The Bridge of Peace was built in Tbilisi, Georgia to connect the old town with the newer district across the river. It was actually a bit controversial, as its modernist architecture felt out-of-place next to the historic buildings in the old town. Still, it's an interesting shape, and I wondered if it was possible to come up with an equation to approximate the surface that covers the bridge.

While standing on the bridge, I noticed that one set of cross-sections had traces that were parabolic, as seen in the right-hand image above. The simplest surface that satisfies this requirement is a parabolic cylinder, which can be given by z = -x^2:

However, the problem here is that a parabolic cylinder isn't wavy in the same way that the bridge was. The waviness looked somewhat like a sine or cosine function, and since the bridge exhibited symmetry with a peak in the middle, this more closely fit the cosine function. Therefore, I modified the equation to get z = cos(y) - x^2:

But then I decided to make one more edit to the domain in the xy-plane, since the ends of the bridge were rounded. I decided to make the domain an elliptical region to get the rounded ends, so when I restricted the domain to {(x,y)|x^2 + y^2/16 < 1}, I got a nicer image:

For a more interactive rendering of this surface, check out the graph on Desmos.

Sample Problems

1. In my calculations, I claimed that the waviness looked sinusoidal. Instead, suppose that the waviness is polynomial. Can you edit the equation of the function to create a similar waviness using a polynomial function instead of a trigonometric function? (Note that there are many correct answers.)

2. Find other architectural shapes and attempt to find an equation to model their surfaces. There are lots of places to look: bridges, buildings, sculptures, car parts, chairs, etc. Many modern designs are very geometrical and tend to be modeled by three-dimensional surfaces.