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.
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👉 Skew lines in the context of Parallel lines
In geometry, parallel lines are coplanar infinite straight lines that do not intersect at any point. Parallel planes are infinite flat planes in the same three-dimensional space that never meet. In three-dimensional Euclidean space, a line and a plane that do not share a point are also said to be parallel. However, two noncoplanar lines are called skew lines. Line segments and Euclidean vectors are parallel if they have the same direction or opposite direction (not necessarily the same length).
Parallel lines are the subject of Euclid's parallel postulate. Parallelism is primarily a property of affine geometries and Euclidean geometry is a special instance of this type of geometry.In some other geometries, such as hyperbolic geometry, lines can have analogous properties that are referred to as parallelism.The concept can also be generalized to non-straight parallel curves and non-flat parallel surfaces, which keep a fixed minimum distance and do not touch each other or intersect.
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Skew lines in the context of Parallel projection
In three-dimensional geometry, a parallel projection (or axonometric projection) is a projection of an object in three-dimensional space onto a fixed plane, known as the projection plane or image plane, where the rays, known as lines of sight or projection lines, are parallel to each other. It is a basic tool in descriptive geometry. The projection is called orthographic if the rays are perpendicular (orthogonal) to the image plane, and oblique or skew if they are not.
Skew lines in the context of Line-line intersection
In Euclidean geometry, the intersection of a line and a line can be the empty set, a single point, or a line (if they coincide). Distinguishing these cases and finding the intersection have uses, for example, in computer graphics, motion planning, and collision detection.
In a Euclidean space, if two lines are not coplanar, they have no point of intersection and are called skew lines. If they are coplanar, however, there are three possibilities: if they coincide (are the same line), they have all of their infinitely many points in common; if they are distinct but have the same direction, they are said to be parallel and have no points in common; otherwise, they have a single point of intersection, denoted as singleton set, for instance .
Skew lines in the context of Coplanar
In geometry, a set of points in space are coplanar if there exists a geometric plane that contains them all. For example, three points are always coplanar, and if the points are distinct and non-collinear, the plane they determine is unique. However, a set of four or more distinct points will, in general, not lie in a single plane.
Two lines in three-dimensional space are coplanar if there is a plane that includes them both. This occurs if the lines are parallel, or if they intersect each other. Two lines that are not coplanar are called skew lines.