Constructing the major/minor axes of an ellipse inscribed in a quadrilateral

The following steps show how to construct the major/minor axes of the ellipse. Previously we showed how to construct the points of tangency between the ellipse and the quadrilateral. The goal of the construction is to draw the ellipse tangent to the four points \(A, B, C, D\) (in the pictures below, this ellipse is drawn in orange to make it easier to visualise, but we do not know this in advance, we want to construct it).

The construction requires some knowledge of the geometry an ellipse, in particular the notion of a pair of conjugate diameters. Sometime (hopefully soon!) I will add some more explanation of this as well as proofs of some of the basic theorems used below.

Construct the centre of the ellipse

Choose two of the points on the ellipse for which we know the tangent (in the diagram below we choose \(A\) and \(B\)) and mark the midpoint (call it \(M\)) and mark the point where the tangent lines intersect (in the diagram below this point is \(X\)). Now draw a line from the intersection of the tangents through \(M\). This line will pass through the centre of the ellipse. (This is Corollary 1 on p108 of Constructive geometry of plane curves by Eagles and the proof is contained in the two propositions immediately preceding Corollary 1.)

Now repeat the construction with a different pair of points on the ellipse to obtain another line passing through the centre. The diagram below shows this for the line passing through \(Z\) and the midpoint of \(CD\). The intersection of these two lines is the centre of the ellipse.

Construct a pair of conjugate diameters of the ellipse

Now draw a line through \(O\) which is parallel to \(AB\). This is the diameter of the ellipse which is conjugate to \(OX\). In the same way, construct the line through \(O\) which is parallel to \(CD\). This is conjugate to the diameter \(OZ\).

Construct the involution relating pairs of conjugate diameters

The conjugate diameters of an ellipse form conjugate pairs of lines in the pencil of lines through the centre \(O\). These lines are related by an involution on this pencil. An involution is determined by two pairs of lines, and so we can construct this involution from the two pairs of conjugate diameters constructed above.

There are a number of ways to construct this involution. For the next step (see below) it will be useful to construct this involution on a circle. Draw a circle passing through the point \(O\). For convenience we will choose \(X\) as the centre of the circle, but any point will do.

Now let \(S_1\) and \(S_2\) be the intersection points of one pair of conjugate diameters with the circle, and let \(T_1\) and \(T_2\) be the intersection points of the other pair.

The centre \(R\) of the involution is the intersection of \(S_1 S_2\) and \(T_1 T_2\). Therefore, given any line through \(R\) intersecting the circle at \(U_1\) and \(U_2\), the lines \(OU_1\) and \(OU_2\) are a pair of conjugate diameters.

Construct the major and minor axes

Now the major/minor axes are the pair of conjugate diameters which intersect at right angles (they are unique in the general case where the ellipse is not a circle). If two lines intersect the point \(O\) on the circle at right angles, then the other intersection points \(M_1\) and \(M_2\) on the circle must lie on opposite sides of a diameter of the circle. Since \(M_1\) and \(M_2\) are related by the involution constructed above, then this diameter must pass through \(R\) (the centre of involution). Therefore, the diameter is the line \(RX\) and the points \(M_1\) and \(M_2\) are the intersection points of \(RX\) with the circle.

Therefore we have shown that the lines \(OM_1\) and \(OM_2\) are the major and minor axes of the ellipse.