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MEDIAN CURVE OF TWO CURVES

Other name: diametral curve of two curves. |

Cartesian equation of the median curve along Oy
of the two curves
and : .
Polar equation of the median curve with pole O
of the two curves
and : . |

The median (curve) of two curves
and *along
a line* (*D*) is the locus of the middle of the points
*M*_{1}
on and
*M*_{2}
on , while
(*M*_{1} *M*_{2}
) remains parallel to (*D*).

Examples:

- the median curve of two lines, along
a third one, intersecting the others, is a line, passing by the intersection
point between the two lines (and it is indeed the median of the triangle
formed by the three lines).

- the median curve of a conic and
itself, along a given direction, is always a line, called the *diameter*
of this conic (and it is a real diameter in the case of a circle).

- more generally, the median curve
of an algebraic curve of degree *n* and itself is a curve of degree
*n*(*n
– *1)/2*.*

- the median curve of two conics with
a common axis, along a line perpendicular to this axis, is a polyzomal
curve.

- the median curve, along *Oy*,
of the two exponential curves:
and is
the catenary: .

- the median curve along *Ox*
of the semicircle
and the tractrix
is the convict's curve: .

See also the trident
of Newton.

The median (curve) of two curves
and *with
pole O* is the locus of the middle of the points *M*_{1}
on (G_{1}) and
*M*_{2}
on , while
(*M*_{1} *M*_{2}
) passes by *O*; this notion is very similar to that of cissoid
of two curves and the previous one in fact corresponds to the case
where the pole *O* is at infinity.

Compare to the mediatrix
curve.

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© Robert FERRÉOL 2017