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I am looking for a way to calculate the lengths of both semi-axes and the rotation angle of the ellipse in the image as shown in this picture. Six points are given, with two pairs of points being equidistant from the center and exactly in opposition. The other two points have the same x-position and their distance A1-A2 is also known. The diameters of the ellipse that end at the given points are conjugate as you can see in this question. Thank you for your help.

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atarax42
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2 Answers2

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Your task is a special case of the more general calculate ellipse from points, so my answer there applies here as well.

Since you have additional information, and even one more point than you'd need, you may make use of that information somehow. But I'd say the first step, the one for determining the rotation angle (using eigenvectors or a quadratic equation resulting from the application of rotation matrices), is always a good start. Once you have that, you can make use of the fact that you essentially already know the center, and don't have to read that off the equation of the ellipse.

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MvG
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Thank you. I found a much easier way to get the properties of the ellipse using the Rytz's construction method.

atarax42
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  • Rytz's construction requires that the given diameters are "conjugate". (That is, tangents at the endpoints of one are parallel to the other.) This property is not assumed in your setup. – Blue May 12 '16 at 03:59
  • The tangents at P2 and P4 are parallel. Also the tangents at P1 and P3. Or am I wrong? – atarax42 May 12 '16 at 04:50
  • The tangents at $P_2$ and $P_4$ are indeed always parallel to each other; conjugate-ness requires that these tangents be parallel to $\overline{P_1P_3}$. (Likewise, tangents at $P_1$ and $P_3$ need to be parallel to $\overline{P_2P_4}$; but, if it works one way, then you get the other way for free.) If the ellipse happened to be a circle, this condition would become "The given diameters are perpendicular." In your problem, the given diameters have no special relationship to each other (apart from having the same midpoint, of course). – Blue May 12 '16 at 05:47
  • The relationship of the diameters is explained in more detail in this post. There you can see that they are parallel to the tangents. – atarax42 May 12 '16 at 06:07
  • Okay, I agree. Since the diameters you really want to discuss arise from an affine transformation of the axes of some other ellipse, the diameters inherit conjugate-ness from those axes. So, Rytz's construction indeed applies. The problem as stated above, however, doesn't provide this important information; you should edit to add a note that you're assuming the diameters are conjugates. (You should also link back to the original question for context.) – Blue May 12 '16 at 06:19
  • ---------- Done. – atarax42 May 12 '16 at 07:37