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Norwich Spiral

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NorwichSpiral

The Norwich spiral is a noncircular plane curve whose radius of curvature equals its distance from a fixed point (Zwikker 1963, pp. 138-140). It is a particular involute of a circle involute, and hence a second involute of a circle.

With the fixed point at the origin, a scale a>0, and a real number t, its parametric equations are

x(t)=a[(1-t^2)cost+2tsint]
(1)
y(t)=a[(1-t^2)sint-2tcost].
(2)

A corresponding parametrization in polar coordinates, with a continuous choice of polar angle, is

r(t)=a(1+t^2)
(3)
theta(t)=t-2tan^(-1)t.
(4)

The radius of curvature is

rho(t)=a(1+t^2)=r(t),
(5)

and the signed arc length measured from t=0 is

s(t)=a(t+(t^3)/3).
(6)

Its evolute is the circle involute generated by the circle of radius 2a centered at the origin (Ferréol). Other choices of the integration constant in the second involute construction generally give different curves.

See also

Circle Involute, Evolute, Involute, Radius of Curvature, Spiral

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References

Ferréol, R. "Norwich Spiral." MathCurve. https://mathcurve.com/courbes2d.gb/sturm/spirale_sturm.shtml.Shikin, E. V. Handbook and Atlas of Curves. Boca Raton, FL: CRC Press, pp. 337-338, 1995. https://doi.org/10.1201/9781498710671.Zwikker, C. The Advanced Geometry of Plane Curves and Their Applications. New York: Dover, pp. 138-140, 1963.

Cite this as:

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

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