{"id":133,"date":"2017-08-23T09:04:15","date_gmt":"2017-08-23T09:04:15","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=133"},"modified":"2017-08-30T06:30:26","modified_gmt":"2017-08-30T06:30:26","slug":"two-dimensional-representations-of-three-dimensional-objects","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/two-dimensional-representations-of-three-dimensional-objects\/","title":{"rendered":"Two-Dimensional Representations of Three-Dimensional Objects"},"content":{"rendered":"<div class=\"twelve columns\" style=\"margin-top: 10%;\">\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/classifying-two-and-three-dimensional-solids\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/geometry-spatial-reasoning\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/triangles\">Next Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Two-Dimensional Representations of Three-Dimensional Objects<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson, you will study the way two-dimensional drawings in one- or two-point perspective can also\u00a0represent three-dimensional objects.<\/p>\n<h4>Previously Covered:<\/h4>\n<ul>\n<li>A<em><strong> polyhedron\u00a0<\/strong><\/em>is three-dimensional figure with faces that are all polygons.<\/li>\n<li>Any polyhedron satisfies <strong><em>Euler\u2019s\u00a0Formula<\/em><\/strong>:\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s3_p1_clip_image002.gif\" width=\"86\" height=\"11\" name=\"graphics2\" align=\"BOTTOM\" border=\"0\" \/>,\u00a0where <em>F\u00a0<\/em>= number of\u00a0faces, <em>E <\/em>=\u00a0number of edges, and <em>V\u00a0<\/em>= number of\u00a0vertices.<\/li>\n<\/ul>\n<section>\n<h3><strong>Using two-point perspective to represent a cube<\/strong><\/h3>\n<p><strong>Step 1 <\/strong>Draw one edge of the polyhedron and the\u00a0horizon line. Choose two points on the horizon line. We call these\u00a0<em><strong>vanishing points<\/strong><\/em>.<\/p>\n<p align=\"CENTER\"><strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/Math%20Mod%204.1%20Art%20038.JPG\" alt=\"edge of a polyhedron and the horizon line &gt;&gt;\" width=\"164\" height=\"113\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/strong><\/p>\n<p><strong>Step 2 <\/strong>Draw a dashed line connecting each of\u00a0the two endpoints with each vanishing point.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/Math%20Mod%204.1%20Art%20039.JPG\" alt=\"dashed line connecting each of the two endpoints with each vanishing point\" width=\"164\" height=\"106\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p><strong>Step 3 <\/strong>Draw two segments parallel to the\u00a0original edge between the dashed lines on each side. Connect the\u00a0top of each new segment to the opposite vanishing point, as shown.<\/p>\n<p align=\"CENTER\"><strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/Math%20Mod%204.1%20Art%20040.JPG\" alt=\"Connect the top of each new segment to the opposite vanishing point\" width=\"164\" height=\"106\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/strong><\/p>\n<p><strong>Step 4 <\/strong>Fill in the necessary dashed line\u00a0segments to complete the visible faces and erase the remaining\u00a0dashed lines.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/Math%20Mod%204.1%20Art%20041.JPG\" alt=\"Figure without dashed lines\" width=\"164\" height=\"106\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Which of the drawings below is in two-point perspective?<\/p>\n<p align=\"LEFT\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/MAth%20Mof%204.1%20Art%20042.JPG\" alt=\" Four drawings in one or two point perspective\" width=\"241\" height=\"448\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct answer is C. C shows a figure with two vanishing\u00a0points. Answers A, B, and D are in one-point perspective and C is\u00a0in two-point perspective.<\/p>\n<\/div>\n<\/section>\n<\/section>\n<p><!-- CONTENT ENDS HERE --><\/p>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/classifying-two-and-three-dimensional-solids\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/geometry-spatial-reasoning\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/triangles\">Next Lesson \u27a1<\/a><\/div>\n<p><a class=\"backtotop\" href=\"#title\">Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2b05 Previous Lesson\u00a0Workshop Index\u00a0Next Lesson \u27a1 Two-Dimensional Representations of Three-Dimensional Objects Objective In this lesson, you will study the way two-dimensional drawings in one- or two-point perspective can also\u00a0represent three-dimensional objects. Previously Covered: A polyhedron\u00a0is three-dimensional figure with faces that are all polygons. Any polyhedron satisfies Euler\u2019s\u00a0Formula:\u00a0,\u00a0where F\u00a0= number of\u00a0faces, E =\u00a0number of edges, and [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-133","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/133","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/comments?post=133"}],"version-history":[{"count":6,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/133\/revisions"}],"predecessor-version":[{"id":492,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/133\/revisions\/492"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=133"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}