Finding the points of intersection of two ellipses

In summary, the points of intersection of two ellipses can be found by using the substitution method to solve the equations of the ellipses simultaneously. Two ellipses can intersect at a maximum of four points and can be graphed by plotting their equations on the same coordinate plane. Even if the ellipses have different centers, they can still intersect, but the points of intersection may not lie on the same line as the centers. There is a formula for finding the points of intersection, which involves solving a system of equations and using the quadratic formula.
  • #1
Physt
49
1
Does anyone know where I can find an algorithm for the points of intersection of two ellipses existing with arbitrary center points and rotations and having 0, 1, 2, 3 or 4 points of intersection?
 
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  • #2
If you have the equations of two conic sections, you can find the intersection points by finding the zeros of a 4th degree resultant (in one variable) as shown here following the paragraph starting with "For the intersections of two conics"

I havn't quality checked the linked page, but at least this method seems sound.
 

Related to Finding the points of intersection of two ellipses

1. How do you find the points of intersection of two ellipses?

To find the points of intersection of two ellipses, you can use the substitution method. This involves solving the equations of the two ellipses simultaneously to find the coordinates of the points where they intersect.

2. Can two ellipses intersect at more than two points?

Yes, it is possible for two ellipses to intersect at more than two points. In fact, they can intersect at a maximum of four points. However, it is also possible for them to not intersect at all.

3. How do you graph two intersecting ellipses?

To graph two intersecting ellipses, you can plot the equations of the ellipses on the same coordinate plane and identify the points of intersection. You can also use a graphing calculator or software to graph the ellipses and find their points of intersection.

4. What if the two ellipses have different centers?

If the two ellipses have different centers, it is still possible for them to intersect. However, the points of intersection will not necessarily lie on the same line as the centers of the ellipses.

5. Is there a formula for finding the points of intersection of two ellipses?

Yes, there is a formula for finding the points of intersection of two ellipses. It involves solving a system of equations and using the quadratic formula to find the coordinates of the points of intersection.

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