Exploring 3D space with a computer – Part 4: Scale models
Adrian Oldknow
Now we have met the basic toolkit we will see if we can use it to construct scale models of real objects. There is an interesting project in Design and Technology (D&T) called `e-scape’ based at Goldsmith’s College, London, looking at ICT in assessing project work in Design and Technology: www.qca.org.uk/downloads/DT_update_summer2005.pdf . One of the projects was to design a container for a light-bulb which would be useful in its own right. One student submitted a design which consisted of a plastic case with pentagonal ends. Once you have enough of these, they can be glued together to make a light shade – whose external form is that of a dodecahedron, but with a hole at the centre where the live bulb will go. So there’s a challenge for Cabri 3D. First we will make an approximate scale model of light-bulb.
The bulb shown is about 10cm long. It looks as if one end is well approximated by (most of) a sphere of radius 30cm and the other by a cylinder of radius 20cm and depth 15cm. Between them we could approximate the shape by a section of cone with a depth of 30 cm, overlapping the sphere by 5 cm, and with a diameter at the narrower end of 25cm.
When we open up a new figure in Cabri 3D we see a set of three mutually orthogonal vectors from the middle of the ground plane. If we print the scene out in plan view the ground vectors OX and OY are each 1cm. So here is a basic starting point for a scale model.
We take O’ as any point on the ground plane, and translate it by vector OX to give X’. The ray OX’ is drawn, and a point S taken on it. The vector OS is constructed, and the ray is hidden. We take OS as our scaled unit of distance, corresponding, say, to 10cm. Now S becomes a `slider’ which we can use to scale up or down the size of any models we build.
Using `Central symmetry’ and `Midpoint’ we can construct the string of points along the central axis.
Using the `Circle’ and `Perpendicular’ tools we can find the points which define the base, and the points where the conical section meets the spherical section. Now that we have a plane view we can start to build up the full 3D model. The cylinder and the sphere will be no problem – but we cannot (yet) create a portion of a cone – so we will have to build a polygonal approximation.
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The green vector is used to define the axis to create the end circle. The vector and this circle then define the cylinder. We note that ends of the cylinder are open – and that there is not (yet) a circular disc in the toolkit – so we shall have to approximate the circle by the roundest available polygon – the dodecagon. |
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The approximation to the conical section is going to have to be built up out of quadrilaterals (trapezia) like the one shown. We can use a combination of rotations and reflections to make all twelve side panels. |
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Then we can tidy up all the unwanted points and lines by a mixture of `masking’ and selecting the point and border styles of polygons to be `empty’. Finally we can get the colours correct and make all surfaces opaque. Here is the nearly complete bulb showing our polygonal approximation to the truncated cone. |
The final version is available for you to download - `
bulb complete.cg3’.
You can also manipulate it online below. See the effect of dragging S.
