How to solve the shortest distance from a point on a circle to a straight line using Python and NumPy?

Python and NumPy solve the minimum distance between a point on a circle and a straight line

This article explores how to use Python and NumPy libraries to calculate the minimum distance between a point on a circle in three-dimensional space and give the coordinates of that point. We will deal with the situation where the circle is not coplanar with the straight line.

Problem description

It is known that the center of the circle o = (0.3501, -0.0881, -4.8466) , the normal vector n = (0.4163, -0.8326, -0.3653) , the radius of the circle r = 1.34954 , and the two points a = (3.1932, -0.9005, 0.8082) and b = (1.9885, -0.9691, -0.8353) on the line. The goal is to find a point p on the circle, minimize the distance to the straight line ab , and calculate the coordinates of p .

Algorithm Principles

Since circles and straight lines may not be coplanar, the minimum distance is not the perpendicular distance from the center to the straight line. we need to:

1. Calculate the direction vector of a straight line: obtained by b - a .
2. Calculate the direction vector of a straight line: normalize it to obtain the unit direction vector.
3. Calculate the distance vector from the center of the circle to the straight line: find the vector from the center of the circle to a point on the straight line, and project it on the direction vector of the straight line, so as to obtain the distance from the center of the circle to the straight line.
4. Calculate the vector from the center of the circle to the vertical foot: Using the result of step 3, calculate the vector from the center of the circle to the vertical foot.
5. Calculate the projection of a vector from the center of the circle to the perpendicular foot of the line on a plane perpendicular to the line: This projection vector points to the point on the circle that is closest to the line.
6. Calculate the nearest point on the circle: Normalize the projection vector, multiply the radius of the circle, and then add it to the vertical foot of the straight line to get the coordinates of the nearest point on the circle.

Python code implementation

 import numpy as np

# Enter data o = np.array([0.3501, -0.0881, -4.8466])
n = np.array([0.4163, -0.8326, -0.3653])
r = 1.34954
a = np.array([3.1932, -0.9005, 0.8082])
b = np.array([1.9885, -0.9691, -0.8353])

# Calculate the straight line direction vector v = b - a
v = v / np.linalg.norm(v) # Normalization# Calculate vector oa = o - a

# Calculate the distance from the center of the circle to the straight line d = np.dot(oa, v)

# Calculate the vertical foot f = ad * v

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

# Calculate the vector projected onto a plane perpendicular to a straight line Proj = of - np.dot(of, v) * v

# Calculate the nearest point on the circle p = fr * proj / np.linalg.norm(proj)

print("The coordinates of the point closest to the line on the circle are:", p)

Code explanation

The code clearly implements the above algorithm steps. numpy library provides efficient vector operations and simplifies the calculation process. The final output is the coordinates of a point p on the circle, and the distance from this point to the line ab is the smallest.

This improved version is more concise and efficient, and provides a clearer explanation of the algorithm steps.

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