Skew lines

In geometry, two lines are called skew if they are not cut or are parallel. This is possible only in three-dimensional space.

To prove that two lines g and h are skewed, it suffices to show that a direction vector of g, a direction vector of h and a displacement vector of a point on g to a point on h are linearly independent. Equivalent, one can show that there is no plane which contains both lines.

Calculating the distance between two skew lines

The uniquely determined route smallest length that connects two skew lines, called Gemeinlot or minimum transversal. It is the only circuit which is perpendicular to the two straight lines. The length of this route is the distance d between the two lines.

Consider the skew lines g and h with the support points A and B and the support vectors and the direction vectors and. Then the parameters of the linear equations are

Being considered and the three vectors must be linearly independent.

The normal vector on the two direction vectors and perpendicular, can be calculated over the cross product

The calculation of the distance is made possible by the orthogonal projection of the connection vector of the bases in the normal vector. To the normal vector is brought to the length 1. The distance between the two skew lines is then

Notation with determinants

Advertised are the two linear equations

The distance between the two skew lines with the help of the determinant is det then

Determination of Lotfußpunkte

The nadir point Fh is obtained by setting up an auxiliary level e. The point A lies on the auxiliary plane, and tension the Help Bene.

The intersection of the E and H results in the nadir point Fh:

Are obtained analogously Fg with the plane and its intersection with g:

In this method, the distance d must not be calculated.

Remark

• In Handbook of Mathematics of i.n. Bronstein and K. A. Semendjajew is " cruising " cited as a synonym for " askew ".