Line-line intersection in the context of "Skew lines"

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⭐ Core Definition: 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 .

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👉 Line-line intersection in the context of 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.

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Line-line intersection in the context of Intersection (aviation)

In aviation, an intersection is a virtual navigational fix that helps aircraft maintain their flight plan. It is usually defined as the intersection (in the geometrical sense) of two VOR (VHF Omnidirectional Range) radials. They are usually identified as major airway intersections where aircraft, operating under instrument flight rules, often change direction of flight while en route. According to the Federal Aviation Regulations, some intersections are designated as mandatory reporting points for pilots who are not in radar contact with air traffic control.

Intersections also play an important role in departure and approach procedures. All intersections have an alphabetical or alphanumeric designation. Near major airports, the intersection designation code typically consists of three letters followed by the runway number. Most other intersection designations consist of five-letter combinations that are either pronounceable or chosen for their mnemonic value, since either air traffic control or the flight plan may require the pilot to announce the designation. In the terminal procedure or approach plate example to the right, note that two of the intersections are called DONUT and KOFFE. Many intersections are named because of local points of interest. In the case of Carroll County Airport, there is a popular diner located on the field to which many pilots fly.

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Line-line intersection 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.

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Line-line intersection in the context of Concurrent lines

In geometry, lines in a plane or higher-dimensional space are concurrent if they intersect at a single point.

The set of all lines through a point is called a pencil, and their common intersection is called the vertex of the pencil.

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LINKED FROM (PARENTS)
← Intersection (aviation) ← Coplanar ← Coplanarity ← Skew lines ← Concurrent lines
LINKS TO (CHILDREN)
Euclidean geometry → Empty set → Point (geometry) → Line (geometry) → Intersection (Euclidean geometry) → Computer graphics → Motion planning → Collision detection → Euclidean space → Coplanarity → Skew lines → Infinity → Parallel (geometry) → Singleton set →