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 = [画像:(h-z)/hrcos(az)]

(1)

y = [画像:(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) = [画像:1/2tsqrt(1+r^2(1+a^2t^2))+(1+r^2)/(2ar)sinh^(-1)((art)/(sqrt(1+r^2)))]

(7)

kappa(t) = [画像:(arsqrt(4+a^2t^2+r^2(2+a^2t^2)^2))/([1+r^2(1+a^2t^2)]^(3/2))]

(8)

phi(t) = [画像:(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 Resource. https://mathworld.wolfram.com/ConicalSpiral.html

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