San Francisco

How to describe San Francisco? It is a city built on 43 hills and surrounded by sea, a sprawling metropolis of a million inhabitants, a city that incites love at first sight.

And what images does ‘San Francisco’ bring to mind? The catastrophic earthquakes and the fallen freeway spans? Or the archaic cable cars which feature in so many movies? Or the moody, foreboding former prison island of Alcatraz? Or perhaps Fisherman’s Wharf, with its amusements, souvenir shops and various animated entertainers. For many though, the symbol of San Francisco is the beautiful Golden Gate Bridge.

The Golden Gate Bridge

The Golden Gate Bridge, built in 1937, has a central span of some 1300 m – one of the longest in the world. It took 4.5 years to build, and in the middle, the roadway is 79 m above the water, a height requested by the Navy to allow its battleships to pass underneath. The main cables are nearly a metre in diameter. The bridge was designed to be able to withstand winds of over 160 km/hour, and can tolerate a swing at the centre of as much as 8 m. The pylons are as high as a 65 storey building, and there are actually129 000 km of wire in the bridge’s cables! Each cable is composed of 27 572 wires. Pedestrians are able to walk across; the round trip takes about an hour. A crew of more than 40 is employed full time to maintain the structure, using some 22 730 litres of paint each year. The colour is ‘international orange’, and is the most easily visible in fog.

The mathematical interest of this bridge, in common with all suspension bridges, is that the main supporting cables assume the shape of a parabola.
We give an optional proof of this.

The parabola in coordinates

In general, the parabola is the set of points (x, y) which satisfy a quadratic equation: y = ax2 + bx + c (a 0). The simplest example is where a = 1 and b = c = 0. This gives the equation y = x2, the graph of which is illustrated here.

 Investigate  On squared or graph paper, reproduce the illustrated graph by plotting a number of points (x, y) where y = x2, and joining them up with a smooth curve. Alternatively, produce the curve with the help of a graphics calculator.

Now do the same with the curve y = x2 + c for c = ... 2, 1, –1, –2, ... . What effect does the value of c have on the graph? Where does the vertex lie?

Finally, look at the graph with equation y = x2 + 2x. Where does the vertex lie? How would you describe the graph with equation y = x2 + 4x? y = x2 + bx?

The path of a projectile

We easily see that the path of a bullet or of a thrown cricket ball is a parabola. For, the second equation of motion shows that the vertical distance y (= s) above the ground at time t is given by y = ut1/2.gt2. Here, g is the acceleration due to gravity, and u is the initial vertical velocity.

If h is the initial horizontal velocity, then the horizontal distance travelled in time t is x = ht.

Focus and directrix

 Investigate  Eliminate t from these two equations to obtain an equation relating y and x. Is this the equation of a parabola?  The term in x2 has a negative coefficient. What does this mean in terms of the graph?

 Investigate The parabola occurs in other interesting ways. By hand, or using a computer drawing program, reproduce this figure of lines and circles. The lines are equally spaced and parallel to the given line d. The circles are all centred at the point F, and are also equally spaced, the spacing being the same as for the lines. Now, plot a set of points such that for each point the distance from F is the same as the distance from d. Join your points up with a smooth curve.

Let F be a fixed point and d a fixed line not through F. The set of points which are equidistant from F (the focus) and d (the directrix) is a parabola. The parabola is symmetric about the horizontal line through the focus – the axis.      

Properties of the parabola

The parabola is a curve which has many interesting and surprising properties.

Investigate Trace a parabola at an oblique angle on a sheet of lined paper. Some of the lines intercept the parabola in parallel chords. Mark in the midpoints of these chords. What do you notice? Would this property hold true for a circle? For other curves?

You may know that the reflector in a car or bicycle headlight has a cross-section which is a parabola. The bulb is placed at the focus of the parabola. But why does the headlight give such a bright beam? A property of light is that when it reflects off a polished surface, the ‘angle of incidence’ (i.e the angle at which the ray of light strikes the surface) and the angle of reflection at a point are equal. An important property of the parabola is that light being emitted from the focus is reflected out parallel to the axis of the parabola.

Investigate On a parabola, carefully measure some angles of incidence and reflection. Do they appear to be equal?

Now let the bulb is placed a little above the focus. Assuming the angles of incidence and reflection remain equal, what happens to the reflected rays? How is this property used in car headlights?  

Conic section

The parabola is known as a conic section, because it can be obtained as the intersection of a plane and a double circular cone. To obtain the parabola, the plane needs to be placed parallel to one of the lines which lies in the surface of the cone and passes through the vertex. We can also obtain the circle, ellipse and hyperbola as conic sections.

Further reading ...

Humphrey, D. (1954), Intermediate mechanics, Longmans, Green and Co.

Lockwood, E. H. (1961), A book of curves, Cambridge University Press.

Plowden, D. (1974), Bridges: the spans of North America, Viking Press.

San Francisco, Berlitz Pocket Guide (1993), Berlitz.