如何在3D中找到圆上两个点之间的点？

Question

I have a circle in 3D (defined through a center, radius, and normal to the plane where the circle lays in), and two points, p1 and p2, that lay on the circle. How can I find the two points, p3 and p4, that lay exactly (ie equal distance) between the two given points, but on the circle?

My approach is shown below. For some points, like in the given example, I'm getting the expected results (p3 = [45.87, 38.43, 38.97] and p4 = [54.13, 41.57, 21.03]), but this is not always the case and might have to do with the signs/directions and/or when p1 and p2 are in line with the center. I still haven't figured out what the problem is. n1 is not used in my calculation.

import numpy as np

p1 = np.array([42.96, 46.23, 33.42])
p2 = np.array([52.91, 32.21, 35.55])
center = np.array([50, 40, 30])
radius = 10
n1 = np.array([0.64233475, 0.53814965, 0.54571148])

def FindIntermPtsOnCircle(p1, p2, center, radius):

    # Calculate the direction vectors of the given points from the center
    v1 = p1-center
    v1 = v1/np.linalg.norm(v1)
    v2 = p2-center
    v2 = v2/np.linalg.norm(v2)

    # Use bisecting line to find direction of the points of interest
    va1 = v1+v2
    va1 = va1/np.linalg.norm(va1)
    va2 = -va1

    # Multiply direction with radius to find relative distance and add to center
    p3 = va1*radius + center
    p4 = va2*radius + center

    return p3, p4

Answer 1

the system is under determined when p1,2 are exactly opposite, then your v1,2 add to 0 so there is no "bisector" va1 from your equation

you can use a conditional ( check cross product v1,2 == 0 or just |va1 + va2| == 0 ) and then use cross product with the normal to generate a "patch" for that case or use the normal, cross product to construct a basis for the circle's plane from the start to make a formula that doesn't need the conditional

    def FindIntermPtsOnCircle(p1, p2, center, normalv):

    # Calculate the vectors of the given points from the center
    v1 = p1 - center
    v2 = p2 - center
    # Use bisecting line to find direction of the points of interest
    # check for special case of v1,2 colinear, opposite sides, adding to Zero    
    va1 = v1 + v2
    if  np.vdot(va1, va1) == 0:     # if v1,2 colinear, opposite use
        va1 = np.cross(normalv, v1) # norlmalv to calculate
                                    # a "bisector" in circle plane
    else:
        va1 = va1 * (np.vdot(v1, v1) / np.vdot(va1, va1)**(1/2))

    # reverse direction of va1 to get other bisector: va2
    va2 = -va1

    # add to center
    p3 = va1 + center
    p4 = va2 + center

    return p3, p4