Superellipse is a figure that many people see every day, but do not know about it.

Greetings! Today we will talk about an interesting curve, which has its own name – superellipse.

Take a look around. What shapes do you see? Most likely, most figures have either straight (acute, obtuse) corners or ellipsoidal (not always perfectly round) borders. However, there are also hybrids. Superellipses are a family of curves that lie somewhere between ellipses and rectangles.

First, let’s remember what an ordinary ellipse is – a second-order curve:

The coefficients a and b determine the “flatness” of the ellipse horizontally and vertically, respectively:

But what if the second degrees are replaced by something else? If you raise to a cube, to the fourth power, etc.?

In 1959, a competition was announced in Swedish Stockholm to design a roundabout for one of the city squares. This area was located where two broad highways intersected in the city center, so it was necessary not to block traffic flows, and even provide a pedestrian zone at a level below:

Architect Pete Hein I was looking for the most aesthetic, but also effective form, about which I wrote:

Man is an animal that draws lines that he then stumbles over. There have always been two trends in civilization: one towards straight lines and rectangular patterns, and the other towards circular lines. There are reasons for both tendencies, mechanical and psychological. Things made with straight lines fit well together and save space. And we can easily move – physically or mentally – through objects made with round lines.

But we are in a straitjacket, forced to take this one, then the other, when often some intermediate form would be better. Drawing something by hand – like the multi-level ring road they tried in Stockholm – is not good. The superellipse solved the problem. It is neither round nor rectangular, but somewhere in between. However, it is fixed, it is definite – there is a unity in it.

The superellipse equation in general form is written as follows:

Depending on the ratio of the parameter n, you can get a whole scattering of figures.

As n approaches zero, the curve degenerates into two straight intersecting lines along the axes.

With 0

When n = 1, you get a rhombus with vertices on the coordinate axes.

With 1

When n = 2, you get an ellipse (or, if a and b are equal, circles).

If the value of n > 2, you get a superellipse. (when designing the area, which I mentioned above, the parameter 2.5 was used).

As n approaches infinity, we get a rectangle.

It’s worth noting that while superellipses may look like they have straight sides connected by curves, they are actually curved all around. Even where a segment looks straight, it is actually slightly curved, and the curvature changes continuously throughout.

If you rotate the superellipse around the main axis, you get a superegg that is stable relative to its biological counterparts:

Architectural objects, interior items, logos of brands and teams have a superellipse shape:

However, the vast majority of people saw the super ellipse for the first time…when they picked up the Iphone! The fact is that starting from the sixth version of iOs, designers stopped using rounded rectangles, and switched to superellipses:

It feels like a superellipse is used here with a factor of n = 5..7. If this is not enough, then know that the superellipse soup is a figure that many people see every day, but do not know about it. The ellipse appeared in the article “Geometric modeling and hydrodynamic analysis of floating spermatozoa”, it’s a pity that this publication not publicly available.

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