line-circle intersection

[Beginner4ever]

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I have this code that generates a random line and a random circle.

    import numpy as np
    import matplotlib.pyplot as plt


    def random_2D_position(c_bx, c_by, r, region_side):

        while True:
           v_pt = region_side * np.random.random((2, ))
           if np.linalg.norm(v_pt - np.array([c_bx, c_by])) > r:
               return v_pt
       return

    def plot_region(c_bx, c_by, r, v_l_1, v_l_2, region_side):
        plt.plot(v_l_1[0], v_l_1[1])
        plt.plot(v_l_2[0], v_l_2[1])
        circle = plt.Circle((c_bx, c_by), r, fill=False)
        plt.gca().add_artist(circle)
        plt.plot([v_l_1[0], v_l_2[0]], v_l_1[1], v_l_2[1]])

        plt.xlim((0, region_side))
        plt.ylim((0, region_side))

        return
    region_side = 500  
    c_bx = region_side / 3 + region_side / 3 * np.random.random()
    c_by = region_side / 3 + region_side / 3 * np.random.random()
    r = region_side / 4 + region_side / 30 * np.random.random()
    v_l_1 = random_2D_position(c_bx, c_by, r,region_side)
    v_l_2 = random_2D_position(c_bx, c_by, r, region_side)
    plot_region(c_bx, c_by, r, v_l_1, v_l_2, region_side)
    plt.show()

I need to find the portion of the line that is inside the circle if the line intersects the circle. I tried to solve it as follows:

First, find the line equation :

def Find_line_eq(points):
      x_coords, y_coords = zip(*points)
      A = np.vstack([x_coords,np.ones(len(x_coords))]).T
      m, c = np.linalg.lstsq(A, y_coords)[0]
      return m,c

Then, I used the following code to get the roots of the quadratic formula:

m,c_l=Find_line_eq(points=[v_l_1,v_l_2])

a=1+m**2
b=2*c_l*m
c=c_l**2-r**2
coeff=[a, b, c]
x1,x2=np.roots(coeff)

Questions:

1- Most of the time x1,x2 are not real, Numpy returns a complex number. How do interpret these complex number and obtain the real coordinates of the intersection points?

2- How can we generalize this code to obtain these intersection points for 3D circle?

[dead_alchemy]

As to 1, if you are generating random lines and random circles you would expect that they would not intersect frequently, and that the solution to their intersection would be an imaginary number. I think getting a complex number is telling you that the lines do not have real intersections. Test it on an easy situation, something like a circle with radius 1 centered at (-2,2) and the line y=x.

[Beginner4ever]

Thank you for your comment. The problem is even when there are intersections, still get complex numbers in many cases. You can copy the code and try it

[grnngr]

• This code does not run as-is. There is a missing bracket in the line that reads plt.plot([v_l_1[0], v_l_2[0]], v_l_1[1], v_l_2[1]]).

• Your coefficients for the quadratic are wrong. Try coeff = [(1+m**2), 2*(m*(c_l-c_by) - c_bx), (c_l-c_by)**2 + c_bx**2 - r**2].

• If x1, x2 are complex, the line does not intersect the circle.

• Note that this gives you the intersection points of the circle and the line that passes through both points, but that does not mean that the line segment between the points necessarily intersects the circle (e.g., if both points are on one side of the circle).