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ISOTEL CURVE


Notion studied by Lepiney in 1909.
From the Greek iso "same" and têle "far away".

 
The isotel curve of a plane curve (G0) with respect to a point O is the locus of the points located at equal distance from O and (G0), in other words, the equidistance curve of (G0) and O. If M0 describes (G0), then the isotel curve is the locus of the intersection points between the perpendicular bisector of [OM0] and the normal at M0; therefore, it is also the locus of the centres of the circles passing by O and tangent to the curve (G0); the isotel is thus none other than the curve the orthotomic of which is the initial curve, in other words, its "anti-orthotomic".
Since the orthotomic is the image of the pedal by a homothety with centre O and ratio 2, the isotel is none other than the image of the negative pedal by a homothety with centre O and ratio 1/2; the notion therefore became obsolete.

Conclusion: up to homothety, isotel = negative pedal = orthocaustic.

Examples:

    - the parabola is the isotel of its directrix with respect to its focus.

    - a centred conic is the isotel of the directrix circle at one focus with respect to the other focus.

    - the isotel curve of the ellipse with respect to its centre is the Talbot curve.
 

For other examples, see negative pedal.
 
 
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© Robert FERRÉOL  2017