Distance Between Two Lines in 3D: A full breakdown
The distance between two lines in 3D is a fundamental concept in three-dimensional geometry with wide-ranging applications in mathematics, engineering, computer graphics, and robotics. Whether you're designing a 3D model, navigating a robot, or solving a physics problem, understanding how to calculate the shortest distance between two lines in space is essential. This guide will walk you through the theory, formulas, and practical steps needed to compute this distance, ensuring you grasp both the mathematical principles and their real-world relevance.
Counterintuitive, but true Small thing, real impact..
Understanding the Concept
In three-dimensional space, two lines can either intersect, be parallel, or be skew (neither intersecting nor parallel). The distance between two lines refers to the shortest path connecting them, which is always along a line segment perpendicular to both. Here's a breakdown of the scenarios:
The official docs gloss over this. That's a mistake.
- Intersecting Lines: The distance is zero since they share a common point.
- Parallel Lines: The distance is constant and can be calculated using vector projections.
- Skew Lines: The distance is the length of the unique common perpendicular segment between them.
Mathematical Approach to Calculate Distance
Case 1: Skew Lines (Non-Parallel, Non-Intersecting)
For two skew lines, the distance is determined using the cross product and dot product of vectors. Let’s define the lines parametrically:
- Line 1: $\mathbf{r}_1 = \mathbf{a}_1 + t\mathbf{b}_1$
- Line 2: $\mathbf{r}_2 = \mathbf{a}_2 + s\mathbf{b}_2$
Where:
- $\mathbf{a}_1$ and $\mathbf{a}_2$ are position vectors of points on each line.
- $\mathbf{b}_1$ and $\mathbf{b}_2$ are direction vectors of the lines.
- $t$ and $s$ are scalar parameters.
The formula for the distance $d$ between these lines is:
$ d = \frac{|\mathbf{(\mathbf{a}_2 - \mathbf{a}_1}) \cdot (\mathbf{b}_1 \times \mathbf{b}_2)|}{|\mathbf{b}_1 \times \mathbf{b}_2|} $
Case 2: Parallel Lines
If the lines are parallel (i.e., $\mathbf{b}_1 \times \mathbf{b}_2 = \mathbf{0}$), the distance simplifies to:
$ d = \frac{|\mathbf{(\mathbf{a}_2 - \mathbf{a}_1}) \cdot \mathbf{n}|}{|\mathbf{n}|} $
Where $\mathbf{n}$ is the direction vector of the lines (since they share the same direction).
Step-by-Step Calculation
Let’s walk through an example to illustrate the process.
Example:
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