Archimedean spiral

It is the locus corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line that rotates with constant angular velocity.

Equivalently, in polar coordinates (r, θ) it can be described by the equation

Changing the parameter b controls the distance between loops.

From the above equation, it can thus be stated: position of the particle from point of start is proportional to angle θ as time elapses.

[1] A physical approach is used below to understand the notion of Archimedean spirals.

Suppose a point object moves in the Cartesian system with a constant velocity v directed parallel to the x-axis, with respect to the xy-plane.

If the xy plane rotates with a constant angular velocity ω about the z-axis, then the velocity of the point with respect to z-axis may be written as:

As shown in the figure alongside, we have vt + c representing the modulus of the position vector of the particle at any time t, with vx and vy as the velocity components along the x and y axes, respectively.

The Archimedean spiral has the property that any ray from the origin intersects successive turnings of the spiral in points with a constant separation distance (equal to 2πb if θ is measured in radians), hence the name "arithmetic spiral".

For large θ a point moves with well-approximated uniform acceleration along the Archimedean spiral while the spiral corresponds to the locations over time of a point moving away from a fixed point with a constant speed along a line which rotates with constant angular velocity[2] (see contribution from Mikhail Gaichenkov).

As the Archimedean spiral grows, its evolute asymptotically approaches a circle with radius ⁠|v|/ω⁠.

One method of squaring the circle, due to Archimedes, makes use of an Archimedean spiral.

Archimedes also showed how the spiral can be used to trisect an angle.

Both approaches relax the traditional limitations on the use of straightedge and compass in ancient Greek geometric proofs.

Scroll compressors, used for compressing gases, have rotors that can be made from two interleaved Archimedean spirals, involutes of a circle of the same size that almost resemble Archimedean spirals,[4] or hybrid curves.

The coils of watch balance springs and the grooves of very early gramophone records form Archimedean spirals, making the grooves evenly spaced (although variable track spacing was later introduced to maximize the amount of music that could be cut onto a record).

[5] Asking for a patient to draw an Archimedean spiral is a way of quantifying human tremor; this information helps in diagnosing neurological diseases.

Archimedean spirals are also used in digital light processing (DLP) projection systems to minimize the "rainbow effect", making it look as if multiple colors are displayed at the same time, when in reality red, green, and blue are being cycled extremely quickly.

[7] They are also used to model the pattern that occurs in a roll of paper or tape of constant thickness wrapped around a cylinder.

[10] The Archimedean Spiral cannot be constructed precisely by traditional compass and straightedge methods, since the arithmetic spiral requires the radius of the curve to be incremented constantly as the angle at the origin is incremented.

But an arithmetic spiral can be constructed approximately, to varying degrees of precision, by various manual drawing methods.

The common traditional construction uses compass and straightedge to approximate the arithmetic spiral.

First, a large circle is constructed and its circumference is subdivided by 12 diameters into 12 arcs (of 30 degrees each; see regular dodecagon).

Starting with the horizontal diameter and the innermost concentric circle, the point is marked where its radius intersects its circumference; one then moves to the next concentric circle and to the next diameter (moving up to construct a counterclockwise spiral, or down for clockwise) to mark the next point.

After all points have been marked, successive points are connected by a line approximating the arithmetic spiral (or by a smooth curve of some sort; see French Curve).

Depending on the desired degree of precision, this method can be improved by increasing the size of the large outer circle, making more subdivisions of both its circumference and radius, increasing the number of concentric circles (see Polygonal Spiral).

A mechanical method for constructing the arithmetic spiral uses a modified string compass, where the string wraps and winds (or unwraps/unwinds) about a fixed central pin (that does not pivot), thereby incrementing (or decrementing) the length of the radius (string) as the angle changes (the string winds around the fixed pin which does not pivot).

Such a method is a simple way to create an arithmetic spiral, arising naturally from use of a string compass with winding pin (not the loose pivot of a common string compass).

The string compass drawing tool has various modifications and designs, and this construction method is reminiscent of string-based methods for creating ellipses (with two fixed pins).

Instead of the central pin and string of the string compass, this device uses a non-rotating shaft (column) with helical threads (screw; see Archimedes’ screw) to which are attached two slotted arms: one horizontal arm is affixed to (travels up) the screw threads of the vertical shaft at one end, and holds a drawing tool at the other end; another sloped arm is affixed at one end to the top of the screw shaft, and is joined by a pin loosely fitted in its slot to the slot of the horizontal arm.