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

## Main Question or Discussion Point

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)...

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

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

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

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

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

mathwonk
Homework Helper
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.