Question 250579: What is the mathematical expression for a spiral (corkscrew)?
Thank you in advance,
Robert Mc
Found 2 solutions by richwmiller, solver91311:
Answer by richwmiller(17219)   (Show Source):
Answer by solver91311(24713)  (Show Source):
You can put this solution on YOUR website!
That is a very interesting question. Unfortunately, it does not have a simple answer. In fact, it raises many questions about the character of the spiral you are trying to create. The information provided here only scratches the surface of the subject.
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In mathematics, a spiral is a curve which turns around some central point or axis, getting progressively closer to or farther from it, depending on which way one follows the curve.
Spiral - Two-dimensional spirals
A two-dimensional spiral may be described using polar coordinates by saying that the radius r is a continuous monotonic function of θ. The circle would be regarded as a degenerate case (the function not being strictly monotonic, but rather constant).
Some of the more important sorts of two-dimensional spirals include:
The Archimedean spiral: r = a + bθ
The Cornu spiral or clothoid
Fermat's spiral: 
The hyperbolic spiral: r = a/θ
The lituus: 
The logarithmic spiral:  ; approximations of this are found in nature
Spiral - Three-dimensional spirals
As in the two-dimensional case, r is a continuous monotonic function of θ.
For simple 3-d spirals, the third variable, h (height), is also a continuous, monotonic function of θ.
For example, a conic helix may be defined as a spiral on a conic surface, with the distance to the apex an exponential function of θ.
For compound 3-d spirals, such as the spherical spiral described below, h increases with θ on one side of a point, and decreases with θ on the other side.
The helix and vortex can be viewed as a kind of three-dimensional spiral.
For a helix with thickness, see spring (math).
Spiral - Spherical spiral
A spherical spiral (rhumb line) is the curve on a sphere traced by a ship traveling from one pole to the other while keeping a fixed angle (but not a right angle) with respect to the meridians of longitude, i.e. keeping the same bearing. The curve has an infinite number of revolutions, with the distance between them decreasing as the curve approaches either of the poles.
An Archimedean spiral (also arithmetic spiral) is a curve which in polar coordinates (r, θ) can be described by the equation
with real numbers a and b. Changing the parameter a will turn the spiral, while b controls the distance between the arms.
This Archimedean spiral is distinguished from the logarithmic spiral by the fact that successive turnings of the spiral have a constant separation distance (equal to 2πb if θ is measured in radians), while in a logarithmic spiral these distances form a geometric progression.
Note that the Archimedean spiral has two arms, one for θ > 0 and one for θ < 0. The two arms are smoothly connected at the origin. Only one arm is shown on the accompanying graph. Taking the mirror image of this arm across the y-axis will yield the other arm.
One method of squaring the circle, by relaxing the strict limitations on the use of straightedge and compass in ancient Greek geometric proofs, makes use of an Archimedean spiral.
Sometimes the term Archimedean spiral is used for the more general group of spirals
The normal Archimedean spiral occurs when x = 1. Other spirals falling into this group include the hyperbolic spiral, Fermat's spiral, and the lituus. Virtually all static spirals appearing in nature are logarithmic spirals, not Archimedean ones. Many dynamic spirals (such as the Parker spiral of the solar wind, or the pattern made by a St. Catherine's wheel) are Archimedean.
See also
Archimedes
Hyperbolic spiral
Fermat's spiral
Logarithmic spiral
Triple spiral symbol
Google any of the above for more information.
Logarithmic spiral - Notes
The logarithmic spiral can be distinguished from the Archimedean spiral by the fact that the distances between the turnings of a logarithmic spiral increase in geometric progression, while in an Archimedean spiral these distances are constant.
Any straight line through the origin will intersect a logarithmic spiral at the same angle α, which can be computed (in radians) as arctan(1/ln(b)). The pitch angle of the spiral is the (constant) angle the spiral makes with circles centered at the origin. It can be computed as arctan(ln(b)). A logarithmic spiral with pitch 0 degrees (b = 1) is a circle; the limiting case of a logarithmic spiral with pitch 90 degrees (b = 0 or b = ∞) is a straight line starting at the origin.
Logarithmic spirals are self-similar in that they are self-congruent under all similarity transformations (Scaling them gives the same result as rotating them). Scaling by a factor b2π gives the same as the original, without rotation. They are also congruent to their own involutes, evolutes, and the pedal curves based on their centers.
Starting at a point P and moving inwards along the spiral, one has to circle the origin infinitely often before reaching it; yet, the total distance covered on this path is finite. This was first realized by Torricelli even before calculus had been invented. The total distance covered is r/cos(α), where r is the straight-line distance from P to the origin.
One can construct approximate logarithmic spirals with pitch about 17.03239 degrees using Fibonacci numbers or the golden mean as is explained in those articles. Similarly, the exponential function exactly maps all lines not parallel with the real or imaginary axis in the complex plane, to all logarithmic spirals in the complex plane with centre at 0. (Up to adding integer multiples of 2πi to the lines, the mapping of all lines to all logarithmic spirals is onto.) The pitch angle of the logarithmic spiral is the angle between the line and the imaginary plane.
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Not sure if any of that will help.
John
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