Skew Lines - Skew Flats in Higher Dimensions

Skew Flats in Higher Dimensions

In higher dimensional space, a flat of dimension k is referred to as a k-flat. Thus, a line may also be called a 1-flat.

Generalizing the concept of skew lines to d-dimensional space, an i-flat and a j-flat may be skew if i + j < d. As with lines in 3-space, skew flats are those that are neither parallel nor intersect.

In affine d-space, two flats of any dimension may be parallel. However, in projective space, parallelism does not exist; two flats must either intersect or be skew. Let I be the set of points on an i-flat, and let J be the set of points on a j-flat. In projective d-space, if i + jd then the intersection of I and J must contain a (i+jd)-flat. (A 0-flat is a point.)

In either geometry, if I and J intersect at a k-flat, for k ≥ 0, then the points of IJ determine a (i+jk)-flat.

Read more about this topic:  Skew Lines

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