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From the wiki article about eccentric anomaly follows:

$$\cos E = \frac{x}{a}$$ $$\sin E = \frac{y}{b}$$

where E - eccentric anomaly, a - semi-major axis, b - semi-minor axis, P = P(x,y) a point on elleptic orbit. So if you know an E, you can find a position on an orbit:

$$x = a \cos E$$ $$y = b \sin E$$

However, from article on Kepler's eqution follows:

$$x = a (\cos E - e)$$ $$y = b \sin E$$

Please, help to understand why these formaluae differ? I expect them to be the same.

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Both formulae are correct. The discrepancy is because the formula from the eccentric anomaly article uses the centre of the ellipse as the origin, but the formula from the Kepler's equation article uses a focus of the ellipse (i.e, the central gravitating body, eg the Sun) as the origin. Note that $c = ae$ is the distance from the ellipse centre to a focus

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    $\begingroup$ Thank you, now it's clear. Can't upvote the answer since I don't have enough rep. $\endgroup$
    – Jonas
    Dec 27, 2019 at 17:27
  • $\begingroup$ @Jonas I've upvoted both question and answer, we'll get you there soon! $\endgroup$
    – uhoh
    Dec 28, 2019 at 1:46
  • $\begingroup$ See my answer here for a diagram showing the eccentric anomaly. $\endgroup$
    – PM 2Ring
    Apr 12, 2021 at 11:46

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