How to solve the shortest distance and coordinates of a point on a circle in three-dimensional space to a straight line?

Solve the minimum distance from a point on a circle in three-dimensional space and its coordinates

This article discusses how to calculate the minimum distance between a point on a circle in three-dimensional space and gives the coordinates of that point. The problem is described as follows: the center O of the circle (0.3501, -0.0881, -4.8466), the normal vector n (0.4163, -0.8326, -0.3653), the radius r=1.34954; the straight line AB is determined by point A (3.1932, -0.9005, 0.8082) and point B (1.9885, -0.9691, -0.8353). The goal is to find a point P on the circle so that it has a minimum distance to line AB.

Since the circle and the straight line may not be in the same plane, directly calculating the distance from the center of the circle to the straight line does not result in the minimum distance. The correct solution requires projection: project the circle onto a plane containing the straight line AB and is perpendicular to the circle normal vector n, and then solve the closest point between the projection of the circle and the straight line AB in this plane, and then calculate the point P on the circle based on the radius of the circle.

The following Python code uses the NumPy library to implement this calculation:

 import numpy as np

# Parameters of the circle o = np.array([0.3501, -0.0881, -4.8466]) # Center of the circle n = np.array([0.4163, -0.8326, -0.3653]) # Normal vector r = 1.34954 # Radius# Two points on the line a = np.array([3.1932, -0.9005, 0.8082])
b = np.array([1.9885, -0.9691, -0.8353])

# Calculate the direction vector of a straight line and normalize d = (b - a) / np.linalg.norm(b - a)

# Calculate the vertical foot of the center of the circle to the straight line t = np.dot(o - a, d)
f = at * d

# Calculate the vector from the center of the circle to the hang foot of the foot of = f - o

# Calculate the vector proj from the center of the circle to the projection point = np.dot(of, n) / np.linalg.norm(n)**2 * n

# Calculate projection point p_proj = o proj

# Calculate the distance from the projection point to the center of the circle dist_to_center = np.linalg.norm(p_proj - o)

# Calculate the distance between the projected point and the point on the circle dist_to_point = np.sqrt(r**2 - dist_to_center**2)

# Calculate the coordinates of point P on the circle# Use cross multiplication to calculate the vector perpendicular to n and d, and then scale to obtain the point v on the circle = np.cross(n, d)
p = p_proj (dist_to_point * v / np.linalg.norm(v))


print("Coordinates of point P on the circle:", p)

This code first calculates the projection point from the center of the circle to the straight line, and then uses the radius of the circle and the projection point to calculate the coordinates of the point P on the circle that is closest to the straight line. Through this method, we effectively solve the problem of the shortest distance from a point on a circle to a straight line in three-dimensional space.

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