and 
: 
.
Polar equation of the median curve with pole O of the two curves
and 
: 
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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