The Spirale of Archimedes is the curve of polar equation following:

$\backslash \; rho=\; has\; \backslash \; theta\; \backslash ,$

The spiral of Archimedes is the curve followed by a point in uniform displacement on a line in uniform rotation itself around a point. The furrow of the discs vinyls is a spiral of Archimedes.

The spiral drawn opposite is a spiral defined for positive angles. The spiral of equation $r\; =\; -\; T\; \backslash \; pi$ definite for negative angles would be the image of the preceding one by a symmetry of axis (OX). In short, it would have the same form but would turn in the other direction.

The polar curve of equation:

$\backslash \; rho=\; has\; \backslash \; theta\; +b\; \backslash ,$

is also a spiral of Archimedes. It is the preceding spiral having undergone a rotation of angle - b/a.

Mechanical engineering

One can consider a mechanical engineering of a spiral of Archimedes by posing the sheet of paper on a base provided with a uniform rotation movement around a vertical axis passing by O. the pencil, him, moves away from the center O according to a uniform rectilinear motion. The two movements can be dependant by a system of endless screw.

Law of the surfaces

the surface swept by a ray on the interval $$ is

$\backslash \; frac\; \{a^2\; \backslash \; theta^3\}\; \{6\}$

Attention, that does not correspond to the surface of the spiral because the ray is likely to sweep several times the same portion of plan.

Famous problems

Trisection of the angle

A spiral of Archimedes makes it possible to solve the problem of the Trisection of the angle: for an angle $\backslash \; theta$ given, it is possible to build with the rule and the compass the angle $\backslash \; theta/3$. It is enough to locate the point M of the spiral associated with the angle $\backslash \; theta$, to build a circle of center O and OM/3. This circle cuts the spiral in a point P associated with the angle $\backslash \; theta/3$.

Correction of the circle

The correction of the circle is a problem similar to its squaring. To seek the Quadrature of the circle, it is to seek the square which with the same surface as a given circle. To seek the correction of the circle is to seek a segment of right-hand side which has even length that the perimeter of the circle. In one of the cases (squaring) it is a question of representing $\backslash \; sqrt\; \{\backslash \; pi\}$ by a length, in the other case (correction), it acts to represent $\backslash \; pi$ by a length. The spiral of Archimedes makes it possible to carry out the second construction.

One uses the property of the tangent to the spiral at the point M associated with the angle $\backslash \; theta$. One can show that the angle $\backslash \; alpha$ which this tangent with line (OM) forms is not constant, as it is the case in a Spirale logarithmic curve, but varies according to $\backslash \; theta$ according to the following law:

$\backslash \; tan\; (\backslash \; alpha)\; =\; \backslash \; theta\; \backslash ,$.

It is then enough to trace the tangent with the spiral at the point M associated with $\backslash \; pi$. It meets line (OY) in P. One obtains then the report/ratio

$\backslash \; pi\; =\; \backslash \; frac\; \{COp\}\; \{OM\}$

Unsolved problems

The two preceding paragraphs could let believe that Archimedes, thanks to its spiral, would have solved the two traditional problems of the trisection of the angle and the quadrature of the circle. But it of it is nothing. The mathematicians of the time sought methods of resolutions to the rule and the compass and scorned the mechanical resolutions . This is why the spiral of Archimedes was not regarded as a tool of resolution and was rejected like were to it other quadratrixes and the other trisecting ones.

Moreover, the layout of the tangent to the spiral did nothing but move the problem.

External bond

Spiral of Archimedes

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