# Source code for pyleecan.Functions.Geometry.circle_from_3_points

```from numpy import sqrt

[docs]def circle_from_3_points(Z1, Z2, Z3):
"""Return the center and radius of a circle defined by 3 points

Parameters:
-----------
Z1 : complex
First point coordinates
Z2 : complex
Second point coordinates
Z3 : complex
Third point coordinates

Returns:
--------
R : float
Zc : complex
Coordinate of the circle center
"""

if Z1.imag == Z2.imag and Z1.imag == Z3.imag:
raise Exception("Error: The 3 points are aligned !")
if Z1.real == Z2.real and Z1.real == Z3.real:
raise Exception("Error: The 3 points are aligned !")

# Avoid div by 0
if Z3.imag == Z1.imag:
# Change Z1 and Z2
tmp = Z2
Z2 = Z1
Z1 = tmp

X1 = Z1.real
Y1 = Z1.imag

X2 = Z2.real
Y2 = Z2.imag

X3 = Z3.real
Y3 = Z3.imag

# All three points are on the circle :
# (X1-Xc)² + (Y1-Yc)² - R² =0  (Eq1)
# (X2-Xc)² + (Y2-Yc)² - R² =0  (Eq2)
# (X3-Xc)² + (Y3-Yc)² - R² =0  (Eq3)

# (Eq1-Eq2)
# (X1² -2*X1*Xc + Xc² + Y1² -2*Y1*Yc + Yc² - R²) - (X2² -2*X2*Xc + Xc² + Y2² -2*Y2*Yc + Yc² - R²) = 0
# X1² - X2² + 2*(X2-X1)*Xc + Y1² - Y2² + 2*(Y2-Y1)*Yc = 0

# (Eq1-Eq3)
# X1² - X3² + 2*(X3-X1)*Xc + Y1² - Y3² + 2*(Y3-Y1)*Yc = 0
# Yc = (-X1² + X3² - 2*(X3-X1)*Xc - Y1² + Y3²)/2*(Y3-Y1)

# Yc into (Eq1-Eq2)
# X1² - X2² + 2*(X2-X1)*Xc + Y1² - Y2² + 2*(Y2-Y1)*(-X1² + X3² - 2*(X3-X1)*Xc - Y1² + Y3²)/2*(Y3-Y1)= 0
# A = (Y2-Y1)/(Y3-Y1)
# -X1² + X2² - Y1² + Y2² = 2*(X2-X1)*Xc  + A(-X1² + X3² - 2*(X3-X1)*Xc - Y1² + Y3²)
# -X1² + X2² - Y1² + Y2² + A(X1² -X3² +Y1² -Y3²) = 2*(X2-X1)*Xc - 2*A*(X3-X1)*Xc
A = (Y2 - Y1) / (Y3 - Y1)
Xc = (
-(X1 ** 2)
+ X2 ** 2
- Y1 ** 2
+ Y2 ** 2
+ A * (X1 ** 2 - X3 ** 2 + Y1 ** 2 - Y3 ** 2)
) / (2 * (X2 - X1) - 2 * A * (X3 - X1))

# Eq1-Eq3
Yc = (-(X1 ** 2) + X3 ** 2 - 2 * (X3 - X1) * Xc - Y1 ** 2 + Y3 ** 2) / (
2 * (Y3 - Y1)
)

# Eq1
R = sqrt((X1 - Xc) ** 2 + (Y1 - Yc) ** 2)

return (R, Xc + 1j * Yc)
```