Why do ancient buildings have curved roofs in Japan and China? Mathematical justification of this form

In the comments to the previous article Why do ancient buildings have curved roofs in Japan and China? a number of readers wrote about comparing the shape of the roof with the curve of the fastest descent - brachistochron.

It is an inverted cycloid curve. If we take the trajectory of the body along this curve, then along it, under the conditions of gravity, it will travel its way in the least amount of time. The problem of finding such a curve was dealt with by mathematicians of the 17th century.

The minimum travel time is related to the initial acceleration. The body will accelerate faster along the cycloid trajectory. But what does this have to do with the topic of the article about the shape of the roofs of ancient buildings in Japan and China (and throughout East and Southeast Asia)?

The fact is that a common explanation for the use of such a form of roofs is the task of diverting the flow of water as quickly and further as possible during tropical showers. And since this shape of the roof is as close as possible to the brachistochrone, the water flow will also have a maximum speed at the lower point of separation of the flow from the roof. This means that the stream will fly further away from the building. Animation from one experience:

On the third, steepest curve, the body, although at first gets more acceleration, but because of its too long a body travels a longer path and is inferior in time to a body that moves along cycloid. The body moves in a straight line for the longest time.

As written above on the slide, there is another feature of the brachistochrone: if you place the body in any point on the curve of this cycloid, then the time taken to descend to the horizon will be the same with each points.

In a tropical downpour, gutters do not make sense, they overflow and do not have time to drain the water. Therefore, the further the water is removed due to its inertia, the better. It turns out that it is precisely this trajectory along the cycloid that has the greatest efficiency. And it determined the choice of the shape of the roof.

Is it possible that the ancient Japanese and Chinese architects possessed such knowledge that appeared in Europe only in the 17th century?

Of course, there are other explanations that deserve attention. A lot of them were suggested in the comments in the first part. The main explanations that readers suggested in the comments are:

In Buddhism, straight lines and angles are avoided. According to their ideas, the energy flows in straight lines are not the same. So that evil spirits do not settle in the corners (they move in straight lines). To prevent the energy in the corners of the roof from getting into people passing under it. This was represented in the form of dragons, which again took off into the sky. In general, some principles of feng shui and superstition.

From the more mundane, this version: a system of beams was made in the corners, which dampen vibrations during earthquakes, but also lift the corner of the roof up.

***

Subscribe to the channel, add it to your browser bookmarks (Ctrl + D). There is a lot of interesting information ahead.