Archimedesspiral

Archimedesspiral, also called Archimedes' spiral, is a planar curve named after Archimedes of Syracuse. In polar coordinates it is described by r(θ) = a + b θ, where a and b are real constants. The parameter a sets the starting radius, and b controls the spacing between successive turns.

Geometric properties: The distance between consecutive turns is constant and equal to 2π b. As θ increases, r grows linearly, producing evenly spaced arms. The curve passes through the origin when a = 0 (for θ = 0) and can be extended indefinitely for θ → ±∞.

Equations and representation: In Cartesian coordinates, x(θ) = (a + b θ) cos θ and y(θ) = (a + b θ) sin θ. The area enclosed by the spiral from θ1 to θ2 is A = (1/2) ∫_{θ1}^{θ2} (a + b θ)^2 dθ.

Origin and generation: The spiral is associated with Archimedes, who studied the locus obtained by a point moving away from the center along a rotating radius. This yields r proportional to θ. If a ≠ 0, the spiral does not pass through the origin; if a = 0, it begins at the origin.

Applications and context: The Archimedesspiral is used in some engineering and design contexts, such as spiral staircases, gears, and certain spiral antennas, where constant spacing between turns offers simple geometry. It contrasts with the logarithmic spiral, which exhibits self-similar growth and increasing spacing with radius.