Skew lines
Lines not in the same plane
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Key Takeaways
- In three-dimensional geometry, skew lines are two lines that do not intersect and are not parallel.
- Two lines that both lie in the same plane must either cross each other or be parallel, so skew lines can exist only in three or more dimensions.
- General position If four points are chosen at random uniformly within a unit cube, they will almost surely define a pair of skew lines.
- However, the plane through the first three points forms a subset of measure zero of the cube, and the probability that the fourth point lies on this plane is zero.
- Similarly, in three-dimensional space a very small perturbation of any two parallel or intersecting lines will almost certainly turn them into skew lines.
In three-dimensional geometry, skew lines are two lines that do not intersect and are not parallel. A simple example of a pair of skew lines is the pair of lines through opposite edges of a regular tetrahedron. Two lines that both lie in the same plane must either cross each other or be parallel, so skew lines can exist only in three or more dimensions. Two lines are skew if and only if they are not coplanar.
General position
If four points are chosen at random uniformly within a unit cube, they will almost surely define a pair of skew lines. After the first three points have been chosen, the fourth point will define a non-skew line if, and only if, it is coplanar with the first three points. However, the plane through the first three points forms a subset of measure zero of the cube, and the probability that the fourth point lies on this plane is zero. If it does not, the lines defined by the points will be skew.
Similarly, in three-dimensional space a very small perturbation of any two parallel or intersecting lines will almost certainly turn them into skew lines. Therefore, any four points in general position always form skew lines.
In this sense, skew lines are the "usual" case, and parallel or intersecting lines are special cases.
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