Find Arc/Circle equation given three points in space (3D)

Question

Given 3 points in space (3D): A = (x1, y1, z1), B = (x2, y2, z2) C = (x3, y3, z3); then how to find the center and radius of the circle (arc) that passes through these three points, i.e. find circle equation? Using Python and Numpy here is my initial code ``` import numpy as np A = np.array([x1, y1, z1]) B = np.array([x2, y2, z2]) C = np.array([x3, y3, z3]) #Find vectors connecting the three points and the length of each vector AB = B - A BC = C - B AC = C - A # Triangle Lengths a = np.linalg.norm(AB) b = np.linalg.norm(BC) c = np.linalg.norm(AC) ``` From the Circumradius definition, the radius can be found using: ``` R = (a * b * c) / np.sqrt(2.0 * a**2 * b**2 + 2.0 * b**2 * c**2 + 2.0 * c**2 * a**2 - a**4 - b**4 - c**4) ``` However, I am having problems finding the Cartesian coordinates of the center. One possible solution is to use the "Barycentric Coordinates" of the triangle points to find the trilinear coordinates of the circumcenter (Circumcenter). First (using this source) we…

Accepted Answer

There are two issues with your code.

The first is in the naming convention. For all the formulas you are using to hold, the side of length `a` has to be the one opposite the point `A`, and similarly for `b` and `B` and `c` and `C`. You can solve that by computing them as:

```
a = np.linalg.norm(C - B)
b = np.linalg.norm(C - A)
c = np.linalg.norm(B - A)
```

The second has to do with the note in your source for the barycentric coordinates of the circumcenter: not necessarily homogeneous. That is, they need not be normalized in any way, and the formula you are using to compute the Cartesian coordinates from the barycentric ones is only valid when they add up to one.

Fortunately, you only need to divide the resulting Cartesian coordinates by `b1 + b2 + b3` to get the result you are after. Streamlining a little bit your code for efficiency, I get the results you expect:

```
>>> A = np.array([2.0, 1.5, 0.0])
>>> B = np.array([6.0, 4.5, 0.0])
>>> C = np.array([11.75, 6.25, 0.0])
>>> a = np.linalg.norm(C - B)
>>> b = np.linalg.norm(C - A)
>>> c = np.linalg.norm(B - A)
>>> s = (a + b + c) / 2
>>> R = a*b*c / 4 / np.sqrt(s * (s - a) * (s - b) * (s - c))
>>> b1 = a*a * (b*b + c*c - a*a)
>>> b2 = b*b * (a*a + c*c - b*b)
>>> b3 = c*c * (a*a + b*b - c*c)
>>> P = np.column_stack((A, B, C)).dot(np.hstack((b1, b2, b3)))
>>> P /= b1 + b2 + b3
>>> R
15.899002930062531
>>> P
array([ 13.42073171,  -9.56097561,   0.        ])
```

Find Arc/Circle equation given three points in space (3D)

Solution

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