Isometric Drawing

8403 /www/nrich/html/content/id/8403/ 1 0000-00-00T00:00:00 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <ul id="stemLinks"> <li><a href="http://nrich.maths.org/8396">Warm-up</a></li> <li><a href="http://nrich.maths.org/240">Try this next</a></li> <li><a href="http://nrich.maths.org/895">Think higher</a></li> <li><a href="http://en.wikipedia.org/wiki/Isometric_projection">Read: mathematics</a></li> <li><a href="http://en.wikipedia.org/wiki/3D_modeling">Read: technology</a></li> <li><a href="http://plus.maths.org/content/putting-it-perspective">Explore further</a></li> </ul> <div class="framework">If you are not familiar with Isometric Perspective, start by reading our article <a href="http://nrich.maths.org/8396">3D Drawing</a>.</div> <mdo:image alt="axes on isometric paper" src="isometric_axes-2.png" style="margin: 10px; float: right; width: 250px; height: 264px;"></mdo:image>The diagram on the right shows a set of three coordinate axes on an isometric drawing. What is the angle between each pair of axes? Because the angles between the three axes are all the same, each one must be 120 degrees. Is it possible to keep the distance between all adjacent grid points equal, and have some other angles between the axes? The photo below shows a multilink structure. Draw it on isometric paper. Which properties of the original structure does your drawing preserve, which are not preserved? You should think about: <ul> <li>the relationship between the lengths of the edges of the cubes</li> <li>the angles between them</li> <li>parallel and perpendicular lines</li> </ul> <mdo:image alt="multilink structure" src="multilink2.png" style="width: 250px; height: 260px;"></mdo:image> How does your drawing compare with a drawing of the same view of this structure in <a href="https://nrich.maths.org/8396?part=index#oblique">Oblique Projection</a>? What are the advantages of each, what are the disadvantages? </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><h3>Why do this problem?</h3> Representing 3D objects in two dimensions on paper is a vital skill in the Design Technology curriculum, as well as an aspect of Shape and Space in the Maths curriculum. This problem is part of a set of problems which will help students to understand why there are different ways to represent a 3D object in two dimensions, and what maths lies behind each method. The article <a href="https://nrich.maths.org/8396">3D Drawing</a> was written to support these problems. <h3>Key questions</h3> What are the advantages of this method of 3D drawing? What are the disadvantages? What features of the object are retained in the drawing, which are not? <h3>Possible extension</h3> Students who find isometric drawing straight-forward should be encouraged to tackle the other problems in this set (linked from <a href="http://nrich.maths.org/8396">3D Drawing</a>) and to compare the various methods. <h3>Possible support</h3> Students who find it difficult to draw the multi-link structure shown in the problem could be given simpler multi-link structures to draw, helping them to build up to drawing more complex ones.</mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><mdo:image alt="isometric drawing" src="isometricdrawing2.png" style="width: 197px; height: 225px; margin: 10px 30px; float: left;"></mdo:image>Drawing in isometric perspective preserves: <ul> <li>relative proportions of lengths of edges</li> <li>parallel lines</li> </ul> It does not preserve: <ul> <li>angles between edges - they are actually all right angles, but the isometric grid means all angles are multiples of 60 degrees</li> <li>perpendicular lines, because there are no right angles on an isometric grid</li> </ul> Comparing this isometric representation of the structure with the same structure represented in oblique projection (below left): <ul> <li><mdo:image alt="oblique perspective drawing" src="oblique-2.png" style="width: 225px; height: 225px; margin: 10px 30px; float: left;"></mdo:image>relative proportions of lengths of edges are preserved in the isometric drawing but not in oblique perspective</li> <li>neither preserves angles</li> <li>both preserve parallel lines</li> <li>neither preserve perpendicular lines</li> </ul> One advantage of isometric perspective is that relative proportions of lengths are preserved - if lengths are drawn equal, or if one length is drawn twice as long as another, then that is also the case in reality. A disadvantage is that there is no way of showing a structure from a slightly oblique direction - the drawing above suggests that the structure is positioned symmetrically relative to the observer, whereas the <a href="http://nrich.maths.org/8403">original photo</a> shows the right hand faces facing toward the observer, and the left hand faces facing away from the observer. One advantage of oblique perspective is the relative ease of drawing a structure using this method. A disadvantage is that no information from the original structure is preserved.</mdoxml> 2 3 0 0 1 0 0 Isometric Drawing Explore the properties of isometric drawings. Applications Design Applications Maths Supporting SET 3D Geometry, Shape and Space 2D representations of 3D shapes Applications STEM - design technology