The magazine of the Melbourne PC User Group Sydney Opera House Shells Ken Holmes kholmes@melbpc.org.au

Quick visits to the local libraries will turn up several informative books about this unique construction. Treatments vary but you will find history of the design development, descriptions of construction methods, artists' sketches and simply magnificent photographs. No raytracing enthusiast could resist trying to depict some of its features.

I was fortunate to be working in the vicinity in the early 1960s and joined the hundreds of self-appointed lunchtime supervisors of the early foundation work and the first concrete casting moulds for the roof arch elements. However, moving interstate meant that my next sighting was of the completed building - and what a sight! Most are aware that the design competition winning structure involved a lower profiled, paraboloid shape with changing curvatures. Attempts to formulate construction details were causing considerable frustration, with the promise of considerably inflated costs. Jorn Utzon finally adopted a pure spherical surface so that only a few casting moulds were necessary for the wedge-shaped elements and could be shortened or lengthened to cast many different units. Perhaps we have just become used to it, but I prefer this more pert, upright shape and it does correspond more closely to the sails that cover the harbour on weekends.

The book "Sydney Opera House" by Michael Pomeroy Smith, Catalogue 725.822 contains a diagram, prepared by Mallika Weerakoon, with a bewildering array of circles and ellipses. But, using a ruler and a little trigonometry, it is possible to decipher the exact geometry of the main shells. Figure 1 is a stereo of the geometry of the tallest shell; all shell surfaces have a radius of curvature of 246 feet and were assembled from curved wedges as outlined by the black lines.

Note that these are of different lengths requiring adjustment of the moulds for the hollowed concrete castings. After erection these were covered, of course, by the white tiling. The (green) ridge line is a circle section of the sphere, with its plane indicated by other green lines, offset by 148 feet from the centre, but the (blue) north and (red) south face edges are great circles whose planes pass through the sphere centres.

In profile, the south faces of the tallest and second tallest shells are slightly convex due to the cutting angles of their planes but all other faces of the four shells are slightly concave when viewed from the side.

You may note that the Opera Theatre is indistinguishable in shape from the Concert Hall and is a 0.8 scale replica of it (or 0.5 by volume). Having defined the Concert Hall in POV Ray it is very simple to scale it, rotate it and shift it into position; you have the Opera Theatre! In fact, I don't know exactly how they match in practice. For example, is the radius of curvature of the shells also 246 feet, to use the same moulds, or is it 0.8 of this?

Figure 1 is a bit of overkill for what is really the simplest geometry, but my motives include trying to show how useful a stereo technique might be as an education aid. The mirror approach, despite its deficiencies, could serve doit-yourselfers until more sophisticated hardware becomes readily available. For example, it could very well supplement the blackboard and paper in introducing students to spherical geometry or, with animation, to the operation of engineering mechanisms. Hardware costs might reach $5 for half a mirror tile, a wire coathanger, silicone cement, a rubber band and three inches of fencing wire; and POV Ray is free.

Other hardware ranges from a few hundred dollars for shuttered glasses to many thousands for television systems under development, and somewhat more involved programming is necessary. The DIY approach is also more effective in developing understanding of elements of the process such as perspective versus orthogonal projection.

Reprinted from the November 1999 issue of PC Update, the magazine of Melbourne PC User Group, Australia