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.