{"id":3734,"date":"2026-04-07T10:24:50","date_gmt":"2026-04-07T09:24:50","guid":{"rendered":"http:\/\/www2.ual.es\/neotrie\/?p=3734"},"modified":"2026-04-07T13:09:34","modified_gmt":"2026-04-07T12:09:34","slug":"conics-in-3d","status":"publish","type":"post","link":"https:\/\/www2.ual.es\/neotrie\/conics-in-3d\/","title":{"rendered":"Conics in 3D"},"content":{"rendered":"\n<p>In this post, I would like to revisit a video from five years ago, taking advantage of the fact that we are about to publish an article in <a href=\"https:\/\/researchinformation.co.uk\/research-information-specialist-publications\/the-international-journal-for-technology-in-mathematics-education\/\" title=\"\">IJTME<\/a>, where this and other activities on conic sections are presented.<\/p>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Conics in space\" width=\"1080\" height=\"608\" src=\"https:\/\/www.youtube.com\/embed\/x0kT2yW72sc?feature=oembed\"  allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<p>We next show how to make an ellipse in space, trail by the point P. We follow the classical construction from two focus and one of its focal circles. This is an the last activity included in the book <a href=\"http:\/\/www2.ual.es\/neotrie\/comunidad\/\">1st steps in dynamic geometry in Neotrie VR<\/a>.  <\/p>\n\n\n\n<p>We start from two points F and G that we want to be the focus of the ellipse. That means that any point P in the ellipse must satisfy a given constant 2a=PF+PG (the major diameter of the ellipse). So we can choose A to fix both the plane, and also the constant radius AF=2a of one focal circle.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"576\" src=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-04-12-1024x576.png\" alt=\"\" class=\"wp-image-3739\" srcset=\"https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-04-12-1024x576.png 1024w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-04-12-300x169.png 300w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-04-12-768x432.png 768w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-04-12-1536x864.png 1536w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-04-12-1080x608.png 1080w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-04-12.png 1920w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><figcaption class=\"wp-element-caption\">Then we build the perpendicular direction point H to the plane AFG on F by selecting A, F, G.<\/figcaption><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"576\" src=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-05-33-1024x576.png\" alt=\"\" class=\"wp-image-3740\" srcset=\"https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-05-33-1024x576.png 1024w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-05-33-300x169.png 300w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-05-33-768x432.png 768w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-05-33-1536x864.png 1536w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-05-33-1080x608.png 1080w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-05-33.png 1920w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p>Press on the circle button on the slider tool, to be prepared to make a circle. Then select the center F, the point A, and finally H. This will make a point B moving on a circle centered at F, of radius FA, and perpendicular to FH.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"576\" src=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-06-34-1024x576.png\" alt=\"\" class=\"wp-image-3741\" srcset=\"https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-06-34-1024x576.png 1024w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-06-34-300x169.png 300w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-06-34-768x432.png 768w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-06-34-1536x864.png 1536w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-06-34-1080x608.png 1080w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-06-34.png 1920w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p><\/p>\n\n\n\n<p>Use the trace pencil to draw this point B, to show the focal circle if you like. <\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"576\" src=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-11-54-1024x576.png\" alt=\"\" class=\"wp-image-3742\" srcset=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-11-54-980x551.png 980w, http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-11-54-480x270.png 480w\" sizes=\"(min-width: 0px) and (max-width: 480px) 480px, (min-width: 481px) and (max-width: 980px) 980px, (min-width: 981px) 1024px, 100vw\" \/><\/figure>\n\n\n\n<p>The construction continues by taking the bisection line of G and B on the plane FAG and cutting it FB. There is no specific tool to make that bisection. For that, make the midpoint C of B and G. <\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"576\" src=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-13-02-1024x576.png\" alt=\"\" class=\"wp-image-3743\" srcset=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-13-02-980x551.png 980w, http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-13-02-480x270.png 480w\" sizes=\"(min-width: 0px) and (max-width: 480px) 480px, (min-width: 481px) and (max-width: 980px) 980px, (min-width: 981px) 1024px, 100vw\" \/><\/figure>\n\n\n\n<p>Make the parallel point D of FH on C. <\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"576\" src=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-14-17-1024x576.png\" alt=\"\" class=\"wp-image-3744\" srcset=\"https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-14-17-1024x576.png 1024w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-14-17-300x169.png 300w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-14-17-768x432.png 768w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-14-17-1536x864.png 1536w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-14-17-1080x608.png 1080w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-14-17.png 1920w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p>Make E with the perpendicular tool by touching D, C, G. The line CE is the bisection of GB on the plane FGA.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"576\" src=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-15-28-1024x576.png\" alt=\"\" class=\"wp-image-3745\" srcset=\"https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-15-28-1024x576.png 1024w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-15-28-300x169.png 300w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-15-28-768x432.png 768w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-15-28-1536x864.png 1536w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-15-28-1080x608.png 1080w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-15-28.png 1920w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p>Let $P$ the intersection of the lines FA and CE. <\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"576\" src=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-16-56-1024x576.png\" alt=\"\" class=\"wp-image-3747\" srcset=\"https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-16-56-1024x576.png 1024w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-16-56-300x169.png 300w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-16-56-768x432.png 768w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-16-56-1536x864.png 1536w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-16-56-1080x608.png 1080w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-16-56.png 1920w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p>Paint the point P with the loci pencil. <\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"576\" src=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-17-31-1024x576.png\" alt=\"\" class=\"wp-image-3748\" srcset=\"https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-17-31-1024x576.png 1024w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-17-31-300x169.png 300w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-17-31-768x432.png 768w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-17-31-1536x864.png 1536w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-17-31-1080x608.png 1080w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-17-31.png 1920w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p>The point P draws the desired ellipse with focus F and G on the plane FGA.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"576\" src=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-17-49-1024x576.png\" alt=\"\" class=\"wp-image-3749\" srcset=\"https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-17-49-1024x576.png 1024w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-17-49-300x169.png 300w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-17-49-768x432.png 768w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-17-49-1536x864.png 1536w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-17-49-1080x608.png 1080w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-17-49.png 1920w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"576\" src=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-17-58-1024x576.png\" alt=\"\" class=\"wp-image-3737\" srcset=\"https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-17-58-1024x576.png 1024w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-17-58-300x169.png 300w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-17-58-768x432.png 768w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-17-58-1536x864.png 1536w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-17-58-1080x608.png 1080w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/02\/screen_1920x1080_2020-01-09_18-17-58.png 1543w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p>Note that moving the points A and one of the focus can produce hyperbolas (if 2a&lt;FG) or circles (if F=G). We recommend to disable the paste option between vertices for this dynamic experiment.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>In this post, I would like to revisit a video from five years ago, taking advantage of the fact that we are about to publish an article in IJTME, where this and other activities on conic sections are presented. We next show how to make an ellipse in space, trail by the point P. We [&hellip;]<\/p>\n","protected":false},"author":3,"featured_media":3737,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_et_pb_use_builder":"","_et_pb_old_content":"","_et_gb_content_width":"","om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[12,43],"tags":[61,96],"class_list":["post-3734","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-bachillerato","category-universidad","tag-conicas","tag-curvas"],"aioseo_notices":[],"views":15,"_links":{"self":[{"href":"https:\/\/www2.ual.es\/neotrie\/wp-json\/wp\/v2\/posts\/3734","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www2.ual.es\/neotrie\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www2.ual.es\/neotrie\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www2.ual.es\/neotrie\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/www2.ual.es\/neotrie\/wp-json\/wp\/v2\/comments?post=3734"}],"version-history":[{"count":7,"href":"https:\/\/www2.ual.es\/neotrie\/wp-json\/wp\/v2\/posts\/3734\/revisions"}],"predecessor-version":[{"id":11507,"href":"https:\/\/www2.ual.es\/neotrie\/wp-json\/wp\/v2\/posts\/3734\/revisions\/11507"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www2.ual.es\/neotrie\/wp-json\/wp\/v2\/media\/3737"}],"wp:attachment":[{"href":"https:\/\/www2.ual.es\/neotrie\/wp-json\/wp\/v2\/media?parent=3734"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www2.ual.es\/neotrie\/wp-json\/wp\/v2\/categories?post=3734"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www2.ual.es\/neotrie\/wp-json\/wp\/v2\/tags?post=3734"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}