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February 17

Ellipse angles

How do you find the angle between the major axis and the line made by connecting a point on the ellipse outline with the ellipse focus if you know the angle between the major axis and the line made by connecting the same point on the ellipse outline with the ellipse center and the ellipse shape (major axis, minor axis, eccentricity, whatever) — Preceding unsigned comment added by Someone with a Question (talk • contribs) 07:55, 17 February 2017 (UTC)[reply]

(Thinking "aloud" here...) Let's say the extremes of the ellipse are (±a,0) and (0,±b). First you want to know where the line from the center, at a given angle α, meets the ellipse. One way is to stretch the whole figure so that the ellipse becomes a circle; replace α with β=arctan((a/b) tan(α)), x = a cos(β), y′ = a sin(β); stretch back, y = (b/a) y′ = b sin(β). Next you need the coordinates (c,0) of the focus, which you get from the identity b²+c² = a². The angle sought is arctan(y/(x-c)). —Tamfang (talk) 09:06, 17 February 2017 (UTC)[reply]

You need to find the equation of the ellipse, given the info about it's shape. Then find the equation of a line at the given angle from the major axis line. Find the intersection of those two equations mathematically. That will give you two points. From each you can find the equation of the line between that ellipse intersection point and either focus point. This will give you 4 equations of lines, which will each have an angle with the equation of the major axis line. Of those 4 angles, there will be 2 identical pairs of angles. That's a lot of steps, but each is fairly basic. This could all be automated in a program. StuRat (talk) 16:55, 17 February 2017 (UTC)[reply]

To elaborate a bit: (1) To find the equation of the line from the center with given angle, note that the slope of the line is the tangent of the given angle (assuming you have positioned the ellipse in the standard way with the long axis coinciding with the horizontal axis). (2) Once you have the equation of a line from an ellipse point to a focus, the tangent of the angle between this line and the horizontal axis is simply the slope of the line (so the angle is the arctangent of that). Loraof (talk) 18:29, 18 February 2017 (UTC)[reply]