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NORWICH SPIRAL

Curve studied by Ricatti in 1712, Euler in 1781,  Sturm in 1857, and Sylvester in 1868.
The name of Norwich spiral was given by Sylvester in reference to where a congress was held in 1868 during which he presented his works about the consecutive involutes of a circle.
Other names: Sturm spiral, radioid.
Ref: [Loria] p.168. 

The Norwich spiral is the curve, different from a circle, such that the radius of curvature is, at all points, equal to the distance to a fixed point (hence the name radioid). It is therefore a special case of Sturm spiral.
 
Differential equation:  (p is the pedal radius).
First integral: , the case a = 0 gives the circle, but, in the case a > 0, we get:
Polar equation: , polar parametrization: .
Curvilinear abscissa: , polar tangential angle: .
Pedal angle: , radius of curvature: .
Intrinsic equation 1: .

 
 
On the right, figure illustrating the fact that any point on the Norwich spiral (in red) is at the same distance from the centre of curvature as from O.

In blue, the evolute, locus of the centres of curvature.

It can be noticed, during, for example, the derivation of the intrinsic equation, that this evolute is none other than an involute of a circle (more precisely, of the circle with centre O and radius 2a).

The Norwich spiral is therefore one of the second involutes of a circle.

And the second involutes of a circle are the curves such that the radius of curvature is, at all points, equal to the distance to the centre of the circle, plus a constant.
The Norwich spiral can be characterized among these second involutes by the fact that it intersects with the circle perpendicularly.


 
 
 
The Norwich spiral is the negative pedal of the Galilean spiral,
and it is asymptotic to the Galilean spiral :.

See also the links between the Norwich spiral and the Tschirnhausen cubic, as well as the inverse Norwich spiral.
 
 
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© Robert FERRÉOL  2017