# Conjugate diameters of ellipse

How to find the length of major and minor axis of ellipse given the length of two conjugate diameters and the angle between them?

I am aware about how to construct the ellipse using the above given facts(not by Rytz's Construction). I would like to know, independent of what method of construction one uses, how one can find the length of the axes.

Here's a geometric construction: if $$MN$$ and $$DE$$ are conjugate diameters, draw line $$QQ'$$ through $$N$$ perpendicular to $$DE$$ (see diagram below). Points $$Q$$ and $$Q'$$ must be chosen such that $$NQ=NQ'=OD$$. Principal axes $$IR$$ and $$ST$$ lie then on the bisectors of the angles formed by lines $$OQ$$ and $$OQ'$$, and: $$\tag{1} IR=OQ'+OQ,\quad TS=OQ'-OQ.$$
If $$ON=a$$, $$OD=b$$ and the angle between them is $$\theta$$, then from the cosine rule applied to triangles $$ONQ$$ and $$ONQ'$$ we get: $$OQ^2=a^2+b^2-2ab\sin\theta,\quad OQ'^2=a^2+b^2+2ab\sin\theta.$$ Inserting these into $$(1)$$ we finally obtain: $$OR\cdot OS=ab\sin\theta,\quad OR^2+OS^2=a^2+b^2.$$ These equalities could have been directly derived, as they are well known properties of conjugate diameters (see properties 1. and 2. listed here).
• @amd Fine then, $a,b$ can be recovered by Viète. The SVD-based method is more general, however, since it also determines the ellipse orientation. – Parcly Taxel Aug 12 '19 at 19:10
Suppose the conjugate radii are $$a$$ and $$b$$ and the angle between them is $$\theta$$. Form the following matrix: $$\mathbf A=\begin{bmatrix}a&b\cos\theta\\ 0&b\sin\theta\end{bmatrix}$$ The columns are vectors corresponding to the conjugate radii. Now perform a singular value decomposition $$\mathbf A=\mathbf{U\Sigma V}^T$$. The diagonal entries of $$\bf\Sigma$$, the singular values of $$\bf A$$, are the semi-axis lengths.
This works because any ellipse centred on the origin is a linear transformation, $$\bf A$$ in this case, of the unit circle. The SVD corresponds to decomposing this transformation into a rotation/reflection $$\mathbf V^T$$ (which visually doesn't change anything), a scaling along the coordinate axes $$\bf\Sigma$$ (so that the ellipse semi-axes are its diagonal entries, as above) and another rotation/reflection $$\bf U$$ (which does not change the axis lengths). This is visualised below, with the displayed arrows being conjugate diameters of the resulting ellipse: In fact this method finds more than just the axis lengths. Suppose the columns of $$\bf A$$ represent any pair of conjugate radii vectors of an ellipse. Then the columns of $$\bf U\Sigma$$ are perpendicular conjugate radii, thus semi-axis vectors, for the same ellipse.