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The distance (or perpendicular distance) from a point to a line is the shortest distance from a fixed point to any point on a fixed infinite line in Euclidean geometry. It is the length of the line segment which joins the point to the line and is perpendicular to the line. The formula for calculating it can be derived and expressed in several ways.
The coordinate-independent definition of the square of the line element ds in an n-dimensional Riemannian or Pseudo Riemannian manifold (in physics usually a Lorentzian manifold) is the "square of the length" of an infinitesimal displacement [2] (in pseudo Riemannian manifolds possibly negative) whose square root should be used for computing curve length: = = (,) where g is the metric tensor ...
In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem , and therefore is occasionally called the Pythagorean distance .
Here, p is the (positive) length of the line segment perpendicular to the line and delimited by the origin and the line, and is the (oriented) angle from the x-axis to this segment. It may be useful to express the equation in terms of the angle α = φ + π / 2 {\displaystyle \alpha =\varphi +\pi /2} between the x -axis and the line.
A closed line segment includes both endpoints, while an open line segment excludes both endpoints; a half-open line segment includes exactly one of the endpoints. In geometry , a line segment is often denoted using an overline ( vinculum ) above the symbols for the two endpoints, such as in AB .
The first two values, Δ(1) and Δ(2), refer to the unit line segment and unit square respectively. For the three-dimensional case, the mean line segment length of a unit cube is also known as Robbins constant, named after David P. Robbins. This constant has a closed form, [6]
The arc length (length of a line segment) defined by a polar function is found by the integration over the curve r(φ). Let L denote this length along the curve starting from points A through to point B, where these points correspond to φ = a and φ = b such that 0 < b − a < 2π.
A line can lie in a given plane, intersect that plane in a unique point, or be parallel to the plane. In the last case, there will be lines in the plane that are parallel to the given line. A hyperplane is a subspace of one dimension less than the dimension of the full space. The hyperplanes of a three-dimensional space are the two-dimensional ...