intersection of an ellipsoid and cylindrical plane.

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Just an idea - I don't know if it will fit your problem or not, cause actually you didn't describe it with sufficient level of details.

Let's think in terms of a cylindrical coordinate system - the equations below describe your cylindrical surface, ignoring that fact it has limited height (if I understood your question correctly).

$$x = R \cdot \sin(\phi), y = R \cdot \cos(\phi), z = z$$

Then you can substitute these $x$ and $y$ into the ellipsoid equation:

$$\frac {R^2} {a^2} \cdot \sin^2(\phi) + \frac {R^2}{b^2}\cdot \cos^2(\phi) + \frac{z^2}{c^2} = 1$$

You can express the $z$ as a function of $\phi$ - this will give you the intersection curve equation in cylindrical coordinates.

I hope it'll help.

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Updated on August 01, 2022

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  • sam
    sam over 1 year

    I need to understand if an ellipsoid and a cylindrical arc intersect, what will be the general equation of the cutted ellipse? How can I solve for that equation? I know in 3D, the equation of an cylinder is x^2+y^2=R^2, z=height and equation of an ellipsoid is (x^2)/(a^2)+(y^2)/(b^2)+(z^2)/(c^2)=1, a,b,c,z,R are known. Now if the ellipsoid intersect the cylindrical wall, how can I solve for the equation of the intersected ellipse? and I need the center of the intersected ellipse as well.I have no idea how to solve it! I will appreciate any help. Thank you very much.

    **cylindrical are is a cylinder wall x2+y2=R2, with z height (this is the equation of a cylinder whose center is at (0,0,0).) ** in 3D, the ellipsoid will cut the plane in a circle (if b=c) or ellipse (if b not equal c). I need to solve this as I will use this concept for my numerical code to create an model.

    • David K
      David K about 9 years
      This question badly needs more context. As stated, it makes no sense: what is a "cylindrical plane", and why would you expect the intersection of a cylinder and an ellipsoid to be an ellipse? You tagged this as mathematical-physics; what is the application to physics here? Perhaps that would shed some light on what you are trying to do.
    • sam
      sam about 9 years
      cylindrical plane is a cylinder wall x2+y2=R2, with z height (this is the equation of a cylinder whose center is at (0,0,0).)
    • sam
      sam about 9 years
      in 3D, the ellipsoid will cut the plane in a circle (if b=c) or ellipse (if b not equal c). I need to solve this as I will use this concept for my numerical code to create an model.
    • David K
      David K about 9 years
      Make up your mind: are you looking for intersection with a plane or with a cylinder? Also, "numerical code" and "model" tell us nothing at all about why you must use these particular two shapes or what they mean, so we cannot guess what it is you really want to ask.