6.1.4 Shortest Distances - Lines

Introduction to Shortest Distances

Shortest distance calculations are essential in 3D geometry for finding:

• The perpendicular distance between two lines.

Perpendicular Distance Between Two Lines

Two skew lines in 3D space can have a minimum distance between them. Let:

$$\mathbf{r}_1 = \mathbf{a}_1 + \lambda \mathbf{b}_1, \quad \mathbf{r}_2 = \mathbf{a}_2 + \mu \mathbf{b}_2$$

Perpendicular Distance Formula:

$$d = \frac{|(\mathbf{a}_2 - \mathbf{a}_1) \times (\mathbf{b}_1 \times \mathbf{b}_2)|}{|\mathbf{b}_1 \times \mathbf{b}_2|}$$

Here:

• $\mathbf{b}_1$ and $\mathbf{b}_2$ are the direction vectors of the lines.
• $\mathbf{a}_1$ and $\mathbf{a}_2$ are points on the lines.

Perpendicular Distance from a Point to a Line

Let the line be:

$$\mathbf{r} = \mathbf{a} + \lambda \mathbf{b}$$

and the point be $\mathbf{p}$.

Perpendicular Distance Formula:

$$d = \frac{|(\mathbf{p} - \mathbf{a}) \times \mathbf{b}|}{|\mathbf{b}|}$$

Worked Example

Note Summary