If I am just trying to create an ellipse with the 4 vertices and center, can I just plug the numbers into the equation of an ellipse without worrying about the foci?

racmanovcf 2022-10-22 Answered
Do I need foci to calculate an ellipse?
I have been trying to find an answer, but where I look does not tell me why I need foci if I have all 4 vertices and the center. If I am just trying to create an ellipse with the 4 vertices and center, can I just plug the numbers into the equation of an ellipse without worrying about the foci? I am trying to understand how the foci come into play, as they don't appear in the actual equation of an ellipse. However, I want my ellipse to be correct. I am trying to take a circle, and scale the y axis only, elongating the circle to create the ellipse that still passes through the 4 points, 2 now scaled. It is a vertical ellipse.
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Answers (2)

Kash Osborn
Answered 2022-10-23 Author has 18 answers
Step 1
The general equation of an ellipse is a quadratic expression a x 2 + 2 b x y + c y 2 + d x + e y + f = 0 where the foci do not appear explicitly.
Step 2
In the case of an axis-parallel ellipse, the cross term xy does not exist, and you can use the simplified form ( x x c ) 2 a 2 + ( y y c ) 2 b 2 = 1 where the center and axis lengths are explicit.
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Wyatt Weeks
Answered 2022-10-24 Author has 1 answers
Explanation:
You have 2a,2b from the given points. If you want to calculate an ellipse from relation x 2 a 2 + y 2 b 2 = 1 which has axes parallel to coordinate axes that is taken implied for your exercise, you can use y = b 1 x 2 / a 2 to find y for each x.
(Foci location, latus rectum, eccentricity) are related parameters that is not required for above calculation).
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