# How can I calculate the radius of an ellipse at a specific angle?

## Main Question or Discussion Point

given the major axis length, minor axis length, at the given angle THETA. what's the formula?

nrqed
Homework Helper
Gold Member
given the major axis length, minor axis length, at the given angle THETA. what's the formula?
$$r(\theta) = r_{max} \frac{1-e}{1+ e cos \theta}$$
with r_max = aphelion = a(1+e). Here my angle is chosen to be zero at perihelion (you can check that when theta=0, we recover a(1-e) = perihelion and when theta= 180 degrees, we get the aphelion).

Patrick

arildno
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Gold Member
Dearly Missed
On standard form, with (x,y) being Cartesian points, the equation for the ellipse is:
$$(\frac{x}{a})^{2}+(\frac{y}{b})^{2}=1$$
Now, just make a standard polar coordinate change of (x,y) and gain an equation for the radius r!

wikipedia says it's (1 - e^2) as opposed to (1 - e) in that equation nrqed. Who's right?

D H
Staff Emeritus
Wikipedia is correct. Note well: nrqed is talking about the angle between line segments subtending from one of the foci of the ellipse. If you followed arildno's advice, you would have computed the angle between line segments subtending from the center of the ellipse.

yes i need the radius from the center as opposed to one of the foci...

what is the formula? i'm not sure how to convert to polar coordinates

x = rcos(theta)
y = rsin(theta)

and plug those values into that equation and solve for "r"? or am i doing something wrongly?

nrqed
Homework Helper
Gold Member
Wikipedia is correct. Note well: nrqed is talking about the angle between line segments subtending from one of the foci of the ellipse. If you followed arildno's advice, you would have computed the angle between line segments subtending from the center of the ellipse.
Oh.. yes. Good catch. Sorry, for some reason I was thinking about it as an astronomy question when I posted, not as a math question, so I though it as asking the distance from one focus (eg the Sun). This is clear in my answer since I specified that the distances at zero and 180 degrees are the perihelion and aphelion. Sorry!

so from center... is it:

(ab)/((b^2cos^2t+a^2sin^2t)^(3/2))

or

(ab)/((b^2cos^2t+a^2sin^2t)^(1/2))?

D H
Staff Emeritus
so from center... is it:

(ab)/((b^2cos^2t+a^2sin^2t)^(3/2))

or

(ab)/((b^2cos^2t+a^2sin^2t)^(1/2))?
Obviously not the first one, as the units are wrong (they are 1/length).

Not so obviously, the second one is wrong too. Don't just guess an answer.

Show your work so we can show where you went astray.