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Do determine a circle, you would need at least three coordinates. How many are necessary to determine a sphere?

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Hint: why you need three points to determine a circle. In other words, given three points, how do you determine the circle? –  Patrick Li Nov 4 '12 at 17:16
    
I could take the bisections of two of the three points twice, and the intersection of those two lines would be the centre of the circle. The radius then is just the distance between the centre and one of the points. –  timvermeulen Nov 4 '12 at 17:20
    
So, to determine a sphere, I assume you would need three points again: taking the three bisections, see where those intersect and there's the centre of the circle. However, it doesn't seem logical to me a circle needs the same amount of coordinates to determine as a sphere. –  timvermeulen Nov 4 '12 at 17:24
    
In how many dimensions is your sphere embedded? –  Peter Taylor Nov 4 '12 at 17:26
    
@timjver Note that the bisection of two points in three dimension is a plane, not a line. –  Patrick Li Nov 4 '12 at 18:44

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I believe the answer is 4 (and hopefully someone will correct me if I'm wrong...). Here's why. Any 3 points are coplanar. Consider 3 points on the plane A, B, and C. One can connect each pair of points with a line segment to create a triangle, then circumscribe a circle. The center of said circle is equidistant to all 3 points. Consider a line perpendicular to this plane. It is easy to prove that any point on this line is also equidistant from A, B, and C. Therefore, it must take more than 3 points to determine a sphere.

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You're absolutely right. I at first thought three points are sufficient, because in order to find a point equidistant to the three given point, I would just have to see where the three planes bisecting the three points meet. (so, the bisection of A and B, of B and C, and of A and C.) However, these three planes don't meet in a single point, but in a line. That's why we need the fourth point. –  timvermeulen May 20 '13 at 18:19

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