如何判断3D空间平面内点相对于直线的左右位置

Alright, let's break this down step by step—since all points lie on the same 3D plane, we can frame this as a planar problem with a few 3D-specific checks. First, let's clarify what "left/right" means here: we're defining it from the perspective of standing at point A, facing directly towards point B—your left hand side is "left", right hand side is "right".

Step 1: Gather your vectors and plane normal #

First, let's list out what we need:

• The three points in 3D space: A(x₁,y₁,z₁), B(x₂,y₂,z₂), and the point to test C(x₃,y₃,z₃)
• A unit normal vector $\vec{n}$ for the shared plane. If you don't have this already, you can calculate it by taking the cross product of two non-collinear vectors in the plane (like $\vec{AB}$ and $\vec{AC}$) then normalizing it to length 1.

Next, compute two core vectors:

• Line direction vector: AB = B - A (subtract corresponding coordinates)
• Vector from A to test point: AC = C - A

Step 2: Cross product + dot product trick #

Here's the key calculation:

1. Compute the cross product of $\vec{AB}$ and $\vec{AC}$:
   $$\vec{cross} = \vec{AB} \times \vec{AC}$$
   This cross product will be a vector perpendicular to both $\vec{AB}$ and $\vec{AC}$—so it's parallel to the plane's normal (since we're working entirely within the plane).

2. Take the dot product of this cross product with your plane normal $\vec{n}$:
   $$dot = \vec{cross} \cdot \vec{n}$$

Step 3: Interpret the result #

The sign of the dot product tells you everything:

• If dot > 0: Point C is on the left of line AB (from the A→B perspective)
• If dot < 0: Point C is on the right of line AB
• If dot ≈ 0: Point C lies directly on line AB (we use a small threshold like 1e-9 instead of exact 0 to account for floating-point precision errors)

Quick code example (Python with NumPy) #

import numpy as np

def get_side_of_line(A, B, C, plane_normal):
    AB = np.array(B) - np.array(A)
    AC = np.array(C) - np.array(A)
    cross_product = np.cross(AB, AC)
    dot_result = np.dot(cross_product, plane_normal)
    
    # Handle floating point precision
    epsilon = 1e-9
    if dot_result > epsilon:
        return "Left"
    elif dot_result < -epsilon:
        return "Right"
    else:
        return "On the line"

Important notes to remember #

• Normal vector direction matters: If you flip the direction of $\vec{n}$ (use $-\vec{n}$ instead), your left/right results will flip too. Make sure the normal aligns with the perspective you want.
• All points must be on the same plane: If C isn't on the plane containing A and B, this method doesn't work—"left/right" has no meaning outside the plane in 3D space.
• Floating point precision: Always use a small epsilon value instead of checking for exact 0, since numerical calculations can have tiny errors.

内容的提问来源于stack exchange，提问作者AvishayDev