Archimedean spiral

A plane transcendental curve the equation of which in polar coordinates has the form:

$$\rho=a\phi.$$

Figure: a013150a

It is described by a point $M$ moving at a constant rate along a straight line $d$ that rotates around a point $O$ lying on that straight line. At the starting point of the motion, $M$ coincides with the centre of rotation $O$ of the straight line (see Fig.). The length of the arc between the points $M_1(\rho_1,\phi_1)$ and $M_2(\rho_2,\phi_2)$ is

$$l=\frac a2\left[\phi\sqrt{1+\phi^2}+\ln(\phi+\sqrt{1+\phi^2})\right]_{\phi_1}^{\phi_2}.$$

The area of the sector bounded by an arc of the Archimedean spiral and two radius vectors $\rho_1$ and $\rho_2$, corresponding to angles $\phi_1$ and $\phi_2$, is

$$S=\frac{\rho_2^3-\rho_1^3}{a}.$$

An Archimedean spiral is a so-called algebraic spiral (cf. Spirals). The generalization of the Archimedean spiral is called a neoid, the equation of which in polar coordinates is

$$\rho=a\phi+l.$$

The spiral was studied by Archimedes (3rd century B.C.) and was named after him.

References

[1]	A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)
[a1]	E.H. Lockwood, "A book of curves" , Cambridge Univ. Press (1961)

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