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Concept
Archimedean spiral
Summary
The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. 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
with real numbers a and b. Changing the parameter a moves the centerpoint of the spiral outward from the origin (positive a toward θ = 0 and negative a toward θ = π) essentially through a rotation of the spiral, while b controls the distance between loops.
From the above equation, it can thus be stated: position of particle from point of start is proportional to angle θ as time elapses.
Archimedes described such a spiral in his book On Spirals. Conon of Samos was a friend of his and Pappus states that this spiral was discovered by Conon.
Circular motion
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. Let at time t = 0, the object was at an arbitrary point (c, 0, 0). 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:
Here vt + c is the modulus of the position vector of the particle at any time t, vx is the velocity component along the x-axis and vy is the component along the y-axis. The figure shown alongside explains this.
The above equations can be integrated by applying integration by parts, leading to the following parametric equations:
Squaring the two equations and then adding (and some small alterations) results in the Cartesian equation
(using the fact that ωt = θ and θ = arctan y/x) or
Its polar form is
Given the parametrization in cartesian coordinates
the arc length from to is
or, equivalently:
The total length from to is therefore
The curvature is given by
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".
Official source
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