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COCHLEOID


Curve studied by Wallis in 1685, Peck in 1700, Bernoulli in 1726, Cesaro in 1878 and Falkenburg in 1884.
In its Latin form cochlea, the name is due to Wallis, and in its current form, to Falkenburg.
The Latin term cochlea comes from the Greek kokhlias: shell, snail (cf. the cochlea in the internal ear, and the cuiller, instrument used to eat snails).

 
Polar equation:  .
Cartesian parametrization: .
Cartesian equation: .
Polar tangential angle: .
Transcendental curve.

 
 
The cochleoid is the locus of the extremity of an arc of a circle of fixed length a while the other end is fixed and tangent to a fixed straight line. In concrete terms, one can imagine the curve described by the extremity of a plastic pipe with a circular shape when the other end is fixed.
The arc of a circle in bold keeps a constant length.
The cochleoid is also the barycentric curve of the circle. More precisely, it is the locus of the gravity centre M of a homogeneous massive wire wrapped around the circle with centre O and radius a (one of its extremities being at A with polar coordinates (a, 0) and the other at ) .
The point M is the gravity centre of the arc AM0
The gravity centre of the circular sector delimited by [OA] and [OM0] also is a homothetic cochleoid with ratio 2/3: .
The cochleoid is also the inverse with respect to O and with power of inversion a2 of the quadratrix of Dinostrate.
When a point N describes the hyperbolic spiral, the point  describes the cochleoid.
In other words, the cochleoid is a part of the strophoidal curve of the hyperbolic spiral; the other part, described by  describes a "syncochleoid", with polar equation .

The cochleoid is one of the conical perspectives of a circular helix, the centre of perspective being a point on the helix and the projection plane being perpendicular to its axis. As a result, the inferior part of the cochleoid represents the view, in perspective, of a helix traced on a cylindrical tube, when looking from the ceiling.
 

the tube when looking from the ceiling: we get the cochleoid

looking from the centre: we get the hyperbolic spiral

The contact points of the tangents to the cochleoid inclined by an angle  with respect to (Ox) are located on the strophoid, which is a right strophoid when  and is reduced to a circle for .
 

the points with horizontal tangent are located on a circle

the points with vertical tangent are located on a right strophoid

the points with given oblique tangent are located on an oblique strophoid.

 
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© Robert FERRÉOL 2017