# distance

Distance between two points is the length of the shortest connection from to${\ displaystyle d}$${\ displaystyle A}$${\ displaystyle B}$

The distance , also the distance or the distance between two points, is the length of the shortest connection between these points.

In Euclidean space this is the length of the straight line between the two points. The distance between two geometric objects is the length of the shortest connecting line between the two objects, i.e. the distance between the two closest points. If the points of two objects that are closest to one another are not considered, this is stated explicitly or results from the relationship, such as the distance between the geometric centers or the centers of gravity.

The metric is the part of math that deals with measuring distance.

The interval , the distance , the distance between two values ​​of a quantity or between two points in time is determined by forming the absolute value of their difference , that is, by subtracting them from one another and forming the absolute value from the result. The measured distance is independent of the selected reference point of the coordinate system , but not of its scaling ( see also scale factor ).

In observational astronomy , the apparent distance in the sky between two celestial objects is given as an angular distance .

The distance between two sets in Euclidean space (or more generally in a metric space ) can be defined using the Hausdorff metric .

## Euclidean distance

In the Cartesian coordinate system , the distance (Euclidean distance) between two points is calculated using the Pythagorean theorem :

The distance between two points in the plane
${\ displaystyle d (A, B) = {\ sqrt {\ sum _ {i = 1} ^ {n} (a_ {i} -b_ {i}) ^ {2}}} {\ text {, where} } A = (a_ {1}, \ dotsc, a_ {n}) \ in \ mathbb {R} ^ {n} {\ text {and}} B = (b_ {1}, \ dotsc, b_ {n} ) \ in \ mathbb {R} ^ {n}}$

For the level ( ): ${\ displaystyle A, B \ in \ mathbb {R} ^ {2}}$

${\ displaystyle d (A, B) = {\ sqrt {(a_ {1} -b_ {1}) ^ {2} + (a_ {2} -b_ {2}) ^ {2}}}}$

For three-dimensional space ( ): ${\ displaystyle A, B \ in \ mathbb {R} ^ {3}}$

${\ displaystyle d (A, B) = {\ sqrt {(a_ {1} -b_ {1}) ^ {2} + (a_ {2} -b_ {2}) ^ {2} + (a_ {3 } -b_ {3}) ^ {2}}}}$

The distance of a point from a straight line or a flat surface is the distance from the base point of the perpendicular fell on it ; that of a curved line is always a distance from one of its tangents .

Calculation options for the distances from points to straight lines or planes are listed in the formulary analytical geometry .

## Distance measurement on curved surfaces

On the spherical surface , the distance is determined along great circles and given in degrees or radians . To calculate the distance, see Orthodromes .

The geodetic line or the normal section is used on the earth's ellipsoid or other convex surfaces .

In geodesy and geosciences one speaks more of distance or distance, which is given metrically.