I was recently on holiday in the small coastal town of Umhlanga, a quaint but lively village north of the major city of Durban on the Indian Ocean (west coast) of South Africa.  One of the striking features of the town is a pier located on a popular promenade, a civil project completed in 2007 and the winner of the South African National Award for Outstanding Civil Engineering Achievement.   I made contact with the town council to access further information and took many photographs of the pier to bring to light some interesting mathematical aspects of the project.

 

The purpose of the pier is to collect storm water runoff and deliver it approximately 85 metres into the ocean into a deep-water channel.  This simple drainage system assists to keep the beaches in the area clean during periods of high rainfall.

 

 

 

            Figure 2:  Photo By Chris

 

 

When investigating the aesthetics of this structure (original Architect drawing shown in Figure 1), the whale-bone design, combined with a ships-hull style decking structure, creates an interesting dual-oceanic theme and constitutes the most striking feature of its design.

 

A common method of generating rib-type shapes of natural (e.g. animal and man) as well as man-made (e.g. shipbuilding) splines or ‘ribs’ for architectural drawings and computer design is through the use of Bezier curves. 

 

 

Bezier curves are named after Pierre Bezier, a French engineer who publicized and used them extensively in vehicle design at the car company Renault in the 1960’s.  Although now named after Bezier, they were originally developed by Paul de Casteljau in 1959, a French mathematician and physicist who worked for rival French car manufacturer Citroen at that time.

 

Bezier curves are commonly used in computer modelling and computer medical applications relating to sternotomy as they give an easy and efficient way to generate rib-type curves from minimal amounts of data points.    This results in a minimal drain on computer processor power for design applications, a matter than is of course much less of a problem today than it was in 1960.  However, in more advanced applications of Bezier curves such as computer gaming, processor power and use remains an important concern for the developer. 

 

One application of Bezier curves that we unknowingly encounter in everyday life is in the use of TrueType fonts on our computers, where each font is defined as a certain arrangement of control points and splines.  These fonts were originally developed by Apple Corp to avoid paying licensing fees, but were later licensed by Microsoft.  They typically use second degree Bezier Curves to define the outline of the font and each character contained therein.

 

Thus, the strength of Bezier curves results from them being defined by only a few data points related to each other via time segmentation splits.  This is a superior method of curve generation for computers, as compared to either interpolative methods or approximation methods.   Figure 2 shows my own photograph of this interesting civil project.

 

 

 

Below is shown an approximation of the whale-bone structure (Figure 3) as well as the ships-hull structure (Figure 4) using simple Bezier control parameters that I determined using a free online applet.  Each curve uses just three control points to determine its highly unique shape (for the hull it is 3 control points mirrored).  From these three control points (A, B, C) one determines two lines (line A-B and line B-C) and then advances Point A to B concurrent with advancing Point B to C, while simultaneously advancing the point that defines the Bezier line along line A-B as time advances.  This results in a uniquely smooth shape defined only by the three starting points and their relationship to each other as A approaches B and B approaches C.  Of further interest is the fact that this shape can be easily scaled to make the pier larger or smaller without losing quality. 

 

As can be seen clearly in Figure 4 (points B specifically), the Bezier curve does not necessarily end at each point but is merely “influenced” by its position.   Each of the shown curves would correctly be called a simple Bezier “spline” (i.e. 3 control points), while higher order (i.e. more control points) Bezier shapes would be termed Bezier “curves” or “surfaces” accordingly.

 

 

   

     Figure 3:  Bezier Spline Whale-Bone

 

 

 

Below in Figure 5 the complete structure is shown, including the whale-bone rib structure as well as the hull decking design, as illustrated by my own Bezier curves shown above in Figure 4 and Figure 5.

 

 

 

 

 

The above demonstrates some interesting links between mathematics and the design and aesthetics of a civil engineering project.  While the aesthetic design of this project revolves around the twin oceanic themes of ocean wildlife and ship structure, the mathematics revolves around how to turn these interesting shapes into drawings that can be used, interpreted, and ultimately built by mankind.