Concho-Spiral
ConchoSpiral
A concho-spiral, also known as a conchospiral, is a space curve with parametric equations
r = mu^ua
(1)
theta = u
(2)
z = mu^uc,
(3)
where mu, a, and c are fixed parameters.
For positive a, c, and mu, the arc length, curvature, and torsion are given by
s(u) = [画像:((mu^u-1)sqrt(a^2+(a^2+c^2)(lnmu)^2))/(lnmu)]
(4)
kappa(u) = [画像:(amu^(-u)sqrt(1+(lnmu)^2))/(a^2+(a^2+c^2)(lnmu)^2)]
(5)
tau(u) = [画像:(cmu^(-u)lnmu)/(a^2+(a^2+c^2)(lnmu)^2).]
(6)
See also
Conical Spiral, SpiralExplore with Wolfram|Alpha
WolframAlpha
More things to try:
References
Boyadzhiev, K. N. "Spirals and Conchospirals in the Flight of Insects." Coll. Math. J. 30, 23-31, 1999.Referenced on Wolfram|Alpha
Concho-SpiralCite this as:
Weisstein, Eric W. "Concho-Spiral." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Concho-Spiral.html