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GEOMETRY:
HOW SMART DO YOU HAVE TO BE?
CHRIS WILLIAMS
Chris Williams is a structural engineer known for his innovative work on the British Museum Great Court and the Savill Building at Windsor Great Park. A pioneer of design computation, he worked on the digital analysis and physical model testing of gridshell structures with Frei Otto and Ted Happold. His work integrating computation as a design tool influenced the creation of Smartgeometry (SG). He was a tutor for many of the early SG events and his work remains an important source of inspiration and knowledge in the community. Known for tackling design and computation problems from first principles, here Williams offers a mathematical discussion of parametric descriptions of geometry for design. He addresses not only how âsmartâ geometry needs to be, but how much geometry we need to know and use as designers.
Throughout human history, new technology has made old skills and knowledge redundant. Bows and arrows made spears obsolete and the people who made or threw spears either had to adapt or find that they had no job. Expert spear makers and throwers would bemoan the loss of skills and the fact that young people no longer respected their special abilities.
However, it is not always the case that new technology replaces the old. The French painter Paul Delaroche (1797â1856) was premature when he said âFrom today painting is dead!â upon seeing an early photograph in 1839. Bicycles exist alongside cars and a person might own and use both under different circumstances. In an inner city the older technology â bicycles â is more practical. Powered boats and ships have replaced sail for practical purposes, but people love to sail. The physical pleasure of cycling and sailing means that they will never die out.
All technologies require people with different skills; the person who makes the best spears is almost certainly not the best thrower. The making of buildings, bridges, cars and aeroplanes requires many skills, creative, intellectual, physical and organisational, and it is unlikely that they will be combined in one person. Even if they were, the client would not be prepared to wait for this one person to do all the work on their own.
Separation of âcreativeâ and âintellectualâ abilities is arbitrary, but it is intended to differentiate between the flash of inspiration and the long and painstaking task of preparing drawings and making sure that everything will work. Stereotomy is the application of three-dimensional geometry to architecture, originally the cutting of stone and timber. Thus in the creation of a cathedral we can imagine the architect, the âstereotomerâ and the stonemason working closely together, each respecting, but at times being irritated by the others.
Computers are no longer a new technology, but their implications for the ways in which people will work are still unclear. Up until about 20 years ago it was necessary for structural engineers to be able to construct an intellectual model of their proposed designs in order to make sure they worked. This model was in their minds and sketches, and they decided how they wanted their structures to function. Now computers are invariably used for structural analysis, and however illogical the structural layout, analysis is not a problem, at least if the mode of structural action fits within the limited palette of commercial software packages. Thus it could be said that structural engineering as an intellectual discipline is dead. However the increasing complexity of codes, standards and legislation means that civil engineers wonât be out of a job: the regulations will provide the work, as they do for lawyers and accountants.
Architects are lucky in that one would imagine that the creative aspects of design are the least likely parts to be taken over by computers.
THREE-DIMENSIONAL GEOMETRY
It is possible to do three-dimensional geometry by projection onto a two-dimensional drawing board, but it is difficult. It is also difficult to achieve the required level of accuracy â perhaps a few millimetres over a distance of 100 metres, so fractions of a millimetre at drawing scale. This means that one has to use analytic geometry to calculate lengths, angles and so on. Points in space are specified by Cartesian coordinates, lengths are calculated by Pythogorasâs theorem and angles are calculated using the scalar product or vector product as appropriate. Naturally these calculations are done by a computer program and invariably the user is not the person who wrote the program. Thus the user does not have to know Pythagorasâs theorem, because a tame mathematician has programmed it into the software for them.
DIFFERENTIAL GEOMETRY
Differential geometry is the study of curved things, lines, surfaces and the space-time of general relativity theory.1 A curve cannot be described by a single equation and it is usual to specify the Cartesian coordinates in terms of a parameter. A typical point on a helix would be described by
in which r is the radius seen in plan and p is the pitch or increase in height per revolution. θ is the parameter and as θ varies the point moves along the curve. However in the language of âparametric designâ p and r would be described as parameters controlling the shape and size of the curve.
The unit tangent to any curve can be found by differentiation
Curvature is defined as the rate of change of t per unit length, which can be found by differentiating again. Surfaces can be defined by a single equation. Thus,
specifies an ellipsoid, or a sphere if the constants a, b and c are all equal. We can obtain z from x and y:
but we have the problem that there are negative and positive values of z for given values of x and y. Instead we can use the parametric form
in which the parameters Ď and θ would correspond to the latitude and longitude on the Earth. Ď and θ are also referred to as surface coordinates. We can choose whatever symbols we like for the surface coordinates; u and v, are commonly used:
Lines of constant u and lines of constant v form a net on the surface. We can get a different net on the same surface by instead writing
to produce the spiralling surface coordinate net shown in figure 1. Note that to cover the ellipsoid fully (u + v) has to tend to plus and minus infinity.
Before going any further let us change the symbols for the parameters or surface coordinates from u and v to θ1 and θ2. Note that θ1 and θ2 are two separate variables, not θ to the power 1 and θ squared. This seems very confusing and annoying to begin with and it is only gradually that the reason for doing so becomes clear. Thus we now have
Dirk J Struikâs Lectures on Classical Differential Geometry is probably the most easily read book on differential geometry, and he uses u and v surface coordinates.2 In their book Theoretical Elasticity, Albert E Green and Wolfgang Zerna use θ1 and θ2 for their surface coordinates and as well as doing geometry they cover the theory of shells.3 It is here that the curvilinear tensor notation using subscripts and superscripts comes into its own. Note that the Cartesian ...
