# Given 3 points how to find the centre and radius of a circle in 3 D?

1. May 19, 2008

### ukmj

Given 3 points how to find the centre and radius of a circle in 3 D??

Say the 3 points are (a,b,c),(a1.b1,c1) and (a2,b2,c2)...

2. May 19, 2008

### exk

Do you mean a sphere? Are these arbitrary points anywhere in the sphere or on the surface?

3. May 19, 2008

### Crosson

Here is the equation of the points {x_i,y_i} for i = 1,2,3 in terms of the variables x and y:

$$\left| \begin{array}{llll} x^2+y^2 & x & y & 1 \\ x_1^2+y_1^2 & x_1 & y_1 & 1 \\ x_2^2+y_2^2 & x_2 & y_2 & 1 \\ x_3^2+y_3^2 & x_3 & y_3 & 1 \end{array} \right|=0$$

4. May 19, 2008

### Staff: Mentor

Three points are not enough to define the sphere. Four will do the trick (unless they will not, but you need four at least). Points inside the sphere do not belong to the sphere. IMHO question is very precise.

5. May 19, 2008

### exk

Borek you are absolutely right. I just wanted to make sure.

6. May 19, 2008

### mathwonk

one might find the plane through the three points, then choosing two pairs of points, find the plane perpendicular bisector of the segments they determine.

these three planes should meet at the center of the circle.