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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 (G_{1}) and (G_{2}) *along a line* (*D*) is the locus of the middle of the points *M*_{1} on (G_{1}) and *M*_{2} on (G_{2}), 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 (G_{1}) and (G_{2}) *with pole O* is the locus of the middle of the points *M*_{1}
on (G_{1}) and *M*_{2}
on (G_{2}), 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