Elliptic Integrals: Arc Length of Ellipses and Elliptic Curves

In summary: The arc length of an ellipse can be found by taking the derivative of the function sn(x) with respect to x, and solving for x.
  • #1
amcavoy
665
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Taken from http://en.wikipedia.org/wiki/Elliptic_integral:

In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse and were first studied by Fagnano and Leonhard Euler.

Is it just me, or does it seem like there is an easier way to find the arc length of an ellipse? I thought elliptic integrals arose in giving the arc length of elliptic curves, which as far as I know are a lot different than ellipses.
 
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  • #2
I thought elliptic integrals arose in giving the arc length of elliptic curves, which as far as I know are a lot different than ellipses.

But an ellipse is an elliptic curve! :)
 
  • #3
I suspect apmcavoy was thinking of "elliptic functions", as in number theory, which are quite different. In any case, he is wrong. Elliptic integrals did, indeed, arise from trying to find the arc length of an ellipse which is NOT as simple as he seems to think. The arclength of an ellipse cannot be written in any simple formula.

(The area is very simple. The area of the ellipse [itex]\frac{x^2}{a^2}+ \frac{y^2}{b^2}= 1[/itex] is just [itex]\pi ab[/itex]. The distance around (circumference?) an ellipse is not.)
 
  • #4
HallsofIvy said:
I suspect apmcavoy was thinking of "elliptic functions", as in number theory, which are quite different. In any case, he is wrong. Elliptic integrals did, indeed, arise from trying to find the arc length of an ellipse which is NOT as simple as he seems to think. The arclength of an ellipse cannot be written in any simple formula.

(The area is very simple. The area of the ellipse [itex]\frac{x^2}{a^2}+ \frac{y^2}{b^2}= 1[/itex] is just [itex]\pi ab[/itex]. The distance around (circumference?) an ellipse is not.)


Elliptic functions are the inverse functions to elliptic integrals.

[tex]sn^{-1}(x) = \int^x_0 \frac{dt}{\sqrt{(1-t^2)}\sqrt{(1-k^2t^2)}}[/tex], etc.
 

1. What are elliptic integrals?

Elliptic integrals are mathematical integrals that involve elliptic functions. They are used to calculate the arc length of ellipses and elliptic curves.

2. How are elliptic integrals different from other types of integrals?

Elliptic integrals are different from other types of integrals because they involve elliptic functions, which are more complex and cannot be expressed in terms of elementary functions like polynomials and trigonometric functions.

3. What is the significance of the arc length of an ellipse or elliptic curve?

The arc length of an ellipse or elliptic curve is an important geometric property that is used in various applications, such as calculating the perimeter of an ellipse or determining the path of a planet in an elliptical orbit.

4. How are elliptic integrals used in real-world problems?

Elliptic integrals are used in various fields of science and engineering, such as physics, astronomy, and mechanical engineering. They are used to solve problems involving curved surfaces, such as calculating the volume of a torus or the surface area of an ellipsoid.

5. Are there different types of elliptic integrals?

Yes, there are several types of elliptic integrals, including incomplete elliptic integrals, complete elliptic integrals, and Jacobi elliptic integrals. Each type has its own specific formula and properties.

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