In order to draw an ellipse you hammer in two nails on a plain table.
You tie the ends of a string to the two nails and you let the tip of a pencil slide along the string while you keep it in tension.
In an ellipse the sum of the distances of each its points from the two fixed points on the plain (foci) is constant:
Then if you let a pencil slide along a string of the length of K that links the two foci, you can trace a series of points that have all the conditions that allow them to be called the points of an ellipse.
This method is called “ the gardener’s” because it is used to draw flowerbeds.

The tool for drawing a parabola consists of two simmetric parts, that allow you to draw two arcs of parabola.
A set square, consisting of two perpendicular rulers "A" and "B", is free to slide with the "A" side alone a straight line "S".
"O" is a fixed point on the plane and "A" is a fixed point on "B".
An inextensible thread whose lenghts is l=AH has the ends fixed on the "A" and "O" points.
With a pencil you join the thread auction.
By moving the pencil tip while keeping the thread near the ruler, you can drow an arc of the parabola.

This is an instrument used to draw hyberbolas.
it consists in a rectangular board on which two sticks of the same length
are hinged. Two unelastic strings of the same length are connected to the
two sticks.
This instrument is based on the property of hyberbolas according to which
A hyperbola is the set of all points where the difference between the distances ofthe points from the two foci is constant.

Ellipsis which roll on one another

Two equal ellipsis are connected by two rigid pivots fixed in their foci.
It's possible to rotate one ellipse around the other and never lose the contact.
Ellipsis will stay tangential
It's not only a mathematical curiosity, elliptical gears are used in mechanics.
Elliptical gears can transform a uniform rotary motion into a rotary motion with a variable speed in the same spin.

Parabolic microphone

A parabolic microphone is a microphone that uses a parabolic receiver to collect and to concentrate sound waves onto a receiver.

The sound waves are reflected on the parabolic surface and are concentrated on the focus where the microphone is.

The parabolic trajectory is a type of motion determined by the combination of two straight motions:
The point trajectory of the point is a geometric curve described by the object during his motion.
The motion of a ball that gets kicked (with no friction) describes a parabola.
There is an initial force that lasts only for the initial moment, then the ball
continues its motion thanks to the action of the initial kick.
In this motion we can clearly distinguish the two-dimensional trajectory:
- the x axis: there is no force, therefore the motion will be uniform straight motion;
- the y axis: along the vertical axis there is the gravity force that is opposite to the initial vertical push.


How to make a
Parabola with origami

Origami is the art of paper folding, without the use of glue, scissors or  other material.
For building a parabola you must follow this procedure: first mark a  point a few inches from the edge of the sheet, then fold the paper at  least 20 times in various positions, so that the folded edge touches the  point that we marked. The signs of the folds pinpoint the tangents to the  parabola and form the envelope.
The marked point is the focus of the parabola and the edge of the paper  is its angle bisector.

How to make an ellipse with origami

Take a sheet of paper, draw a circle and mark a point that you will call P inside the circle.
Then fold the sheet so that point A on the edge of the circle overlaps with the point P that you have marked.
If you repeat this process you will see that an ellipse will be formed by the lines of the folds; more precisely an ellipse as the envelope of the lines represented by the folds.

Origami of an hyperbole

You can obtain a hyperbole by tracing a circumference on a paper sheet, taking a random point external to it and folding the sheet so that different points of the circumference would coincide with the chosen point.

We can recognize a hyperbole traced by the folds of the sheet after folding it a lot of times along the majority of the circumference.


the mirascope is a device that forms the imageof an object.

the object looks as it was hanging in the hole of the lid whereas it is on the bottom.

the device consists in two parabolic mirrors that are equal and overlapping.

each of them has it's focus in the vertex of the other

Put a ball on the elliptical table;
hit the ball with the billiard cue to make it pass over one of the two
foci of the ellipse; after hitting the side the ball will continue along a trajectory that
passes over the other focus.

The dish spins with constant speed.
Tip of the pencil moves on uniform rectilinear motion.
Combination of these two motions makes a spiral track: the Archimede's spiral.
Archimede's spiral was discovered by a great syracuse mathematician and it showed in its treatise for the first time.,
The spiral has got lots application in real life: form of some galaxies is ugual about Archimede's spiral. The vinil dishìs track is on Archimedeìs spiral.
Scroll pumps are formed by two ugual spirals for gas and liquids.

A small parabolic mirror that reflects the sun beams and concentrates these at a point called "Focus". In this point the temperature reaches such a high level as to burn a small paper or a cigarette. The proprieties of the focus of a parabola has been know since old times. There is the logend of Archimede, who destroyed the enemy's fleet by using a big solar lighter.


The envelope of a family curves is a curve that tendes to only one curve of this family tangentially in every point.