Archimedean spiral
The Archimedean spiral (also arithmetic spiral or Archimedean spiral) was named in memory of the Greek mathematician Archimedes, who lived in the 3rd century BCE. C. It is defined as the locus of a point moving at constant speed on a line that rotates about a fixed point of origin at constant angular speed. Equivalently, in polar coordinates (r,θ) the Archimedean spiral can be described by the following equation:
- r=a+bθ θ {displaystyle ,r=a+btheta }
where a and b are real numbers. When the parameter a changes, the spiral moves in the X axis, while b controls the distance between successive turns.
Archimedes described this spiral in his book On Spirals.
Features
The Archimedean spiral can be traced within a circle and as it grows one arc moves away from another.
The polar subnormal of an Archimedean spiral is constant.
This curve is distinguished from the logarithmic spiral by the fact that successive turns of it have constant separation distances (equal to 2πb if θ is measured in radians), while in a logarithmic spiral the separation is given by a geometric progression. (The distances referred to are measured on a straight line that passes through the center of the spiral)
Note that the Archimedean spiral has two arms, one for θ > 0 and another for θ < 0. The two arms are discretely connected at the origin and only one of them is shown on the graph. Taking the reflected image in the Y axis we will produce the other arm.
Sometimes, the term is used for a more general group of spirals.
- r=a+bθ θ 1/x.{displaystyle r=a+btheta ^{1!/!x}. !
The normal spiral occurs when x = 1. Other spirals that fall into the group include the hyperbolic spiral, the Fermat spiral, and the Lituus. Virtually all static spirals that appear in nature are logarithmic, not Archimedean, spirals. Many dynamic spirals (such as the Parker spiral of the solar wind, or the pattern produced by a Catherine wheel) are from the Archimedean group.
Applications and uses
The Archimedean spiral has a plethora of applications. For example, compression pumps or rotary compressors (scroll pumps), made of two Archimedean spirals of the same size sandwiched together, are used to compress liquids and gases. This is a common mechanism in air conditioning machines with low noise emissions.
The grooves of early gramophone (vinyl record) recordings form an Archimedean spiral, making the grooves equally spaced and maximizing the recording time that could be accommodated within the recording (although this was later changed to increase quality sound).
Asking a patient to draw an Archimedean spiral is one way to quantify human tremor; this information helps in the diagnosis of neurological diseases. These spirals are also used in DLP projection systems to minimize the rainbow effect, which simulates a display of several colors at the same time, when in reality cycles of red, green and blue are projected rapidly.
A method for the square of the circle, relaxing the strict limitations in the use of a rule and a compass in the geometric tests of ancient Greece, makes use of the Spiral of Archimedes. There is also a method to trisecte angles based on the use of this spiral. As the length of the subtangent is St=rθ θ {displaystyle ,St=rtheta } can be used to rectify the circumference.
