Two Line Cases

Topics: Geometry - Line - 3D Space

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Given two lines in ${\mathbb{R}}^3$, we can have four different cases:

1. Both lines are the same line
2. Both lines are parallel
3. Both lines intersect on a single point
4. Both lines never intersect, but one passes above the other

Determining the Case §

Given two lines in ${\mathbb{R}}^3$, $l_1=\{P+su:s\in \mathbb{R}\}$ and $l_2=\{Q+tv:t\in \mathbb{R}\}$, we can determine their specific case by calculating the following:

1. If $u\times v=0$, $(P-Q)\times u=0$, then $l_1=l_2$
2. If $u\times v=0$, $(P-Q)\times u\ne 0$, then $l_1$ and $l_2$ are parallel (but $l_1\ne l_2$)
3. If $u\times v=0$, $(P-Q)\cdot (u\times v)=0$, then $l_1$ and $l_2$ intersect on a single point
4. If $(P-Q)\cdot (u\times v)\ne 0$, then $l_1$ and $l_2$ never intersect, but one passes above the other (we say they cross each other)

We might find it convenient to check for the fourth condition first, then moving upwards, since we can reuse several of our calculations.