Exploring 3D space with a computer – Part 4: Scale models
Adrian Oldknow
Tower Bridge Model Part 1
So we have made an approximate model of a real domestic object, and used our computer model to test out the best size for a tapering pentagonal prism as its container.
or the next model we are going to change size considerably. After all, in the words of the TV adverts for a certain make of French cars: " size matters ". But also, according to recent versions of the advert, their cars are actually Anglo-French. So I thought I would take a London landmark as a model, and leave you a French one as a challenge! I will find out as much as I can about the proportions of Tower Bridge in London, and start to build a Cabri 3D scale model - and I'll leave you with the challenge to do similar for a well known Parisian landmark - the Arc de Triumphe in the Place de l'Etoile.
Above is a family snap of some younger Oldknows in front of Tower Bridge. I tried searching using Google to find information on the bridge’s dimensions, with hopefully a plan view or scale diagram. The best it came up with was some detailed information from Geoffrey Hartwell: http://www.hartwell.demon.co.uk/tbpic.htm. The central span is 205 ft, the towers are 293 ft tall from their foundations, the walkways are 110ft above the roadway, and the headroom at high tide is 29ft when closed and 135ft when open. Then I had the brain-wave of searching on “scale model Tower Bridge” – which took me to: http://www.fiddlersgreen.net/misc/towerbrg/towerbr.htm . The image of the completed model is from their website, and, after processing my $2, I was able to download the Adobe pdf file and print out the “cut out” templates.
Now I admit that I am not a very practical person with my hands - as you can see in the rather `wonky' attempt to build one of the towers from paper. But the main thing is that I now have a way to start to work out the important dimensions we need to build a scale model. The important measure for doing the conversions is that the paper walkways are 75mm long. So my Cabri 3D units will correspond to 1cm on the paper model, which will represent 27.33 feet or about 8.4 metres. So my arbitrary scale is 1:8400 . Here are my preliminary sketches, then.
The tricky part is going to build a tower with a hole through it! The technique is going to be to approximate the curved roof with polygons (again) and then to use these to make prisms which fit together to make the `solid' part of the tower. So down to the Cabri 3D drawing board! The vector v = O'S defines the unit equivalent to 10mm on paper, or 8.4m on Tower Bridge . The centre point O of the ground plane has been translated by v to give points at 1, 2, 3 and 4 units from O . Using the `midpoint' tool twice we can construct points at 3.5 and 3.75 units from O .
We can construct a circle in the ground plane with this last point as centre and with the vector v as radius. Using parallel lines we can now start to build up the ground plan of
one tower. Here we use a unit sphere and a perpendicular to the ground plane.
Now we have a vertical scale started we can use `central symmetry' to extend it upwards, and downwards. In this model the ground plane will actually represent the surface of the River Thames at high tide - so the towers will extend downwards as well as upwards. Here the green rectangle has been made into a prism using the downward pointing vector, which is the cuboid making the base of the tower - shown hatched.
The rectangle has also been translated by the 5 unit long upward vector and further polygons (also rectangles) have been drawn to show two elevations of the tower. Now we can start to construct the outline of the arch on one of these elevations. It has straight sides and then an arc of circle as top. We can construct the whole circle by finding its centre, which is where the perpendicular bisector of a chord from the top point to the top of a straight side meets the vertical line of symmetry.
So now we can start to do the physically impossible! The circular arc has now been approximated by four short segments. Each segment is used to define a vertical polygon (trapezium) which is then used with a vector define a prism like the one shaded in grey. The tower's superstructure can thus be built from six prisms like this one and its mirror image. Thus the "tower with a hole in it" can be built up from three cuboids and two trapezoidal prisms.
The tower is then completed by tidying up all the unnecessary bits - by masking the polygons, and changing the styles of the solids so that their `Point size' and their `Border size' are both empty. You can download the file for the drawing so far:
`
tower-and-arch.cg3' . There is also a version embedded in the webpage below for you to manipulate.
