarkable book endures as a true masterpiece of mathematical exposition. The book is overflowing with mathematical ideas, which are always explained clearly and elegantly, and above all, with penetrating insight. It is a joy to read, both for beginners and experienced mathematicians. Geometry and the Imagination is full of interesting facts, many of which you wish you had known before. The book begins with examples of the simplest curves and surfaces, including thread constructions of certain quadrics and other surfaces. The chapter on regular systems of points leads to the crystallographic groups and the regular polyhedra in $\mathbb{R}^3$. In this chapter, they also discuss plane lattices. By considering unit lattices, and throwing in a small amount of number theory when necessary, they effortlessly derive Leibniz's series: $\pi/4 = 1 - 1/3 + 1/5 - 1/7 + - \ldots$. In the section on lattices in three and more dimensions, the authors consider sphere-packing problems, including the famous Kepler problem. One of the most remarkable chapters is ``Projective Configurations''. In a short introductory section, Hilbert and Cohn-Vossen give perhaps the most concise and lucid description of why a general geometer would care about projective geometry and why such an ostensibly plain setup is truly rich in structure and ideas. The chapter on kinematics includes a nice discussion of linkages and the geometry of configurations of points and rods that are connected and, perhaps, constrained in some way. This topic in geometry has become increasingly important in recent times, especially in applications to robotics. This is another example of a simple situation that leads to a rich geometry. It would be hard to overestimate the continuing influence Hilbert-Cohn-Vossen's book has had on mathematicians of this century. It surely belongs in the pantheon of great mathematics books."/><meta property="og:title" content="Geometry and the Imagination"/><meta property="og:type" content="book"/><meta property="og:site_name" content="Google Books"/><meta property="og:image" content="https://books.google.com.sg/books/publisher/content?id=7WY5AAAAQBAJ&printsec=frontcover&img=1&zoom=1&edge=curl&imgtk=AFLRE73IpZMLTrtQjeRw7FowJJN_583FPTooI9sHemSuVg17OQWac_F0GZ226yAnbvuGdJSx_c0gmAsVuCbzL0Gl1U5C6obwGCwlKX-qxgMbPBlSN4GCCByc9n-M0mfwsUoiyWkE1HuT"/><link rel="image_src" href="https://books.google.com.sg/books/publisher/content?id=7WY5AAAAQBAJ&printsec=frontcover&img=1&zoom=1&edge=curl&imgtk=AFLRE73IpZMLTrtQjeRw7FowJJN_583FPTooI9sHemSuVg17OQWac_F0GZ226yAnbvuGdJSx_c0gmAsVuCbzL0Gl1U5C6obwGCwlKX-qxgMbPBlSN4GCCByc9n-M0mfwsUoiyWkE1HuT"/><script></script><style>#gbar,#guser{font-size:13px;padding-top:1px !important;}#gbar{height:22px}#guser{padding-bottom:7px !important;text-align:right}.gbh,.gbd{border-top:1px solid #c9d7f1;font-size:1px}.gbh{height:0;position:absolute;top:24px;width:100%}@media all{.gb1{height:22px;margin-right:.5em;vertical-align:top}#gbar{float:left}}a.gb1,a.gb4{text-decoration:underline !important}a.gb1,a.gb4{color:#00c !important}.gbi .gb4{color:#dd8e27 !important}.gbf .gb4{color:#900 !important} #gbar { padding:.3em .6em !important;}</style></head><body class=""><div id=gbar><nobr><a target=_blank class=gb1 href="https://www.google.com.sg/search?tab=pw">Search</a> <a target=_blank class=gb1 href="https://www.google.com.sg/imghp?hl=en&tab=pi">Images</a> <a target=_blank class=gb1 href="https://maps.google.com.sg/maps?hl=en&tab=pl">Maps</a> <a target=_blank class=gb1 href="https://play.google.com/?hl=en&tab=p8">Play</a> <a target=_blank class=gb1 href="https://www.youtube.com/?tab=p1">YouTube</a> <a target=_blank class=gb1 href="https://news.google.com/?tab=pn">News</a> <a target=_blank class=gb1 href="https://mail.google.com/mail/?tab=pm">Gmail</a> <a target=_blank class=gb1 href="https://drive.google.com/?tab=po">Drive</a> <a target=_blank class=gb1 style="text-decoration:none" href="https://www.google.com.sg/intl/en/about/products?tab=ph"><u>More</u> »</a></nobr></div><div id=guser width=100%><nobr><span id=gbn class=gbi></span><span id=gbf class=gbf></span><span id=gbe></span><a target=_top id=gb_70 href="https://www.google.com/accounts/Login?service=print&continue=https://books.google.com.sg/books%3Fid%3D7WY5AAAAQBAJ%26q%3Dparabolic%2Bpoints%26source%3Dgbs_word_cloud_r%26hl%3Den&hl=en&ec=GAZACg" class=gb4>Sign in</a></nobr></div><div class=gbh style=left:0></div><div class=gbh style=right:0></div><div role="alert" style="position: absolute; left: 0; right: 0;"><a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=parabolic+points&source=gbs_word_cloud_r&hl=en&output=html_text" title="Screen reader users: click this link for accessible mode. Accessible mode has the same essential features but works better with your reader."><img border="0" src="//www.google.com/images/cleardot.gif"alt="Screen reader users: click this link for accessible mode. Accessible mode has the same essential features but works better with your reader."></a></div><div class="kd-appbar"><h2 class="kd-appname"><a href="/books">Books</a></h2><div class="kd-buttonbar left" id="left-toolbar-buttons"><a id="appbar-view-print-sample-link" href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&printsec=frontcover&vq=parabolic+points&source=gbs_vpt_read"></a><a id="appbar-view-ebook-sample-link" href=""></a><a id="appbar-patents-prior-art-finder-link" href=""></a><a id="appbar-patents-discuss-this-link" href="" data-is-grant=""></a><a id="appbar-read-patent-link" href=""></a><a id="appbar-download-pdf-link" href=""></a></div><div class="kd-buttonbar right" id="right-toolbar-buttons"></div></div><div style="display: none"><ol id="ofe-gear-menu-contents" class="gbmcc"><li class="gbe gbmtc"><a class="gbmt goog-menuitem-content" id="" href="https://www.google.com/accounts/Login?service=print&continue=https://books.google.com.sg/books%3Fop%3Dlibrary&hl=en">My library</a></li><li class="gbe gbmtc"><a class="gbmt goog-menuitem-content" id="" href="http://books.google.com.sg/support/topic/4359341?hl=en-SG">Help</a></li><li class="gbe gbmtc"><a class="gbmt goog-menuitem-content" id="" href="https://books.google.com.sg/advanced_book_search">Advanced Book Search</a></li></ol></div><div id="volume-main"><div id="volume-left"><div id=menu_container ><div id="menu_scroll_wrapper"><div id="menu_scroll" role="navigation"><div id="gb-get-book-container"><a id="gb-get-book-content" href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&sitesec=buy&source=gbs_vpt_read">Get print book</a></div><p id="gb-get-book-not-available">No eBook available<p><h3 class=about_title><a name="buy_anchor"></a></h3><div id=buy class=about_content><div id=buy_v><ul style="list-style-type: none; padding-left: 0; margin: 0;"><li><a style="white-space:normal" href="http://www.ams.org/bookstore-getitem/item=CHEL-87-H" dir=ltr onMouseOver="this.href='http://www.ams.org/bookstore-getitem/item\x3dCHEL-87-H';return false" onMouseDown="this.href='/url?client\x3dca-print-ams\x26format\x3dgoogleprint\x26num\x3d0\x26id\x3d7WY5AAAAQBAJ\x26q\x3dhttp://www.ams.org/bookstore-getitem/item%3DCHEL-87-H\x26usg\x3dAOvVaw2YEwQkQfJX7f-knO2bjyLG\x26source\x3dgbs_buy_r';return true"><span dir=ltr>AMS Bookstore</span></a></li><li><a style="white-space:normal" href="http://www.amazon.com/gp/search?index=books&linkCode=qs&keywords=9780821819982" dir=ltr onMouseOver="this.href='http://www.amazon.com/gp/search?index\x3dbooks\x26linkCode\x3dqs\x26keywords\x3d9780821819982';return false" onMouseDown="this.href='/url?client\x3dca-print-ams\x26format\x3dgoogleprint\x26num\x3d0\x26id\x3d7WY5AAAAQBAJ\x26q\x3dhttp://www.amazon.com/gp/search%3Findex%3Dbooks%26linkCode%3Dqs%26keywords%3D9780821819982\x26usg\x3dAOvVaw3Y0iHDhtaI-80a7Z0JmU79\x26source\x3dgbs_buy_r';return true"><span dir=ltr>Amazon.com</span></a></li><li><a style="white-space:normal" href="http://www.mphonline.com/books/nsearch.aspx?do=detail&pcode=9780821819982" dir=ltr onMouseOver="this.href='http://www.mphonline.com/books/nsearch.aspx?do\x3ddetail\x26pcode\x3d9780821819982';return false" onMouseDown="this.href='/url?client\x3dca-print-ams\x26format\x3dgoogleprint\x26num\x3d0\x26id\x3d7WY5AAAAQBAJ\x26q\x3dhttp://www.mphonline.com/books/nsearch.aspx%3Fdo%3Ddetail%26pcode%3D9780821819982\x26usg\x3dAOvVaw0RHiaCk6IzdEkPlaMbFxIV\x26source\x3dgbs_buy_r';return true"><span dir=ltr>MPH</span></a></li><li><a style="white-space:normal" href="https://www.popular.com.sg/catalogsearch/result/?q=9780821819982" dir=ltr onMouseOver="this.href='https://www.popular.com.sg/catalogsearch/result/?q\x3d9780821819982';return false" onMouseDown="this.href='/url?client\x3dca-print-ams\x26format\x3dgoogleprint\x26num\x3d0\x26id\x3d7WY5AAAAQBAJ\x26q\x3dhttps://www.popular.com.sg/catalogsearch/result/%3Fq%3D9780821819982\x26usg\x3dAOvVaw1cte2Cv9_6z2qwxKdDO3Au\x26source\x3dgbs_buy_r';return true"><span dir=ltr>Popular</span></a></li><li><hr style="margin-right: 20%; color: #666;"></li><li><a style="white-space:normal" href="https://books.google.com.sg/url?id=7WY5AAAAQBAJ&pg=PA202&q=http://www.worldcat.org/oclc/1033654178&clientid=librarylink&usg=AOvVaw2S9ifeUWB5GixjfGCjfuce&source=gbs_buy_r"><span dir=ltr>Find in a library</span></a></li><li><a class="secondary" style="white-space:normal" href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&sitesec=buy&source=gbs_buy_r" id="get-all-sellers-link"><span dir=ltr>All sellers</span> »</a></li></ul></div></div><div class=menu id=menu><div class="menu_content" style="margin-bottom:6px"><div style="margin-bottom:4px"><div class="sidebarnav"><table border="0" cellpadding="0" cellspacing="0"><tr><td><div class="sidebarcover"><a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&printsec=frontcover&vq=parabolic+points" onclick="_OC_Page('PP1',this.href); return false;"><img src="https://books.google.com.sg/books/publisher/content?id=7WY5AAAAQBAJ&printsec=frontcover&img=1&zoom=5&edge=curl&imgtk=AFLRE70xmMfyRaysY4IKq3GesmbbyuzXNWjeE27RSJD5QLom2TgVxZc1fSqS2L361YSr91Osf2PsrJQtyKWw28CdlpqpbBeFrB3yUA7SXN0wb9LEk_siOvsPxJIHZihXCE87YWY2lEeL" alt="Front Cover" title="Front Cover" height=80 border=1 id=summary-frontcover ></a></div></td><td></td></tr></table></div><div style="clear:both"></div></div><div id="volume-info-sidebar"><h1 class="gb-volume-title" dir=ltr>Geometry and the Imagination</h1><span class="addmd">By David Hilbert, Stephan Cohn-Vossen</span></div><div style="margin-bottom:3px"><form action=/books id=search_form style="margin:0px;padding:0px;" method=get> <input type=hidden name="id" value="7WY5AAAAQBAJ"><table cellpadding=0 cellspacing=0 class="swv-table"><tr><td class="swv-td-search"><span><input id=search_form_input type=text maxlength=1024 class="text_flat swv-input-search" aria-label="Search in this book" name=q value="" title="Go" accesskey=i></span></td><td class="swv-td-space"><div> </div></td><td><input type=submit value="Go"></td></tr></table><script type="text/javascript">if (window['_OC_autoDir']) {_OC_autoDir('search_form_input');}</script></form></div><div><p><a id="sidebar-atb-link" href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&vq=parabolic+points&source=gbs_navlinks_s"><span dir=ltr>About this book</span></a></p></div></div></div><div class="sidebar-hr"></div><div class="ebook-promo"><a href="https://play.google.com/store/books"><img border="0" src="/googlebooks/images/ebook_promo.png" /><h3 class="section">Shop for Books on Google Play</h3><p class="ebook-promo-description">Browse the world's largest eBookstore and start reading today on the web, tablet, phone, or ereader.</p><p class="ebook-promo-clickme">Go to Google Play Now »</p></a></div><div class="sidebar-hr"></div><div><div id="navbarContainer" class="gb-navbar"></div><script>_OC_InitNavbar({"child_node":[{"title":"My library","url":"https://books.google.com.sg/books?uid=114584440181414684107\u0026source=gbs_lp_bookshelf_list","id":"my_library","collapsed":true},{"title":"My History","url":"","id":"my_history","collapsed":true}],"highlighted_node_id":""});</script><h3 class=about_title><a name="pub_info_anchor"></a></h3><div id=pub_info class=about_content><div id=pub_info_v><table cellspacing=0><tr><td><a href="https://books.google.com.sg/url?id=7WY5AAAAQBAJ&pg=PA202&q=http://www.ams.org/bookstore&clientid=ca-print-ams&linkid=1&usg=AOvVaw270BAOptGwzIAFZwB1jUPl&source=gbs_pub_info_r" style="text-decoration:none"><img width=175 height=35 border=0 src="https://pagead2.googlesyndication.com/pagead/imgad?id=CMz65v2v1LDEugEQvgEYJjIIEXz9vzia33I" alt="American Mathematical Soc."></a><tr><td style="font-size:84.6%;color:#666666">Pages displayed by permission of <a class=link_aux href="https://books.google.com.sg/url?id=7WY5AAAAQBAJ&pg=PA202&q=http://www.ams.org/bookstore&clientid=ca-print-ams&linkid=1&usg=AOvVaw270BAOptGwzIAFZwB1jUPl&source=gbs_pub_info_r">American Mathematical Soc.</a>. <a style="color:#7777cc;white-space:normal" href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&printsec=copyright&vq=parabolic+points&source=gbs_pub_info_r">Copyright</a>. </table></div></div></div></div></div></div></div><div id="volume-center"><div id="scroll_atb" role="main"><div id="toolbar_container"><div style="float:left;white-space:nowrap"><table cellpadding=0 cellspacing=0><tr><td id="l_toolbar"></td><td class=toolbar-pc-cell><table cellpadding=0 cellspacing=0><tr><td class=no-jump-cell align=right><span id=page_label style="margin-right:.5em">Page 202</span></td><td class=arrow style="padding-right:2px"><a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&pg=PA201&lpg=PA202&focus=viewport&vq=parabolic+points" onclick="_OC_EmptyFunc(this.href); return false;"><div class=pagination><div id=prev_btn alt="Previous Page" title="Previous Page" class="SPRITE_pagination_v2_left"></div></div></a></td><td class=arrow><a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&pg=PA204&lpg=PA202&focus=viewport&vq=parabolic+points" onclick="_OC_EmptyFunc(this.href); return false;"><div class=pagination><div id=next_btn alt="Next available page" title="Next available page" class="SPRITE_pagination_v2_right"></div></div></a></td></tr></table></td><td>  </td><td id=view_toolbar></td><td id=view_new></td></tr></table></div><div style="float:right"><table cellpadding=0 cellspacing=0><tr><td></td><td id="r_toolbar" style="white-space:nowrap"></td></tr></table></div><div style="clear:both"></div></div><div id="search_bar"></div><div class="gback"><div id="viewport" class="viewport" tabindex="0"><a name="page" accesskey="c"></a><table class="viewport-table" id="container" align="center" cellpadding="0" cellspacing="0"><tr><td valign="top" align="center"><noscript><style type=text/css>.imgg { width:575px;height:821px;background:#eee;padding-bottom:25px}</style><div class=imgg><div align=center><table border=0 cellpadding=0 cellspacing=0 width=500 align=center style="margin-top:2em"><tr><td rowspan=2 valign=top style="width:9px;background:#fff url('/googlebooks/bbl_l.gif') top left repeat-y"><img src="/googlebooks/bbl_tl.gif" width=9 height=7 alt=""></td><td style="background:url('/googlebooks/bbl_t.gif') top left repeat-x"><img width=1 height=7 alt=""></td><td rowspan=2 valign=top style="width:10px;background:#fff url('/googlebooks/bbl_r.gif') top right repeat-y"><img src="/googlebooks/bbl_tr.gif" width=10 height=7 alt=""></td></tr><tr><td align=center style="background:#ff9;text-align:center;line-height:1.2em"><div style="margin:1em"><img width=60 height=60 align=absmiddle src="/googlebooks/restricted_logo.gif" alt=""> <span style="font-weight:bold;font-size:1.2em"><br>Restricted Page</span></div><div style="margin:1em" align=left>You have reached your viewing limit for this book (<a href=https://books.google.com.sg/support/answer/43729?topic=9259&hl=en>why?</a>).</div></td></tr><tr><td><img src="/googlebooks/bbl_bl.gif" width=9 height=9 alt=""></td><td style="background:url('/googlebooks/bbl_b.gif') bottom left repeat-x"><img width=1 height=9 alt=""></td><td><img src="/googlebooks/bbl_br.gif" width=10 height=9 alt=""></td></tr></table></div></div></noscript></td></tr></table></div></div><script>_OC_addFlags({Host:"https://books.google.com.sg/", IsBooksUnifiedLeftNavEnabled:1, IsBooksRentalEnabled:1, IsBrowsingHistoryEnabled:1, IsZipitFolderCollectionEnabled:1});_OC_Run({"page":[{"pid":"PP1","flags":32,"order":0},{"pid":"PR1","order":1,"title":"i"},{"pid":"PR2","order":2,"title":"ii"},{"pid":"PR7","order":7,"title":"vii"},{"pid":"PR8","order":8,"title":"viii"},{"pid":"PR9","order":9,"title":"ix"},{"pid":"PR10","order":10,"title":"x"},{"pid":"PA1","order":11,"title":"1"},{"pid":"PA2","order":12,"title":"2"},{"pid":"PA3","order":13,"title":"3"},{"pid":"PA4","order":14,"title":"4"},{"pid":"PA5","order":15,"title":"5"},{"pid":"PA6","order":16,"title":"6"},{"pid":"PA7","order":17,"title":"7"},{"pid":"PA8","order":18,"title":"8"},{"pid":"PA9","order":19,"title":"9"},{"pid":"PA10","order":20,"title":"10"},{"pid":"PA11","order":21,"title":"11"},{"pid":"PA12","order":22,"title":"12"},{"pid":"PA13","order":23,"title":"13"},{"pid":"PA14","order":24,"title":"14"},{"pid":"PA15","order":25,"title":"15"},{"pid":"PA16","order":26,"title":"16"},{"pid":"PA17","order":27,"title":"17"},{"pid":"PA18","order":28,"title":"18"},{"pid":"PA19","order":29,"title":"19"},{"pid":"PA20","order":30,"title":"20"},{"pid":"PA21","order":31,"title":"21"},{"pid":"PA22","order":32,"title":"22"},{"pid":"PA23","order":33,"title":"23"},{"pid":"PA24","order":34,"title":"24"},{"pid":"PA25","order":35,"title":"25"},{"pid":"PA26","order":36,"title":"26"},{"pid":"PA27","order":37,"title":"27"},{"pid":"PA28","order":38,"title":"28"},{"pid":"PA29","order":39,"title":"29"},{"pid":"PA30","order":40,"title":"30"},{"pid":"PA31","order":41,"title":"31"},{"pid":"PA32","order":42,"title":"32"},{"pid":"PA33","order":43,"title":"33"},{"pid":"PA34","order":44,"title":"34"},{"pid":"PA35","order":45,"title":"35"},{"pid":"PA36","order":46,"title":"36"},{"pid":"PA37","order":47,"title":"37"},{"pid":"PA38","order":48,"title":"38"},{"pid":"PA39","order":49,"title":"39"},{"pid":"PA40","order":50,"title":"40"},{"pid":"PA41","order":51,"title":"41"},{"pid":"PA42","order":52,"title":"42"},{"pid":"PA43","order":53,"title":"43"},{"pid":"PA44","order":54,"title":"44"},{"pid":"PA45","order":55,"title":"45"},{"pid":"PA46","order":56,"title":"46"},{"pid":"PA47","order":57,"title":"47"},{"pid":"PA48","order":58,"title":"48"},{"pid":"PA49","order":59,"title":"49"},{"pid":"PA50","order":60,"title":"50"},{"pid":"PA51","order":61,"title":"51"},{"pid":"PA52","order":62,"title":"52"},{"pid":"PA53","order":63,"title":"53"},{"pid":"PA54","order":64,"title":"54"},{"pid":"PA55","order":65,"title":"55"},{"pid":"PA56","order":66,"title":"56"},{"pid":"PA57","order":67,"title":"57"},{"pid":"PA58","order":68,"title":"58"},{"pid":"PA59","order":69,"title":"59"},{"pid":"PA60","order":70,"title":"60"},{"pid":"PA61","order":71,"title":"61"},{"pid":"PA62","order":72,"title":"62"},{"pid":"PA63","order":73,"title":"63"},{"pid":"PA64","order":74,"title":"64"},{"pid":"PA65","order":75,"title":"65"},{"pid":"PA66","order":76,"title":"66"},{"pid":"PA67","order":77,"title":"67"},{"pid":"PA68","order":78,"title":"68"},{"pid":"PA69","order":79,"title":"69"},{"pid":"PA70","order":80,"title":"70"},{"pid":"PA71","order":81,"title":"71"},{"pid":"PA72","order":82,"title":"72"},{"pid":"PA73","order":83,"title":"73"},{"pid":"PA74","order":84,"title":"74"},{"pid":"PA75","order":85,"title":"75"},{"pid":"PA76","order":86,"title":"76"},{"pid":"PA77","order":87,"title":"77"},{"pid":"PA78","order":88,"title":"78"},{"pid":"PA79","order":89,"title":"79"},{"pid":"PA80","order":90,"title":"80"},{"pid":"PA81","order":91,"title":"81"},{"pid":"PA82","order":92,"title":"82"},{"pid":"PA83","order":93,"title":"83"},{"pid":"PA84","order":94,"title":"84"},{"pid":"PA85","order":95,"title":"85"},{"pid":"PA86","order":96,"title":"86"},{"pid":"PA87","order":97,"title":"87"},{"pid":"PA88","order":98,"title":"88"},{"pid":"PA89","order":99,"title":"89"},{"pid":"PA90","order":100,"title":"90"},{"pid":"PA91","order":101,"title":"91"},{"pid":"PA92","order":102,"title":"92"},{"pid":"PA93","order":103,"title":"93"},{"pid":"PA94","order":104,"title":"94"},{"pid":"PA95","order":105,"title":"95"},{"pid":"PA96","order":106,"title":"96"},{"pid":"PA97","order":107,"title":"97"},{"pid":"PA98","order":108,"title":"98"},{"pid":"PA99","order":109,"title":"99"},{"pid":"PA100","order":110,"title":"100"},{"pid":"PA101","order":111,"title":"101"},{"pid":"PA102","order":112,"title":"102"},{"pid":"PA103","order":113,"title":"103"},{"pid":"PA104","order":114,"title":"104"},{"pid":"PA105","order":115,"title":"105"},{"pid":"PA106","order":116,"title":"106"},{"pid":"PA107","order":117,"title":"107"},{"pid":"PA108","order":118,"title":"108"},{"pid":"PA109","order":119,"title":"109"},{"pid":"PA110","order":120,"title":"110"},{"pid":"PA111","order":121,"title":"111"},{"pid":"PA112","order":122,"title":"112"},{"pid":"PA113","order":123,"title":"113"},{"pid":"PA114","order":124,"title":"114"},{"pid":"PA115","order":125,"title":"115"},{"pid":"PA116","order":126,"title":"116"},{"pid":"PA117","order":127,"title":"117"},{"pid":"PA118","order":128,"title":"118"},{"pid":"PA119","order":129,"title":"119"},{"pid":"PA120","order":130,"title":"120"},{"pid":"PA121","order":131,"title":"121"},{"pid":"PA122","order":132,"title":"122"},{"pid":"PA123","order":133,"title":"123"},{"pid":"PA124","order":134,"title":"124"},{"pid":"PA125","order":135,"title":"125"},{"pid":"PA126","order":136,"title":"126"},{"pid":"PA127","order":137,"title":"127"},{"pid":"PA128","order":138,"title":"128"},{"pid":"PA129","order":139,"title":"129"},{"pid":"PA130","order":140,"title":"130"},{"pid":"PA131","order":141,"title":"131"},{"pid":"PA132","order":142,"title":"132"},{"pid":"PA133","order":143,"title":"133"},{"pid":"PA134","order":144,"title":"134"},{"pid":"PA135","order":145,"title":"135"},{"pid":"PA136","order":146,"title":"136"},{"pid":"PA137","order":147,"title":"137"},{"pid":"PA138","order":148,"title":"138"},{"pid":"PA139","order":149,"title":"139"},{"pid":"PA140","order":150,"title":"140"},{"pid":"PA141","order":151,"title":"141"},{"pid":"PA142","order":152,"title":"142"},{"pid":"PA143","order":153,"title":"143"},{"pid":"PA144","order":154,"title":"144"},{"pid":"PA145","order":155,"title":"145"},{"pid":"PA146","order":156,"title":"146"},{"pid":"PA147","order":157,"title":"147"},{"pid":"PA148","order":158,"title":"148"},{"pid":"PA149","order":159,"title":"149"},{"pid":"PA150","order":160,"title":"150"},{"pid":"PA151","order":161,"title":"151"},{"pid":"PA152","order":162,"title":"152"},{"pid":"PA153","order":163,"title":"153"},{"pid":"PA154","order":164,"title":"154"},{"pid":"PA155","order":165,"title":"155"},{"pid":"PA156","order":166,"title":"156"},{"pid":"PA157","order":167,"title":"157"},{"pid":"PA158","order":168,"title":"158"},{"pid":"PA159","order":169,"title":"159"},{"pid":"PA160","order":170,"title":"160"},{"pid":"PA161","order":171,"title":"161"},{"pid":"PA162","order":172,"title":"162"},{"pid":"PA163","order":173,"title":"163"},{"pid":"PA164","order":174,"title":"164"},{"pid":"PA165","order":175,"title":"165"},{"pid":"PA166","order":176,"title":"166"},{"pid":"PA167","order":177,"title":"167"},{"pid":"PA168","order":178,"title":"168"},{"pid":"PA169","order":179,"title":"169"},{"pid":"PA170","order":180,"title":"170"},{"pid":"PA171","order":181,"title":"171"},{"pid":"PA172","order":182,"title":"172"},{"pid":"PA173","order":183,"title":"173"},{"pid":"PA174","order":184,"title":"174"},{"pid":"PA175","order":185,"title":"175"},{"pid":"PA176","order":186,"title":"176"},{"pid":"PA177","order":187,"title":"177"},{"pid":"PA178","order":188,"title":"178"},{"pid":"PA179","order":189,"title":"179"},{"pid":"PA182","order":192,"title":"182"},{"pid":"PA183","order":193,"title":"183"},{"pid":"PA185","order":195,"title":"185"},{"pid":"PA189","order":199,"title":"189"},{"pid":"PA190","order":200,"title":"190"},{"pid":"PA192","order":202,"title":"192"},{"pid":"PA193","order":203,"title":"193"},{"pid":"PA194","order":204,"title":"194"},{"pid":"PA196","order":206,"title":"196"},{"pid":"PA197","order":207,"title":"197"},{"pid":"PA198","order":208,"title":"198"},{"pid":"PA201","flags":8,"order":211,"title":"201"},{"pid":"PA202","flags":8,"order":212,"title":"202"},{"pid":"PA204","flags":8,"order":214,"title":"204"},{"pid":"PA205","flags":8,"order":215,"title":"205"},{"pid":"PA206","order":216,"title":"206"},{"pid":"PA208","order":218,"title":"208"},{"pid":"PA211","order":221,"title":"211"},{"pid":"PA212","order":222,"title":"212"},{"pid":"PA213","order":223,"title":"213"},{"pid":"PA214","order":224,"title":"214"},{"pid":"PA216","order":226,"title":"216"},{"pid":"PA218","order":228,"title":"218"},{"pid":"PA219","order":229,"title":"219"},{"pid":"PA221","order":231,"title":"221"},{"pid":"PA222","order":232,"title":"222"},{"pid":"PA223","order":233,"title":"223"},{"pid":"PA226","order":236,"title":"226"},{"pid":"PA227","order":237,"title":"227"},{"pid":"PA228","order":238,"title":"228"},{"pid":"PA229","order":239,"title":"229"},{"pid":"PA230","order":240,"title":"230"},{"pid":"PA231","order":241,"title":"231"},{"pid":"PA233","order":243,"title":"233"},{"pid":"PA234","order":244,"title":"234"},{"pid":"PA235","order":245,"title":"235"},{"pid":"PA236","order":246,"title":"236"},{"pid":"PA237","order":247,"title":"237"},{"pid":"PA239","order":249,"title":"239"},{"pid":"PA240","order":250,"title":"240"},{"pid":"PA241","order":251,"title":"241"},{"pid":"PA242","order":252,"title":"242"},{"pid":"PA244","order":254,"title":"244"},{"pid":"PA245","order":255,"title":"245"},{"pid":"PA246","order":256,"title":"246"},{"pid":"PA248","order":258,"title":"248"},{"pid":"PA249","order":259,"title":"249"},{"pid":"PA251","order":261,"title":"251"},{"pid":"PA254","order":264,"title":"254"},{"pid":"PA255","order":265,"title":"255"},{"pid":"PA257","order":267,"title":"257"},{"pid":"PA259","order":269,"title":"259"},{"pid":"PA263","order":273,"title":"263"},{"pid":"PA264","order":274,"title":"264"},{"pid":"PA266","order":276,"title":"266"},{"pid":"PA268","order":278,"title":"268"},{"pid":"PA270","order":280,"title":"270"},{"pid":"PA271","order":281,"title":"271"},{"pid":"PA272","order":282,"title":"272"},{"pid":"PA275","order":285,"title":"275"},{"pid":"PA276","order":286,"title":"276"},{"pid":"PA279","order":289,"title":"279"},{"pid":"PA280","order":290,"title":"280"},{"pid":"PA281","order":291,"title":"281"},{"pid":"PA283","order":293,"title":"283"},{"pid":"PA286","order":296,"title":"286"},{"pid":"PA287","order":297,"title":"287"},{"pid":"PA288","order":298,"title":"288"},{"pid":"PA290","order":300,"title":"290"},{"pid":"PA292","order":302,"title":"292"},{"pid":"PA294","order":304,"title":"294"},{"pid":"PA295","order":305,"title":"295"},{"pid":"PA296","order":306,"title":"296"},{"pid":"PA297","order":307,"title":"297"},{"pid":"PA298","order":308,"title":"298"},{"pid":"PA299","order":309,"title":"299"},{"pid":"PA300","order":310,"title":"300"},{"pid":"PA301","order":311,"title":"301"},{"pid":"PA302","order":312,"title":"302"},{"pid":"PA303","order":313,"title":"303"},{"pid":"PA304","order":314,"title":"304"},{"pid":"PA306","order":316,"title":"306"},{"pid":"PA307","order":317,"title":"307"},{"pid":"PA312","order":322,"title":"312"},{"pid":"PA316","order":326,"title":"316"},{"pid":"PA317","order":327,"title":"317"},{"pid":"PA319","order":329,"title":"319"},{"pid":"PA323","order":333,"title":"323"},{"pid":"PA324","order":334,"title":"324"},{"pid":"PA325","order":335,"title":"325"},{"pid":"PA327","order":337,"title":"327"},{"pid":"PA328","order":338,"title":"328"},{"pid":"PA329","order":339,"title":"329"},{"pid":"PA330","order":340,"title":"330"},{"pid":"PA331","order":341,"title":"331"},{"pid":"PA333","order":343,"title":"333"},{"pid":"PA335","order":345,"title":"335"},{"pid":"PA336","order":346,"title":"336"},{"pid":"PA337","order":347,"title":"337"},{"pid":"PA339","order":349,"title":"339"},{"pid":"PA341","order":351,"title":"341"},{"pid":"PA343","order":353,"title":"343"},{"pid":"PA344","order":354,"title":"344"},{"pid":"PA345","order":355,"title":"345"},{"pid":"PA346","order":356,"title":"346"},{"pid":"PA347","order":357,"title":"347"},{"pid":"PA348","order":358,"title":"348"},{"pid":"PA349","order":359,"title":"349"},{"pid":"PA350","order":360,"title":"350"},{"pid":"PA351","order":361,"title":"351"},{"pid":"PA352","order":362,"title":"352"},{"pid":"PA353","order":363,"title":"353"},{"pid":"PA354","order":364,"title":"354"},{"pid":"PA355","order":365,"title":"355"},{"pid":"PA356","order":366,"title":"356"},{"pid":"PA357","order":367,"title":"357"},{"pid":"PA359","order":369,"title":"359"}],"prefix":"https://books.google.com.sg/books?id=7WY5AAAAQBAJ\u0026lpg=PA202\u0026vq=parabolic+points"},{"fullview":false,"page_width":575,"page_height":821,"font_height":12,"first_content_page":11,"disable_twopage":false,"initial_zoom_width_override":685,"show_print_pages_button":false,"title":"Geometry and the Imagination","subtitle":"","attribution":"By David Hilbert, Stephan Cohn-Vossen","additional_info":{"[JsonBookInfo]":{"BuyLinks":[{"Seller":"AMS Bookstore","Url":"http://www.ams.org/bookstore-getitem/item=CHEL-87-H","TrackingUrl":"/url?client=ca-print-ams\u0026format=googleprint\u0026num=0\u0026id=7WY5AAAAQBAJ\u0026q=http://www.ams.org/bookstore-getitem/item%3DCHEL-87-H\u0026usg=AOvVaw2YEwQkQfJX7f-knO2bjyLG","IsPublisher":true},{"Seller":"Amazon.com","Url":"http://www.amazon.com/gp/search?index=books\u0026linkCode=qs\u0026keywords=9780821819982","TrackingUrl":"/url?client=ca-print-ams\u0026format=googleprint\u0026num=0\u0026id=7WY5AAAAQBAJ\u0026q=http://www.amazon.com/gp/search%3Findex%3Dbooks%26linkCode%3Dqs%26keywords%3D9780821819982\u0026usg=AOvVaw3Y0iHDhtaI-80a7Z0JmU79"},{"Seller":"MPH","Url":"http://www.mphonline.com/books/nsearch.aspx?do=detail\u0026pcode=9780821819982","TrackingUrl":"/url?client=ca-print-ams\u0026format=googleprint\u0026num=0\u0026id=7WY5AAAAQBAJ\u0026q=http://www.mphonline.com/books/nsearch.aspx%3Fdo%3Ddetail%26pcode%3D9780821819982\u0026usg=AOvVaw0RHiaCk6IzdEkPlaMbFxIV"},{"Seller":"Popular","Url":"https://www.popular.com.sg/catalogsearch/result/?q=9780821819982","TrackingUrl":"/url?client=ca-print-ams\u0026format=googleprint\u0026num=0\u0026id=7WY5AAAAQBAJ\u0026q=https://www.popular.com.sg/catalogsearch/result/%3Fq%3D9780821819982\u0026usg=AOvVaw1cte2Cv9_6z2qwxKdDO3Au"}],"AboutUrl":"https://books.google.com.sg/books?id=7WY5AAAAQBAJ","PreviewUrl":"https://books.google.com.sg/books?id=7WY5AAAAQBAJ","allowed_syndication_flags":{"allow_disabling_chrome":true},"TocLine":[{"Title":"CHAPTER ","Pid":"PA1","PgNum":"1","Order":11},{"Title":"4 ","Pid":"PA19","PgNum":"19","Order":29},{"Title":"APPENDICES TO CHAPTER I ","Pid":"PA25","PgNum":"25","Order":35},{"Title":"CHAPTER II ","Pid":"PA32","PgNum":"32","Order":42},{"Title":"Crystals as Regular Systems of Points ","Pid":"PA52","PgNum":"52","Order":62},{"Title":"Plane Motions and their Composition Classification of ","Pid":"PA59","PgNum":"59","Order":69},{"Title":"The Crystallographic Groups of Motions in the Plane ","Pid":"PA70","PgNum":"70","Order":80},{"Title":"Crystallographic Classes and Groups of Motions in Space ","Pid":"PA81","PgNum":"81","Order":91},{"Title":"The Regular Polyhedra ","Pid":"PA89","PgNum":"89","Order":99},{"Title":"Perspective Ideal Elements and the Principle of Duality ","Pid":"PA112","PgNum":"112","Order":122},{"Title":"Ideal Elements and the Principle of Duality in Space ","Pid":"PA119","PgNum":"119","Order":129},{"Title":"Comparison of Pascals and Desargues Theorems ","Pid":"PA128","PgNum":"128","Order":138},{"Title":"Reyes Configuration ","Pid":"PA134","PgNum":"134","Order":144},{"Title":"Regular Polyhedra in Three and Four Dimensions ","Pid":"PA143","PgNum":"143","Order":153},{"Title":"Enumerative Methods of Geometry ","Pid":"PA157","PgNum":"157","Order":167},{"Title":"Schl盲flis DoubleSix ","Pid":"PA164","PgNum":"164","Order":174},{"Title":"CHAPTER IV ","Pid":"PA171","PgNum":"171","Order":181},{"Title":"Space Curves ","Pid":"PA178","PgNum":"178","Order":188},{"Title":"The Spherical Image and Gaussian Curvature ","Pid":"PA193","PgNum":"193","Order":203},{"Title":"Developable Surfaces Ruled Surfaces ","Pid":"PA204","PgNum":"204","Order":214},{"Title":"The Twisting of Space Curves ","Pid":"PA211","PgNum":"211","Order":221},{"Title":"Bendings Leaving a Surface Invariant ","Pid":"PA232","PgNum":"232","Order":242},{"Title":"Hyperbolic Geometry and its Relation to Euclidean and ","Pid":"PA242","PgNum":"242","Order":252},{"Title":"Stereographic Projection and CirclePreserving Trans ","Pid":"PA248","PgNum":"248","Order":258},{"Title":"CHAPTER V ","Pid":"PA272","PgNum":"272","Order":282},{"Title":"An Instrument for Constructing the Ellipse and its Roul ","Pid":"PA283","PgNum":"283","Order":293},{"Title":"Polyhedra ","Pid":"PA290","PgNum":"290","Order":300},{"Title":"OneSided Surfaces ","Pid":"PA302","PgNum":"302","Order":312},{"Title":"The Projective Plane as a Closed Surface ","Pid":"PA313","PgNum":"313","Order":323},{"Title":"Topological Mappings of a Surface onto Itself Fixed ","Pid":"PA324","PgNum":"324","Order":334},{"Title":"Conformal Mapping of the Torus ","Pid":"PA330","PgNum":"330","Order":340},{"Title":"APPENDICES TO CHAPTER VI ","Pid":"PA340","PgNum":"340","Order":350}]}},"table_of_contents_page_id":"PR7","max_resolution_image_width":1280,"max_resolution_image_height":1827,"num_toc_pages":3,"quality_info":"We think this book has good quality.","volume_id":"7WY5AAAAQBAJ","permission_info":"Pages displayed by permission of \u003ca class=link_aux href=\"https://books.google.com.sg/url?id=7WY5AAAAQBAJ\u0026pg=PA202\u0026q=http://www.ams.org/bookstore\u0026clientid=ca-print-ams\u0026linkid=1\u0026usg=AOvVaw270BAOptGwzIAFZwB1jUPl\u0026source=gbs_pub_info_r\"\u003eAmerican Mathematical Soc.\u003c/a\u003e","is_ebook":false,"volumeresult":{"has_flowing_text":false,"has_scanned_text":true,"can_download_pdf":false,"can_download_epub":false,"is_pdf_drm_enabled":false,"is_epub_drm_enabled":false},"publisher":"American Mathematical Soc.","publication_date":"1999","subject":"Mathematics","num_pages":357,"sample_url":"https://play.google.com/books/reader?id=7WY5AAAAQBAJ\u0026source=gbs_vpt_hover","synposis":"This remarkable book endures as a true masterpiece of mathematical exposition. The book is overflowing with mathematical ideas, which are always explained clearly and elegantly, and above all, with penetrating insight. It is a joy to read, both for beginners and experienced mathematicians. Geometry and the Imagination is full of interesting facts, many of which you wish you had known before. The book begins with examples of the simplest curves and surfaces, including thread constructions of certain quadrics and other surfaces. The chapter on regular systems of points leads to the crystallographic groups and the regular polyhedra in $\\mathbb{R}^3$. In this chapter, they also discuss plane lattices. By considering unit lattices, and throwing in a small amount of number theory when necessary, they effortlessly derive Leibniz's series: $\\pi/4 = 1 - 1/3 + 1/5 - 1/7 + - \\ldots$. In the section on lattices in three and more dimensions, the authors consider sphere-packing problems, including the famous Kepler problem. One of the most remarkable chapters is ``Projective Configurations''. In a short introductory section, Hilbert and Cohn-Vossen give perhaps the most concise and lucid description of why a general geometer would care about projective geometry and why such an ostensibly plain setup is truly rich in structure and ideas. The chapter on kinematics includes a nice discussion of linkages and the geometry of configurations of points and rods that are connected and, perhaps, constrained in some way. This topic in geometry has become increasingly important in recent times, especially in applications to robotics. This is another example of a simple situation that leads to a rich geometry. It would be hard to overestimate the continuing influence Hilbert-Cohn-Vossen's book has had on mathematicians of this century. It surely belongs in the pantheon of great mathematics books.","my_library_url":"https://www.google.com/accounts/Login?service=print\u0026continue=https://books.google.com.sg/books%3Fop%3Dlibrary\u0026hl=en","is_magazine":false,"is_public_domain":false,"last_page":{"pid":"PA359","order":369,"title":"359"}},{"enableUserFeedbackUI":true,"pseudocontinuous":true,"is_cobrand":false,"sign_in_url":"https://www.google.com/accounts/Login?service=print\u0026continue=https://books.google.com.sg/books%3Fid%3D7WY5AAAAQBAJ%26q%3Dparabolic%2Bpoints%26source%3Dgbs_word_cloud_r%26hl%3Den\u0026hl=en","isEntityPageViewport":false,"showViewportOnboarding":false,"showViewportPlainTextOnboarding":false},{"page":[{"pid":"PA202","flags":8,"order":212,"vq":"parabolic points"}]},null,{"number_of_results":13,"search_results":[{"page_id":"PR8","page_number":"viii","snippet_text":"... \u003cb\u003eParabolic Points\u003c/b\u003e . Lines of Curvature and Asymptotic Lines . Um- bilical Points , Minimal Surfaces , Monkey Saddles ..... 183 搂 29. The Spherical Image and Gaussian Curvature .. 搂 30. Developable Surfaces , Ruled Surfaces .... 搂 31\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=7WY5AAAAQBAJ\u0026pg=PR8\u0026vq=parabolic+points"},{"page_id":"PA12","page_number":"12","snippet_text":"... \u003cb\u003epoints\u003c/b\u003e of FIG . 18 FIG . 19 contact of this cone with the surface form a conic . Moreover , the quadrics are the ... \u003cb\u003eparabolic\u003c/b\u003e and the hyperbolic cylinder from the \u003cb\u003eparabola\u003c/b\u003e and the hyperbola by the same procedure . ( See Figs . 18\u0026nbsp;..."},{"page_id":"PA183","page_number":"183","snippet_text":"... \u003cb\u003eParabolic Points\u003c/b\u003e . Lines of Curvature and Asymptotic Lines ; Umbilical Points , Minimal Surfaces , Monkey Saddles In beginning our ... Points Lines of Curvature and Asymptotic Lines bilical Points, Minimal Surfaces, Monkey Saddles."},{"page_id":"PA192","page_number":"192","snippet_text":"... points and at hyperbolic points in yet another way , which will at the same time justify the terms \u0026quot; elliptic \u0026quot; and ... \u003cb\u003eparabolic point\u003c/b\u003e the corresponding process may lead to any one of several types of curves . At an umbilical\u0026nbsp;..."},{"page_id":"PA197","page_number":"197","snippet_text":"... \u003cb\u003eparabolic points\u003c/b\u003e assume a role intermediate between that of the elliptic and the hyperbolic points . We should therefore ex- pect that the Gaussian curvature at \u003cb\u003eparabolic points\u003c/b\u003e is equal to zero . This is readily verified by referring\u0026nbsp;..."},{"page_id":"PA198","page_number":"198","snippet_text":"... \u003cb\u003eparabolic points\u003c/b\u003e , are called the parabolic curves of the surface.2 Of course the presence of parabolic curves is inevitable only on those surfaces on which the Gaussian curvature assumes both signs . This does not happen on any of\u0026nbsp;..."},{"page_id":"PA201","page_number":"201","snippet_text":"... \u003cb\u003eparabolic\u003c/b\u003e circles are parallel ; hence each of the \u003cb\u003eparabolic\u003c/b\u003e circles is mapped into a single \u003cb\u003epoint\u003c/b\u003e of the sphere ... \u003cb\u003epoints\u003c/b\u003e , exactly once . The same is true for the FIG . 211 1 = 31 4 \u0026#39; part of the torus where the curvature is\u0026nbsp;..."},{"page_id":"PA202","page_number":"202","snippet_text":"... \u003cb\u003eparabolic point\u003c/b\u003e surrounded by a region of negative curva- ture ( Fig . 213 ) . This is the monkey saddle , described on page 191 . 1 \u0026#39; = 4 \u0026#39; FIG . 213 Evidently the points of the surface where the normals are parallel are the points\u0026nbsp;..."},{"page_id":"PA204","page_number":"204","snippet_text":"... \u003cb\u003eparabolic\u003c/b\u003e curve has a variable tangent plane , as in the case of the bell surface , then the asymptotic lines have cusps along the \u003cb\u003eparabolic\u003c/b\u003e curve ( Fig . 214 ) . If , on the other hand , all the \u003cb\u003epoints\u003c/b\u003e of a \u003cb\u003eparabolic\u003c/b\u003e curve have a\u0026nbsp;..."},{"page_id":"PA205","page_number":"205","snippet_text":"... points of this straight line , the straight line is composed of \u003cb\u003eparabolic points\u003c/b\u003e of the surface . And since the totality of these straight lines covers the entire surface ( for this reason we call them generators of the surface ) , it\u0026nbsp;..."},{"page_id":"PA271","page_number":"271","snippet_text":"... \u003cb\u003epoints\u003c/b\u003e of G ; by increasing the hyperbolic distance between images of these \u003cb\u003epoints\u003c/b\u003e , we eventually brought the ... \u003cb\u003eparabolic\u003c/b\u003e curves , at which the spherical image might return on itself . Yet the spherical image of Neovius\u0026nbsp;..."},{"page_id":"PA347","page_number":"347","snippet_text":"... parabolic . See Parabolic curvature ; Parabolic curve ; \u003cb\u003eParabolic point\u003c/b\u003e . positive . See Elliptic curvature . radius ... points of , 173-174 , 199 second - order , confocal , 4-6 , 17 , 188- 189. See also Conics . third - order\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=7WY5AAAAQBAJ\u0026pg=PA347\u0026vq=parabolic+points"},{"page_id":"PA353","page_number":"353","snippet_text":"... \u003cb\u003ePoint\u003c/b\u003e , branch , 202 , 271 , 322 Brianchon , 106-107 , 118 of contact , 172 of inflection , 102 , 173-174 , 177 , 183 , 198 , 207 of striction , 207-210 , 286 \u003cb\u003eparabolic\u003c/b\u003e , 184 , 186 , 191 , 197 , 202- 204 umbilical , 187-189 , 192 , 203\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=7WY5AAAAQBAJ\u0026pg=PA353\u0026vq=parabolic+points"}],"search_query_escaped":"parabolic points"},{});</script></div></div></div><script>(function() {var href = window.location.href;if (href.indexOf('?') !== -1) {var parameters = href.split('?')[1].split('&');for (var i = 0; i < parameters.length; i++) {var param = parameters[i].split('=');if (param[0] == 'focus') {var elem = document.getElementById(param[1]);if (elem) {elem.focus();}}}}})();</script>