Catenary: the gravity-dependent curve
In mathematics there are infinite possible curves that
can be drawn in a plane. We can draw lines, circles,
ellipses, parabolas, cubics, and many, many other
unimaginable shapes.
However, an ellipse may appear to be a circle if the
ellipse axes’ lengths are very similar in length. If in the
ellipse the mayor axis MM is equal to the minor axis
mm, then the ellipse becomes a perfect circle.
Some of those shapes are easy to visually identify. There is no
possible confusion between lines and circles, or between
ellipses and parabolas.
However, there are two plane curves that are
so similar in shape that ancient mathematician
and philosophers were confused about the true
shape of each one.
The confusing curves are the catenary and the
parabola. The parabola was known to ancient Greeks,
but not the catenary. The word catenary is derived from
the Latin catena which means cadena, or chain in
English.
The centenary is also known as the
chainette, alysoid, or hyperbolic cosine.
The greatest advantage of the catenary is that when used inverted it can be applied
to complex architecture buildings. Catenary arches are inverted catenaries. Inverted
catenaries are very efficient in carrying heavy loads.
The inverted catenary is used in special buildings construction, in bridges, in
cathedrals, and monuments, like the St. Louis Gateway Arch, designed by the FinishAmerican architect Eero Saarinen. The monument, erected to honor the early
explorers of the American west, resembles a true inverted catenary, but for
architectural considerations it is slightly deviated from the pure catenary.
In Spain, the architect Antonio Gaudí has left a deep footprint in famous building designs
with his novel applications of the upright and inverted catenary.
The catenary has only one shape; it is not a family of curves. So, all hanging
cables shapes are the same. This is similar to circles: all circles are the same
shape. If we look very closely a very big circle, we’ll see a straight line; but we
know that it is not so.
The equation of the catenary is
where
cosh is the hyperbolic cosine function.
When the hyperbolic cosine expression is
expanded as a Taylor series, we obtain
The catenary is shown in
red, the parabola is green.
This corresponds to the equation of the
parabola (
) plus a fourth order term.
That explains why both graphs seem so alike.
Christiaan Hygens (1629-1695).
Dutch mathematician that coined
the term catenary when he solved a
famous problem about this curve.
© E. Pérez http://4DLab.info