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ORTHOTOMIC CURVE


Notion studied by Quételet in 1822 (?)
From the Greek orthos "right" and tomê "to cleave".
Other name: podoid.

 
 
The orthotomic of a plane curve with respect to a point O is the locus of the symmetric images of O about the tangents to the curve . It is therefore the image of the pedal of with respect to O by a homothety with centre O and ratio 2.
It is also the envelope of circles centred on, and passing by, O; see anallagmatic curve.

Its evolute is the caustic by reflection of for a light source placed at O: the orthotomic curve is therefore also a special case of anticaustic (or secondary caustic).

The orthotomic curve can also be considered as a roulette: when the curve rolls without slipping on itself in such a way that the two curves are symmetric images of one another about their common tangent, the trace of the point O of the moving plane on the fixed plane is the orthotomic curve (this is why, for example, the cardioid is, at the same time, the pedal of a circle with respect to one of its point and an epicycloid. See also the construction of the cissoid of Diocles as a roulette).

Examples:
    - the orthotomic of a centred conic with respect to one of its foci is the directrix circle centred on the other focus;
    - the orthotomic of a parabola with respect to its focus is its directrix.

For exhaustive examples; see pedal.

The curve of which a given curve is the orthotomic is the isotel of the initial curve.

See also the notion of symmetric image of a curve  about , which gives the orthotomic when is reduced to a point.
 
 
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© Robert FERRÉOL, Jacques MANDONNET 2017