I've been googling, searching forums and looking in my old algebra/trig books to try to understand how to find the end points to the major and minor axis of an ellipse given the end points of two conjugate diameters (assume ellipse centered at the origin). I want to be able to recast an ellipse given that data into a form acceptable for use in an SVG diagram which requires the major(x) and minor(y) axis radii. I can calculate any rotation necessary from the major axis end point.
See for example the below image. I have conjugate points P and Q and need to find (a) and (b).
Example showing conjugate points P and Q (I couldn't upload an image(rep !> 10 yet)).
I hope someone here can help shed some light on this for me.
Best Answer
Perhaps added since this question was answered, Wikipedia has good information on this problem. There is an interesting geometric construction which contrasts with the algebraic solutions offered here: Rytz's construction.
(I have been told to add information to the answer rather than just posting links. Unfortunately as my rep is less than 50 I can't make comments yet)
The setting in which I found Rytz's construction useful was in drawing the elevation of a circle in a plan oblique projection. In this case, as in the other conjugate tangent problems that arise in parallel projection, the ellipse is tangent to the midpoints of the edges of a parallelogram. This is a slightly more constrained and regular situation than the diagram referenced in the original question, though a tangent parallelogram could easily be constructed around the ellipse shown in that image.
Rytz's construction is apparently the last refinement of a long series of solutions to this problem, starting with Pappus. It relies on the fact that conjugate diameters are affine images of perpendicular diameters of a circle. In particular, the perpendiculars from the foci to any tangent intersect the tangent on the auxiliary circle, the circle centered at the centre of the ellipse with the major axis as diameter. As I understand it Rytz's construction is a carefully minimized (in terms of number of steps) derivative of the earlier techniques, intended for practical use in drafting, etc.