parabola

The word 'parabola' refers to the parallelism of the conic section and the tangent of the conic mantle. Also   the parable ¹⁾ has been derived from the Greek 'parabole'.
The parabola can be seen as an ellipse with one focus in infinity. This means that a parallel light bundle in a parabolic mirror will come together in one point. It had been told that Archimedes did use a parabolic mirror in warfare. It was during the siege of Syracuse (214 - 212 BC) by the Romans, that Archimedes constructed reflecting plates in about the form of a parabola. These plates were used to converge the sunlight onto the Roman ships, and put them in fire. Though this event is discussed by some historians, recently the feasibility of Archimedes' plan has been proved ²⁾.
Other technological parabola shaped objects are the parabolic microphone and the parabolic antenna, used to focus sound and electromagnetic waves, respectively.

Menaichmos (350 BC) found the parabola while trying to duplicate the cube: finding a cube with an area twice that of a given cube ³⁾. In fact he tried to solve the equation x³ = 2. Menaichmos solved the equation as the intersection of the parabolas y = x² and x = ½ y².
Euclid wrote about the parabola, and Apollonius (200 BC) gave the curve its name.
Pascal saw the curve as the projection of a circle.

In the parabola curve the parabola (with its vertex oriented downwards)  is being repeated infinitely.

Some properties of the parabola:

• the parabola is the involute of the semi-cubic parabola
• a path of a parabola is formed when the involute of a circle rolls over a line
• the curve is a specimen of the sinusoidal spiral
• it is the pedal of the Tschirnhausen's cubic

Besides, the following curves can be derived from the parabola:

Pedals of the parabola are given by:

 The Italian Luca Valerio determined the area of a parabola, in 1606; is was called the quadrature of the parabola. But it was Archimedes who first found the value of this area. In his work "Quadrature of a Parabola" he formulated the area as 2/3 from the product of the base and the height of the parabolic sector ³⁾.

At the end of the the Middle Ages the cannon came into use on the battlefield. Therefore it was important to predict the exact location where the projectile would land. Many scientists tried to answer this question, and it was Galileo Galilei who found the relation as first.
That the trajectory of a projectile - neglecting effects of friction - has the form of a parabola, can easily be understood as follows: the x coordinate is proportional to the elapsed time, the y coordinate quadratic (as result of the constant gravitation force).
As a matter of fact, this relation was in the 15th century already known as the 'square law'.
When the air friction is taken into account, a braked parabola emerges, what in fact is a kind of exponential curve.

The 'parabolic velocity' is the minimum velocity to escape from (the gravity forces of) a celestial object.
And an arch (e.g. of stone) is called parabolic when its shape has the form of a ....

Now it's time for an experiment: hang on a horizontal chord some lamps, at equal distances. Neglecting the weight of the chord compared to the lamps, the hanging points of the lamp form a parabola ⁴⁾. This form is independent of the weight of the lamps, and also of the distance between the lamps. In the case of a chord without lamps, instead of a parabola, a catenary is formed.
Jungius (1669) proved that Galileo was wrong to state this curve being a parabola.

On holidays at the Dutch North Sea beach, you can see dunes in the form of a parabola (parabola dune), formed by the sand, thrown by the wind, but hold by the dune plants.
When experimenting, one will often encounter quadratic relations between variables. So is the kinetic energy of a body proportional with the square velocity. In another branch of mathematics we find the iterated parabola, studied for the first time (in the 60s) by Myrberg. In the case of convergence one speaks of an attractor. And the warmth equation, a partial second order differential equation, is also called the parabolic differential equation. Reason is the similarity between the relations ⁵⁾.

notes

1) Para = next to, bole (Gr.) = throw, parabole = what is thrown next to (to compare).

2) Mueller 1985 p. 111.

3) one was looking for a cube with contains twice the volume of a given cube: Waerden 1950 p. 88.

4) with thanks to Chris Rorres (Drexel University).

5) This follows from setting to zero of the force momentum, working on the lowest point of the chord.

6) .