Lines, Skew

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For geometric figures in a plane, two straight lines must either be parallel to one another or must intersect at one point. Skew lines are non-parallel and do not intersect. Skew lines must therefore lie in separate planes from one another. Since skew lines are defined in terms of distinct planes, discussing such lines leads directly to the branch of mathematics called solid geometry.

Solid geometry is the branch of Euclidian geometry (named for Euclid, c. 325 B.C.E.–265 B.C.E.) that examines the relative positions, sizes, shapes, and other aspects of geometric figures that are not in a single plane. Whereas plane geometry is about two-dimensional space described by parameters such as length and width, solid geometry concerns itself with three-dimensional space.

One example of a three-dimensional object is a cube, which has height, length, and width. Another familiar example of a solid (three-dimensional) figure is the pyramid. Figures like these can be used to illustrate skew lines.

The edges of the pyramid above form skew lines. Each of the four faces of the pyramid (as well as its bottom) define a unique plane. Line segments AB, AC, and BC, for instance, define a unique plane, and each plane constitutes one of the four faces of the pyramid. None of the three line segments (AB, AC, BC) can be skew lines relative to one another because they all lie in the same plane. Recall that lines in the same plane either intersect (as do AB, AC, and BC) or are parallel to one another.

There are, however, several pairs of line segments on the pyramid that form skew lines. Line segments AB and CD form a pair of such lines. These two segments are skew to one another because they are neither parallel nor intersecting. Even if the line segments are extended into infinite lines, they still remain skew. Though some of the line segments are hidden from view in this picture, one can envision several other pairs of skew lines formed by the edges of the pyramid.

A cube possesses many combinations of parallel, perpendicular, and skew line segments. The cubes above illustrate different line pairs: parallel, perpendicular, and skew. (The skew segments have been extended to indicate infinite lines.) The shortest distance between the two skew lines is the length of the dashed line segment, which is perpendicular to both of the indicated skew lines. Any other distance measured between the two skew lines will be longer than the dashed line segment.

The spatial orientation of any two skew lines can be described by two quantities: the closest or perpendicular distance between the two lines, and the angle between them. The illustration below shows these two quantities of distance and angle.

The two skew lines are A and B. Line B lies along the intersection of the two shaded planes. The dotted line segment joining lines A and B is perpendicular to both, and is the shortest distance between any two points on lines A and B.

To find the angle between A and B, a line, A_p, is constructed, which is parallel to A and which also intersects line B. The angle between lines A_p and B is designated by the Greek letter alpha (α). Because A_p and A are parallel lines, the angle α is also the angle between lines A and B. Therefore, by determining the distance (dotted line segment) and angle (α) between skew lines A and B, the relative position between the two is uniquely determined.

Ringenberg, Lawrence A., and Presser, Richard S. Geometry. New York: Benziger, Inc. (in association with John Wiley and Sons, Inc.), 1971.

Several of the lines formed by elevated roadways at a highway interchange skew to one another; that is, they neither intersect nor are parallel. However, lampposts and overpass columns are vertical structures and, therefore, all of them must be parallel to one another.

Lines, Parallel and Perpendicular.

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