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Concho-Spiral

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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)=((mu^u-1)sqrt(a^2+(a^2+c^2)(lnmu)^2))/(lnmu)
(4)
kappa(u)=(amu^(-u)sqrt(1+(lnmu)^2))/(a^2+(a^2+c^2)(lnmu)^2)
(5)
tau(u)=(cmu^(-u)lnmu)/(a^2+(a^2+c^2)(lnmu)^2).
(6)

See also

Conical Spiral, Spiral

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References

Boyadzhiev, K. N. "Spirals and Conchospirals in the Flight of Insects." Coll. Math. J. 30, 23-31, 1999.

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Concho-Spiral

Cite this as:

Weisstein, Eric W. "Concho-Spiral." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Concho-Spiral.html

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