Yesterday, after dinner in Venice and with nothing better to do, I applied myself to the problem of finding the perfect spot on the Grand Canal to watch the sunset.
My objective was to find a spot whence the setting sun can be seen reflected in the water of the canal. This is not as easy as it may seem: unlike, say, a west-facing seashore from where you can always see the sun go down over the water, the Grand Canal curves this way and that.
It is possible to find the right place by trial-and-error. But this would involve a lot of walking about (I speak from experience) and is not guaranteed to produce results: the Grand Canal is nearly 4 km long, and not all sections have fondamenti (canal-side paths). Alternatively, you could take a vaporetto (water-bus) ride along the canal, but (a) by the time you realise that you are at the right place, the boat may have taken a turn, or (b) when you are at the right place, the sun may be too high or have already set, or (c) more aggressive tourists may have occupied the best windows. The other option of course is to hire a gondola, but private water transport involves more cash than I am willing to shell out, and is therefore outside the scope of this discussion.
So I did some maths.
The Grand Canal is shaped like an inverted S.
Image created by NASA, used with permission.
If you modelled a mathematical function to approximate its shape, tangents to the curve would make all possible angles to the horizontal. For a given solar azimuth angle, there are at least two spots on the canal where the tangent is parallel to the perpendicular projection on the surface of the vector from your position to the sun.
My maths not being advanced enough to come up with such a function, I cheated a little. I used an online tool to calculate the azimuthal angle of the sun. Then, with the aid of a ruler, I approximated tangents to the curve and determined the points where the angles matched. A spot between the San Stae and Ca' d’Oro vaporetto stops looked promising. So today, for about half an hour up to the predicted time of sunset, and armed with a ticket which gives me unlimited vaporetto rides, I embarked on a succession of rides back and forth between these two stations.
The photos are not out of the ordinary: it wasn’t an especially spectacular sunset, and there were no buildings with interesting silhouettes. But it was fun to do the maths.