How is the Length of an Elliptical Curve Calculated?

In summary, to find the circumference of an ellipse, you can use the formula P = 4a\int_0^{\frac{\pi}{2}}\sqrt{1-e^2\sin^2{t}}dt, where a is the semi-major axis and e is the eccentricity. This can be approximated using a series expansion or computed numerically. Another option is to use the arc-length formula s=\int_a^b \sqrt{1+(\frac{dy}{dx})^2}dx, where y=f(x) and x=a,x=b. There are also various approximations available on websites, such as http://astronomy.swin.edu.au/~pbourke/geometry/ellipsecirc
  • #1
tandoorichicken
245
0
is there a formula to find the circumference of an ellipse?
 
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  • #2
You can approximate it. An exact expression for the perimeter of an ellipse is
[itex] P = 4a\int_0^{\frac{\pi}{2}}\sqrt{1-e^2\sin^2{t}}dt [/itex]
where a is the semi-major axis, the eccentricity [itex] e = \frac{\sqrt{a^2-b^2}}{a} [/itex], and b is the semi-minor axis. This is found writing the equation of the ellipse in parametric form and using the arc-length formula. You can compute the integral numerically or write an approximation using a series expansion.

Here is a website with a number of approximations you can try out:

http://astronomy.swin.edu.au/~pbourke/geometry/ellipsecirc/ [Broken]
 
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  • #3
Length of the curve is given by


[tex] s=\int_a^b \sqrt{1+(\frac{dy}{dx})^2}dx[/tex]
where y=f(x) and x=a,x=b
 

1. What is the formula for finding the circumference of an ellipse?

The formula for finding the circumference of an ellipse is 2π√(a²+b²/2), where a and b are the lengths of the semi-major and semi-minor axes, respectively.

2. How is the circumference of an ellipse different from a circle?

The circumference of an ellipse is different from a circle because it is not a constant distance from the center like a circle. Instead, it varies depending on the length of the semi-major and semi-minor axes.

3. Can the circumference of an ellipse be calculated if only one axis length is given?

Yes, the circumference of an ellipse can still be calculated if only one axis length is given. The formula for finding the circumference can be modified to use only one axis length, as long as the other length is known to be 0.

4. Are there any real-world applications for calculating the circumference of an ellipse?

Yes, there are many real-world applications for calculating the circumference of an ellipse. For example, it is used in engineering and architecture to design curved structures such as bridges and arches. It is also used in astronomy to calculate the orbits of planets and other celestial bodies.

5. Can the circumference of an ellipse ever be greater than the circumference of a circle with the same diameter?

No, the circumference of an ellipse can never be greater than the circumference of a circle with the same diameter. The circumference of a circle is the maximum possible length for any closed curve with the same diameter, meaning that the circumference of an ellipse can only be equal to or less than the circumference of a circle with the same diameter.

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