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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 2017