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

In summary, the formula for finding the radius of an ellipse at a given angle THETA, with major axis length a and minor axis length b, is (ab)/((b^2cos^2t+a^2sin^2t)^(1/2)). This formula can be derived by converting the standard Cartesian equation for an ellipse to polar coordinates and solving for r. It is important to note that the angle THETA is measured from the center of the ellipse, not from one of the foci.
  • #1
bobthebanana
23
0
given the major axis length, minor axis length, at the given angle THETA. what's the formula?
 
Mathematics news on Phys.org
  • #2
bobthebanana said:
given the major axis length, minor axis length, at the given angle THETA. what's the formula?

[tex] r(\theta) = r_{max} \frac{1-e}{1+ e cos \theta} [/tex]
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
 
  • #3
On standard form, with (x,y) being Cartesian points, the equation for the ellipse is:
[tex](\frac{x}{a})^{2}+(\frac{y}{b})^{2}=1[/tex]
Now, just make a standard polar coordinate change of (x,y) and gain an equation for the radius r!
 
  • #4
wikipedia says it's (1 - e^2) as opposed to (1 - e) in that equation nrqed. Who's right?
 
  • #5
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.
 
  • #6
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?
 
  • #7
D H said:
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. :frown: 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!
 
  • #8
so from center... is it:

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

or

(ab)/((b^2cos^2t+a^2sin^2t)^(1/2))?
 
  • #9
bobthebanana said:
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.
 

1. How do I determine the radius of an ellipse at a specific angle?

The radius of an ellipse at a specific angle can be calculated by using the formula:
r = a*b / √[(b*cosθ)^2 + (a*sinθ)^2]
Where a and b are the major and minor axes of the ellipse, and θ is the angle at which the radius is measured.

2. What is the difference between major and minor axes of an ellipse?

The major axis of an ellipse is the longest diameter of the ellipse, while the minor axis is the shortest diameter. The major axis is also the axis of symmetry of the ellipse.

3. Can I use the circumference of an ellipse to calculate the radius at a specific angle?

No, the circumference of an ellipse cannot be used to calculate the radius at a specific angle. The circumference depends on the entire shape of the ellipse, not just one specific angle.

4. How does the eccentricity of an ellipse affect the radius at a specific angle?

The eccentricity of an ellipse is a measure of how elongated or stretched out the ellipse is. The higher the eccentricity, the more elongated the ellipse and the more the radius will vary at different angles.

5. Are there any online tools or calculators available for calculating the radius of an ellipse at a specific angle?

Yes, there are several online tools and calculators available that can help you calculate the radius of an ellipse at a specific angle. Some popular ones include WolframAlpha, Desmos, and GeoGebra.

Similar threads

Replies
6
Views
1K
Replies
1
Views
1K
Replies
4
Views
676
Replies
9
Views
696
Replies
1
Views
2K
Replies
2
Views
2K
Replies
1
Views
1K
  • General Math
Replies
9
Views
1K
Back
Top