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STROPHOIDAL CURVE


Notion studied by Torricelli in 1645, Montucci in 1846, Turrière in 1913.
From the Greek strophos "string, belt, braid". 

 
For an initial curve with polar equation in the frame (O and a point A with coordinates (a, b) in this frame:
Polar equation in this frame: .

 
The strophoid (or strophoidal curve) of a curve with respect to two points O and A is the locus of the points M on a variable line (D) passing by O such that M0M = M0A where M0 is an intersection point (different from O) between the line (D) and the curve .
In other words, it is the locus of the intersection points between a circle centred on M0 on the curve and passing by A, and the line (OM0). 

The strophoid is therefore composed of two branches and  the median with pole O of which is .


 

When the curve is a line, A a point on this line and O a point outside this line, the corresponding strophoidal curves are the strophoids (right strophoids when (OA) is perpendicular to the line ).

When the curve is a circle, A its centre, the strophoid is the conchoid of with pole O and modulus the radius of the circle; in particular, when O is on the circle, we get the trisectrix limaçon.
 
 
Construction of the strophoid of a circle, in the case where A is on the circle and O, A and the centre of the circle are aligned.
When O is on the centre of the circle, the strophoid is Freeth's nephroid. When O goes to infinity, the strophoid goes to a torpedo (see below).

GENERALISATIONS

1)  the point O is placed at infinity.

The strophoid of a curve with respect to a point A and a line direction D is the locus of the intersection points between a circle centred on M0 on the curve and passing by A, and the line parallel to D passing by M0.
The associated transformation is sometimes called "Brocard transformation", since he studied it in the special case of a circle.
 
For a line direction Ax and for an initial curve with polar equation in the frame (A): , the strophoid is the reunion of the two curves with polar equation ans .
Opposite, construction of one of the two branches.

Examples:
 
When the curve is a line, and A a point outisde this line, we get the hyperbolas (see opposite the case where the line is perpendicular to (Ax)).
When the curve is a circle, and A a point on that circle (equation ), the strophoid is the trifolium. See opposite the case where the circle is centred on Ax, which gives the torpedo. When the circle is centred on Ay, we get the regular bifolium.
When the initial curve is a parabola with focus A and parameter p, and the line direction is the axis of the parabola, one of the branches of the strophoid is none other than the directrix while the other branch is a parabola with vertex A and parameter p/2.
When the initial curve is an ellipse with focus A and the line direction is an axis of the ellipse, then the strophoid is composed of two other ellipses, with summits A.

A similar phenomenon occurs for the hyperbolas.

When the initial curve is a hyperbolic spiral , the strophoid "on the right" is a syncochleoid , and that "on the left" (in green), a cochleoid .

2)  the point A is replaced by a curve (idea of Pierre Daniel).

The strophoid of a curve with respect to a curve and a point O is the locus of the intersection points between a circle centred on M0 on the curve and tangent to the curve , and the line (OM0).

Here is a simple special case: the curve is a line passing by O.
 
With the axis Oy as the line and , with polar equation in the frame (O) , as the initial curve, the polar equation of the strophoid is .

Opposite, the example of the parabola ; the equation of the strophoid is ; it is a focal conchoid of the parabola .

Another example:
 
The strophoid of an ellipse (C) with pole one of its foci F, with respect its directrix circle centred on F, is composed of the directrix circle itself and the conchoid with pole F and parameter 2a of the image of the ellipse by the homothety with centre F and ratio 2 (indeed, with the notations of the figure:
FS = FC - CP = 2FC - 2a = FC' - 2a.).

This strophoid is therefore a Jerabek curve.

The same phenomenon occurs in the case of a hyperbola.

 


 
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© Robert FERRÉOL 2017