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

In summary, to find the centre and radius of a circle in 3D with three given points, one can find the plane through the points, find the plane perpendicular bisector of two pairs of points, and find the intersection of these three planes to determine the centre of the circle.
  • #1
ukmj
2
0
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)...
 
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  • #2
Do you mean a sphere? Are these arbitrary points anywhere in the sphere or on the surface?
 
  • #3
Here is the equation of the points {x_i,y_i} for i = 1,2,3 in terms of the variables x and y:

[tex]
\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
[/tex]
 
  • #4
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.
 
  • #5
Borek you are absolutely right. I just wanted to make sure.
 
  • #6
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.
 

1. How do you find the center of a circle given 3 points in 3D?

To find the center of a circle in 3D, you can use the method of solving equations for the center coordinates. This involves setting up and solving three equations using the coordinates of the given points. Once the equations are solved, the resulting values will be the coordinates of the center of the circle.

2. What is the equation for finding the radius of a circle in 3D?

The equation for finding the radius of a circle in 3D is similar to the equation in 2D, which is r = √(x² + y²). However, in 3D, the equation becomes r = √(x² + y² + z²), where z represents the third coordinate of the center of the circle.

3. Is it possible to have more than one center and radius for a circle in 3D?

No, it is not possible to have more than one center and radius for a circle in 3D. A circle can only have one center point and one radius, regardless of the number of points given.

4. Can the center of a circle be located outside of the given points in 3D?

Yes, it is possible for the center of a circle to be located outside of the given points in 3D. This can happen if the given points do not lie on the same plane, and the circle is tilted or oblique to that plane.

5. Can the radius of a circle in 3D be negative?

No, the radius of a circle in 3D cannot be negative. The radius is a measure of the distance from the center of the circle to any point on its circumference, and distance cannot be negative. Thus, the radius will always be a positive value.

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