Conjugate diameters

Each pair of conjugate diameters of an ellipse has a corresponding tangent parallelogram, sometimes called a bounding parallelogram (skewed compared to a bounding rectangle).

In his manuscript De motu corporum in gyrum, and in the 'Principia', Isaac Newton cites as a lemma proved by previous authors that all (bounding) parallelograms for a given ellipse have the same area.

It is possible to reconstruct an ellipse from any pair of conjugate diameters, or from any bounding parallelogram.

For example, in proposition 14 of Book VIII of his Collection, Pappus of Alexandria gives a method for constructing the axes of an ellipse from a given pair of conjugate diameters.

Another method is using Rytz's construction, which takes advantage of the Thales' theorem for finding the directions and lengths of the major and minor axes of an ellipse regardless of its rotation or shearing.

Similar to the elliptic case, diameters of a hyperbola are conjugate when each bisects all chords parallel to the other.

[1] In this case both the hyperbola and its conjugate are sources for the chords and diameters.

In the case of a rectangular hyperbola, its conjugate is the reflection across an asymptote.

The placement of tie rods reinforcing a square assembly of girders is guided by the relation of conjugate diameters in a book on analytic geometry.

[3] Conjugate diameters of hyperbolas are also useful for stating the principle of relativity in the modern physics of spacetime.

The ellipse, parabola, and hyperbola are viewed as conics in projective geometry, and each conic determines a relation of pole and polar between points and lines.

Using these concepts, "two diameters are conjugate when each is the polar of the figurative point of the other.