[RESOLVED] Plot ellipse through apex and two other points

[boops boops]

I don't think the constraint you've said makes complete sense. You want P2 to be as far as possible perpendicularly from line P1P3 (typo), but on "the arc": you're using this constraint to define which arc you mean, so referencing "the arc" here is circular. Perhaps your hope is that among all possible ellipses passing through the three points, there is a unique one where P2 is a point on the ellipse furthest perpendicularly from the line P1P3?

Yes, that's what I meant.Or you could say choose the elliptical arc whose furthest point from line P1P3 is closest to P2. But I suspect this won't be necessary.

In the symmetrical situation which Logophobic illustrated in post #16, the arc can be circular.

I can't find any ambiguity in the asymmetrical situation I illustrated in post #18. Once P1 and P3 are positioned on the ellipse, there seems to be only one position where I can place P2 on the arc so that P2-m-c forms a straight line. I realize this isn't a proof that there is only one solution, but it seems reasonable evidence. What is more, the resulting position of P2 always appears to be (roughly) the furthest point of the arc from line P1P3; so I suspect that this will turn out to be a consequence rather than a constraint!

I'll try to sum up the desired behaviour of the tool (square brackets mean line length):

1. While [P2m] is less than [P1m]
- a. if [P1P2] = [P3P2] the arc is part of the circle through P1, P2 and P3.
- b. otherwise, P2, P2 and P3 are on the arc and the centre of the ellipse c is collinear with P2m and on the side of m opposite to P2.
2. When [P2m] = [P1m], the arc is a semicircle of radius [p1m] through P1, P2 and P3. The centre c coincides with m.
3. While [P2m] > [P1m] the arc is an ellipse through P1, P3 and P3 with P2 at the vertex of the major axis. The centre c coincides with m.

Situation 1b is the only one which needs resolving.

You could also let the user specify the eccentricity of the ellipse (the ratio A/B, say). Eccentricity 1 would give a circle, etc.

That's another one I would save for a more mathematical version of the tool.

regards, BB