Center,Radius of a 3 Dimensional circle given 3 points

• ukmj
In summary, the equation for finding the center of a 3D circle given 3 points is (x,y,z) = (x1 + x2 + x3)/3, (y1 + y2 + y3)/3, (z1 + z2 + z3)/3, and the radius can be calculated by using the distance formula. The center of a 3D circle must lie within the triangle formed by the 3 given points, and if the points are collinear, there is no unique solution. It is possible for the 3D circle to have an infinite radius if all 3 points are the same.

ukmj

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

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ukmj said:
Say the points are (a,b,c),(a1,b1,c1),(a2,b2,c2).

Hi ukmj! Welcome to PF!

Show us what you've tried, and where you're having difficulty, and then we'll know how to help!

The center of a 3-dimensional circle can be determined by finding the point of intersection of the three perpendicular bisectors of the triangle formed by the three given points (a,b,c), (a1,b1,c1), and (a2,b2,c2). The radius of the circle can then be calculated using the distance formula between the center point and any of the three given points. This method ensures that the circle will pass through all three given points, making it a unique solution. It is important to note that in order for a 3-dimensional circle to exist, the three given points must not all lie on the same plane. Otherwise, the points would form a flat circle rather than a 3-dimensional one.

1. What is the equation for finding the center of a 3D circle given 3 points?

The center of a 3D circle can be found by using the formula:
(x,y,z) = (x1 + x2 + x3)/3, (y1 + y2 + y3)/3, (z1 + z2 + z3)/3
Where (x1,y1,z1), (x2,y2,z2), and (x3,y3,z3) are the coordinates of the 3 given points.

2. How do you calculate the radius of a 3D circle using 3 points?

The radius of a 3D circle can be calculated by finding the distance between any of the 3 points and the center of the circle. This can be done using the distance formula:
r = √[(x-xc)^2 + (y-yc)^2 + (z-zc)^2]
Where (xc,yc,zc) is the coordinates of the center point and (x,y,z) is the coordinates of any of the 3 given points.

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

No, the center of a 3D circle must lie within the triangle formed by the 3 given points. This is because the 3 points are the only points that lie on the circle's circumference, and the center must be equidistant from all points on the circumference.

4. What happens if the 3 given points are collinear?

If the 3 given points are collinear, it means they lie on the same line. In this case, there is no unique solution for the center and radius of the 3D circle. This is because any point on the line can be chosen as the center, and the radius would be equal to half the distance between any two of the given points.

5. Is it possible for the 3D circle to have an infinite radius?

Yes, if the 3 given points are all the same point, the 3D circle would have an infinite radius. This is because the distance between any point on the circle and the center point is always 0, resulting in a radius of infinity.