Conjugate_Diameters

Conjugation affinity of the ellipse

Conjugate diameters of an ellipse are directions represented by vectors u=(u₁,u₂),
such that u^tMv=0. M represents here the matrix defining the conic (see Conic_Equation.html ).
Geometrically diameters are defined as the locus of middles of parallel chords of the conics.
Each direction defining a family of parallel chords defines also another direction: that of the corresponding diameter carrying the middles. The conjugate of a diameter of a conic passes through the contact points of the tangents parallel to that diameter (idib).
For ellipses the diameter-conjugation defines an affinity of the ellipse realized through the tangents to an homothetic of it with ratio 1/sqrt(2).

The affinity preserving the conic is defined by associating to a point A of the ellipse the point B, such that OB is the conjugate direction of OA, O being the center of the ellipse and B being the first point on the conjugate diameter in the clockwise sense of rotation.
The claim about the tangency which occurs at the middle M of AB results from the affinity mapping the circle to the ellipse. There is always such an affinity and since this kind of map preserves ratios along a line and parallelity it maps conjugate diameters of the circle to conjugate diameters of the ellipse.
But conjugation on a circle corresponds to a diameter its orthogonal one, the corresponding map A|-->B defining chords which are tangent to a concentric circle with radius r/sqrt(2), if r is the radius of the circle.

Since the squares on AB are the maximal inscribed (regarding area) quadrangles in the circle, their affine images under the map carrying the circle onto the ellipse are the maximal quadrangles inscribed in the ellipse. Thus, the parallelograms of the ellipse having conjugate diameters as diagonals are the quadrangles of maximum area inscribed in an ellipse. All of them have the same area whose ratio to that of the ellipse is the same with the ratio of the area of the circle to the area of the inscribed square (see ParaInscribedEllipse.html ).

Notice that a similar kind of map corresponding intersection points of the conic with conjugate diameters is not possible for the other kinds of conics. For hyperbolas if A is on a branch of the hyperbola then the conjugate diameter OB of OA does not intersect the hyperbola. For parabolas all conjugate directions coincide with the direction of its axis.

Conjugation affinity of the ellipse

See Also