 # CATENARIES Catenaries curves could have the same height but different diameters and the same diameter but different heights.

The cartesian equation of a catenary curve is:
$y=a.cosh\frac{x}{a} \tag{1}$ Where "a" is a parameter. This equation is equivalent to:
$y=\frac{a}{2}(e^\frac{x}{a}+e^\frac{-x}{a})$ Graphical representation of a catenary curve.

The same equation could be written as:
$y=h+a-\frac{a}{2}(e^\frac{x}{a}+e^\frac{-x}{a})$ Where "a" is a parameter and "h" the height. Graphical representation of a vindex catenary curve.

The choice of the catenary as the main form for vindex Architecture is due to three important characteristics of this curve. The first is the possibility of having an infinity of catenaries with the same height but with different diameters and vice versa. That is, an infinity of catenaries with the same diameter but with different heights. This feature allows you to create a variety of spaces with different volumes, yet harmonious, well-connected and spectacular. Two vindex catenaries with the same height but different diameters. Two vindex catenaries with the same diameter but different heights.

The second feature is that the catenary shape is a natural form, ie like hills, sand dunes... So vindex domes integrate perfectly with the natural environment without defacing it. Inside a vindex dome, the geometric center (in red) coincides with the center of gravity of the masses and with that of the pressure forces.

The third feature is related to the stability of the catenary shape. In this case, the center of gravity of the sections of the wall, that is, the geometric center, the center of the masses and the center of the pressure forces coincide. This means that the dome is subject to only a simple pressure force. Consequently, a vindex dome is inherently stable thanks to its shape.

(1): It was the mathematician, physicist and Dutch astronomer Christiaan Huygens who first found its exact equation in 1691 and gave it the name of catenary.