Intersection between rotated & translated ellipse and line

In summary, the conversation discusses finding the intersection points between a rotated ellipse and a segment. There are two suggested methods to solve this problem: either by hand using algebraic equations or by transforming the equations into standard form and using a calculator. The latter method may be more efficient, but the former method is also possible.
  • #1
piercazzo
1
0
I have a rotated ellipse, not centered at the origin, defined by x,y,a,b and angle.
Then I have a segment defined by two points x1,y1 and x2,y2
Is there a quick way to find the intersection points?

I used wolfram alpha equation solver, I tried to insert the equation of a line into the one of a standard non rotated, non translated ellipse,

10how40.gif


and resolving for x this is the result

15qw704.png


which is nice.

Then I took the equation of a rotated and translated ellipse

vngnwi.png


and this is the result

2rnzx1h.png

6p5bat.png


Which is obviously impractical, can anyone suggest a different method?
 
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  • #2
Do it by hand rather than use the calculator.
You can either -
1. Solve for one variable in the line equation and sub into the ellipse equation - solve for the other coordinate ... you know: as usual.

2. transform the ellipse into standard form, apply the same transformation to the line, find solutions for the transformed system, apply the inverse transformation to the solution.
 

Related to Intersection between rotated & translated ellipse and line

1. What is the definition of intersection between a rotated and translated ellipse and a line?

The intersection between a rotated and translated ellipse and a line is the point or points where the ellipse and the line meet or cross each other.

2. How do you determine if an ellipse and a line intersect?

To determine if an ellipse and a line intersect, you can use the algebraic equation of the ellipse and the equation of the line to find the coordinates of the intersection point(s). If the coordinates satisfy both equations, then there is an intersection.

3. Can an ellipse and a line intersect at more than two points?

Yes, an ellipse and a line can intersect at more than two points if the ellipse is degenerate (a circle) or if the line passes through the center of the ellipse.

4. How many solutions are there for the intersection between a rotated and translated ellipse and a line?

There can be 0, 1, 2, or infinite solutions for the intersection between a rotated and translated ellipse and a line, depending on the relative positions of the ellipse and the line. For example, if the line is completely outside the ellipse, there will be no intersection. If the line is tangent to the ellipse, there will be one solution. If the line passes through the ellipse, there will be infinite solutions.

5. Is there a geometric interpretation of the intersection between a rotated and translated ellipse and a line?

Yes, the intersection between a rotated and translated ellipse and a line can be interpreted as the point or points where the line cuts through the boundary of the ellipse. This can be visualized as the point where a straight line touches and crosses the curved edge of an oval-shaped object.

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