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Conical Spiral

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ConicalSpiral

The conical spiral with angular frequency a on a cone of height h and radius r is a space curve given by the parametric equations

x=(h-z)/hrcos(az)
(1)
y=(h-z)/hrsin(az)
(2)
z=z.
(3)

The general form has parametric equations

x=trcos(at)
(4)
y=trsin(at)
(5)
z=t,
(6)

which is essentially a form of the Pappus spiral. In the form above, this curve has arc length function, curvature, and torsion given by

s(t)=1/2tsqrt(1+r^2(1+a^2t^2))+(1+r^2)/(2ar)sinh^(-1)((art)/(sqrt(1+r^2)))
(7)
kappa(t)=(arsqrt(4+a^2t^2+r^2(2+a^2t^2)^2))/([1+r^2(1+a^2t^2)]^(3/2))
(8)
phi(t)=(a(6+a^2t^2))/(4+a^2t^2+r^2(2+a^2t^2)^2).
(9)

See also

Concho-Spiral, Cone, Pappus Spiral, Seashell

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Cite this as:

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

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