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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 M1 on  and M2 on , while (M1 M2 ) 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 M1 on (G1) and M2 on , while (M1 M2 ) 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