Finding the focus of an ellipse with ruler and compass

Question

Given an ellipse E, find his focus with ruler and compass.

I tried to generalize the next theorem:

Given a circle C find his center. The idea for prove this is with an chord and it's bisector line (the ortogonal line in the midle point of the chord)

Ortogonality is fundamental for this theorem but in the ellipse ortogonality doesn't work.

Inspired by the equations and the linear algebra, an ellipse is a circle transformed by a linear transformation, but there are infinite linear transformations...the chord determines a linear transformation if we know the center of the elipse...

No, only the Ellipse is given...I want to find the symmetry axes given the Ellipse... — Gabriel Canedo, Mar 05 '21 at 19:50
@JeanMarie look this https://math.stackexchange.com/questions/2042664/axes-of-symmetry-for-a-general-ellipse then the symmetry axes are buildable in the sense of ruler and compass — Gabriel Canedo, Mar 05 '21 at 19:53
You can find something here: https://math.stackexchange.com/questions/263307/a-construction-using-straightedge-and-compass/388504#388504 — RicardoCruz, Mar 05 '21 at 19:59

Intelligenti pauca · Answer 1 · 2021-03-05T21:35:58.583

6

Draw two parallel chords $AB$ and $CD$ and let $M$, $N$ be their midpoints. Line $MN$ intersects the ellipse at the endpoints of a diameter $PQ$, whose midpoint $O$ is the center of the ellipse. The diameter through $O$ parallel to $AB$ is the conjugate of $PQ$.

Finally, having a pair of conjugate diameters, you can construct the axes as explained in this old answer of mine and from them find the foci.

edited Mar 05 '21 at 21:35

answered Mar 05 '21 at 19:57

Intelligenti pauca

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Thank you, the construction depends of the paralelism...I'm surprised – Gabriel Canedo Mar 05 '21 at 20:00
@GabrielCanedo This is a famous property: https://www.cut-the-knot.org/Curriculum/Geometry/ParallelChordsInEllipse.shtml – Intelligenti pauca Mar 05 '21 at 20:01
I started to think that it makes complete sense that the construction depends on parallelism since in the projective plane it would not make sense to speak of the center of an ellipse, right? – Gabriel Canedo Mar 05 '21 at 20:03

score 3 · Answer 2 · answered Mar 05 '21 at 20:40

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As in previous answer find the center of the ellipse $C$. Draw a circle with center at $C$ which intersects the ellipse in 4 points. The points form a rectangle. Draw through $C$ lines parallel to the sides of the rectangle. They are the main axes of the ellipse.

answered Mar 05 '21 at 20:40

user

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Great! This is a very elegant proof. – Gabriel Canedo Mar 05 '21 at 22:46

Finding the focus of an ellipse with ruler and compass

2 Answers2