# how to find center of ellipse when arc of ellipse given?

i have arc of ellipse with start and end point of ellipse. And also i have radius's of ellipse. using this data how can i find center of ellipse? EX:: arc with (140,50) as start point and (240,100) as end points, (200,50) as radius's of ellipse. from this given data how could i find center of ellipse?

• What is radius of ellipse? Lengths of semi-axis of ellipse? – Shuchang Sep 20 '13 at 5:44
• yes. length of semi mojor axies , and semi -minor axies. – sindhu Sep 20 '13 at 6:00

## 1 Answer

I don't think this is possible. Let $P$ and $Q$ be the given two points. Take an ellipse with given semi-axes $a$ and $b$ that passes through $P$ and $Q$. Imagine $P$ and $Q$ as pins sticking in the floor, and the ellipse is a piece of cardboard that's touching both of them. Clearly you can "slide" the ellipse around, keeping it in contact with the two pins. The sliding motion will change the center.

I'm sure that an argument could be constructed that is more mathematically rigorous. But the physical argument works OK for me.

Here's a picture of two ellipses, both with semi-axes (200,50) that both pass through your two given points. Obviously they have different centers. Using the "sliding" process I described, you can produce an infinite number of ellipses with this same size through these same two points. So, your problem is unsolvable.

The original question (which I answered above) was about the case where the orientation of the ellipse is not known. If the orientation is known/fixed, then the question has been answered already here.

• Assume that ellipse can passes via those points. – sindhu Sep 20 '13 at 6:26
• I did assume that the ellipse passes through the two given points. I still say that this does not fully determine the center. – bubba Sep 20 '13 at 8:02
• yes , it gives infinite number of ellipse. And also i have data is it clockwise or anti clock wise ?and rotational angle of that ellipse. i think using all this data can we able to find center of ellipse? – sindhu Sep 20 '13 at 8:28
• Yes, if you fix the rotational angle, the ellipse may be (essentially) unique. But that's not the question you asked. I answered the question you asked. If you want to ask a different one, go ahead, and maybe I'll answer that one, too. – bubba Sep 20 '13 at 8:43
• ya it has rotational angle and direction(clockwise and anticlockwise) also , could you please give the solution how to solve this?plz – sindhu Sep 20 '13 at 10:58