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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 (G1) and (G2) along a line (D) is the locus of the middle of the points M1 on (G1) and M2 on (G2), 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 (G1) and (G2) with pole O is the locus of the middle of the points M1 on (G1) and M2 on (G2), 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