Curves

Ending a term with the suffix 'oid' means 'having the form of or resembling' - and that's invariably to do with something round or at least curved. This month the talk is round and round in circles... PETER WILLIAMS

I have always been fascinated by geometry and maths and the way that they can describe the world in which we live and the motorcycles we love to ride. The most recognisable geometric continuous shape is the circle. Engineering relies almost completely on the natural shape of the circle simply because so many components rotate and in doing so describe circular paths. The components themselves are made by rotating them and by rotating tools. So we depend on the circle.

Of course there are many other shapes that have a certain elegance through simplicity and symmetry, these are defined by mathematical rules, and often have fantastic names too; the ellipse, parabola and hyperbola for example. Of course there are countless more.

Many years ago, when I was trying to re-enter engineering after my racing career ended, I designed a mechanism for a mobile mirror for articulated lorries, so that the driver could always see the tail of his trailer during any manoeuvre. I felt a peculiar pleasure when I discovered that the correct mathematical term for the path of a point on the tail of the trailer is called a tractrix. Rather fitting I thought for a lorry tractor and trailer. Some other important shapes in engineering are the involute, trochoid, cycloid, epi-cydoid, hypo-cycloid, and even the quasi-hypo-cycloid, etc. The cycloid curve is best described as the snake-like path followed by the logo on a tyre as the bike rolls along the road. The involute curve is described by the path of the end of a piece of string as it is unwound from a stationary round shaft. The curve opens out - a sort of increasing radius spiral.

Gear teeth have an involute curve form because it makes it possible for the contact point between the teeth on the mating gear pair to be always at the same distance from the gear axes and the engaged teeth roll against each other rather than slide - and rub -and wear out. It is fascinating that these shapes are used every day in the machines we use but our eyes never notice them; we are oblivious of them.

The circle is such a natural shape it seems odd that pi (n), the number that is the ratio of the circumference to the diameter, cannot be determined exactly. 3.1416 is only an approximation but what would we do without it? How would I have worked out what speed my John Player Norton was doing - it had only a rev counter - at Daytona, or on Sulby Straight in the Isle of Man without knowing that IT x the diameter of my rear tyre gave me the circumference of my rear tyre?

Although there are simple connections between curves, maths and geometry there are many others. To me, the most amazing one is between the Fibonacci Series numbers and the natural wonders of the world. What an amazing coincidence that the numbers, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34... (each number in the series being the sum of the previous two numbers) describe the patterns on a pineapple, the way branches grow on a tree and also the proportions of beautiful architecture. Perhaps the Fibonacci Series is in itself a Wonder of the World.

You may be wondering by now what these ramblings have to do with motorcycles. Well, it's like this. I think I can say that I taught myself many of the skills and techniques of riding fast motorcycles from riding my bicycle on race tracks that I designed around my parents' back garden and in the woods.

The rest I learned in the space of two seconds from John Cooper.

Somehow I had squeezed into qualifying for the Race of the Year at Mallory Park in 1964. I was having a good dice with another guy and we were both using the cornering method of 'slow in and fast out' - that was the advice everyone was given. Halfway through the race John came diving past us going into The Esses as if we were standing still. He was most certainly using the system of 'fast in and fast out' which, on seeing it, seemed to me by far the best and, clearly, the most effective method!

I saw that he was holding the brakes on far into the corner and I worked out that he did this by gradually easing his braking as he increased his lean angle and his radius of turn became increasingly tighter.

I practiced the technique through the following winter in my imagination. Sitting at my drawing board at Ford's, my hands would get all sweaty and I would be shaking from the unrequited self injection of adrenalin after each of the many imagined laps I did round Brands, Mallory, Snetterton and Oulton using the Cooper method.

Ever since I have tried to work out how the racing line through bends and corners would be described mathematically. It is where the maximum friction is used at every instant but the friction vector normal (radial) to the curve gradually increases as the friction vector tangential to the curve gradually reduces.

A few of my readers have been kind enough to take the trouble to correct me for some occasional rubbish I write in these columns. Thank you, but perhaps someone can give me the maths of the 'racing line'. I suspect that it is an Archemedian spiral.

It would be unlikely, but it would be nice if it could be a new kind of mathematical shape that we could call the Cooper spiral. Though it might not be a spiral in which case it would be the original Cooper Curve.

who is peter williams?

Peter Williams was, indeed to many still is, the patron saint of British motorcycling. During the 60s and 70s, he was consistently the fastest single cylinder man around the TT course with the Wagon Wheels Arter Matchless, beaten only by the all conquering Agostini MV Agusta. He designed and raced the ingenious monocoque John Player Norton and took it to many victories against machines far more powerful, before a serious crash curtailed his racing career. A talented and logical thinking engineer, his talents have been employed by both Cosworth and Lotus. He is presently involved in building two more Monocoque chassis prototype bikes and a TT Zero racer. He lives in Cornwall with his wife, Pam, daughter, Mimi (17) and son, Jack (14)

GPL Product Guide

Bell Jet R-T bubble visor

If you're looking for that added touch of authenticity, Bell has recently introduced a bubble visor for their famous Jet helmet. It looks the part and if you've never used a bubble visor you should • give one a try. You get complete protection from the elements, but still feel like you're riding an open-face helmet. And rain; it just runs off it like no visor you've ever worn. It comes in smoke or clear.

Slylmartin's classic Indian boot

Slylmartin's classic Indian boot

This new boot from Italian maker Stylmartin is simply gorgeous. Made from a waterproof yet breathable leather, the Indian boot features a padded collar and hard ankle protectors as well as an anti-slip Vibram sole. You also get an extra layer of leather on the toe area for changing gear. This boot is truly an object of beauty and we particularly like the fact that it comes in dark brown rather than the usual black. If you want a protective, shortish boot that doesn't make you look as though you're trying too hard, you may just have found it. Just one thing confuses us. The Indian comes with both brown and red laces; we've no idea why. But the red ones do look pretty cool. Sizes: 41, 42, 43, 44, 45, 46 & 47. (STY019) £169.99 | ->f Stylmartin

The Hailwood zip sweatshirt

When we first saw a sample of this 100% Cotton top wc weren't sure whether to wear it or frame it. It's an absolute work of art. And, constructed from 26 separate panels, it's been put together with quality in mind. Starting at the top, the quarter length zip is accompanied by a press-stud fastened collar, which gives it that authentic racer look. Moving down, printed logos and sew-on badges adorn the chest. Whilst down each arm reads the name of the legend himself. It's a fantastic top and a real A tribute to 'Mike the bike'. Mi 6TCIPJ Sizes: S, M, L, XL & XXL.

The famous Belstaff wax cotton Trialmaster jacket

Candidate for the title of most famous jacket in the history of motorcycling is the Trialmaster, which has been reintroduced by Belstaff, albeit to a much higher quality and specification than the original, with heavy duty lOoz wax cotton, a zip-out thermalliner and fitted body armour. As you can see, the Trialmaster is virtually identical to the jacket worn by Steve McQueen when he raced for the American Vase team at the 1964 ISDT in East Berlin. (Ironically, he actually wore a Barbour International but a Barbour doesn't come with armour). It comcs in sizes S (36"), M (38"), L (40"), XL (42"), XXL (44"), XXXI. (46") & XXXXL (48"). For other colours and versions, and indeed the entire Belstaff collection, go to: www.grandprixlegends.com/belstaff

1. Black (BEL007B) £389.99 2. Leather (BEL045) £619.99 3. Gold (BEL007BE) £389.99

1. Black (BEL007B) £389.99 2. Leather (BEL045) £619.99 3. Gold (BEL007BE) £389.99

Website: www.grandprixlegends.com

The Vintage line from Premier helmets

The Premier brand originated in the U.S. back in the fifties, although their helmets are now manufactured in Italy.

Newly released is a terrific range of open-face, Jet style helmets. Obviously, you can get one in plain white or plain black, but in addition there are a number of retro colourschemes that are redolent of the sixties and seventies.

You can't fit a visor peak to one of these lids, but there is a rather neat flip down visor to protect your eyes.

Of course, this doesn't stop you wearing a pair of goggles if that's the look you prefer.

All in all, a great range of top quality helmets made of the very best materials throughout. We think the carbon version is particularly stunning

They come in a range of sizes: XS (53/54cm), S (55/56), M (57/58), L (59/60), XL (61/62).

Captain Virgil Hilts 'Rough-Out' boot

These boots are made by a Mexican company and are stitch perfect replica of the M-43 service boot issued to American GIs during the last war. In truth, they barely qualify as motorcycling footwear but they are the boots that were worn by McQueen throughout The Great Escape. The boots are excellent quality. We've seen cheaper versions and more expensive copies but in terms of accuracy and attention to detail it doesn't get better than these, which is why they were used in films like Saving Private Ryan and Band of Brothers. And if McQueen could jump barbed wire in them, they're good enough for us UK sizes 7, 8, 9, 10 & 11.

1:12 Benelli 750 Sei

Benelli started building bikes in the early 1920s, and before and after the war it was a force to be reckoned with on the race tracks of Europe. In 1972, De Tomaso bought Benelli and rather bravely decided to take on the Japanese manufacturers, reasoning that he could triumph by producing motorcycles full of Italian grace, style and handling. To begin this campaign he needed a showpiece, the result of which was the first ever six-cylinder production motorcycle: the Benelli 750 Sei. It became available for sale two years after its launch in 1964, but it was too expensive to be a huge success. The 750 Sei is rhc latest release in Minichamps' Classic Bike Collection. Although we're not experts on the Benelli, those who know tell % us that Minichamps have

'ULa vfc replicated it very faithfully.

•Replica measures approx. 17cm in length.

Orderline: 0844 887 8888

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