# Perpendicular distance

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{{short description|Distance from one geometric object to another along a line perpendicular to both}}

In [geometry](/source/geometry), the '''perpendicular distance''' between two objects is the [distance](/source/distance) from one to the other, measured along a [line](/source/Line_(geometry)) that is [perpendicular](/source/perpendicular) to one or both.

The [distance from a point to a line](/source/distance_from_a_point_to_a_line) is the distance to the nearest [point](/source/Point_(geometry)) on that line.<ref>{{cite journal |last1=Ballantine |first1=J. P. |last2=Jerbert |first2=A. R. |title=Distance from a Line, or Plane, to a Point |journal=The American Mathematical Monthly |date=April 1952 |volume=59 |issue=4 |pages=242 |doi=10.2307/2306514}}</ref><ref>{{cite journal |last1=Payne |first1=R. W. |title=164. The perpendicular distance from (x′, y′) to ax + by + c = 0 |journal=The Mathematical Gazette |date=May 1968 |volume=52 |issue=380 |pages=152–152 |doi=10.2307/3612683}}</ref> That is the point at which a [segment](/source/line_segment) from it to the given point is perpendicular to the line.

Likewise, the distance from a point to a [curve](/source/curve_(geometry)) is measured by a line segment that is perpendicular to a [tangent line](/source/tangent_line) to the curve at the nearest point on the curve.

The [distance from a point to a plane](/source/distance_from_a_point_to_a_plane) is measured as the length from the point along a segment that is perpendicular to the plane, meaning that it is perpendicular to all lines in the plane that pass through the nearest point in the plane to the given point.<ref>{{cite journal |last1=Bundrick |first1=Charles M. |last2=Sherry |first2=David L. |title=Distance From a Point to a Line and a Point to a Plane Via Synthetic Methods |journal=School Science and Mathematics |date=April 1978 |volume=78 |issue=4 |pages=304–306 |doi=10.1111/j.1949-8594.1978.tb09363.x}}</ref>

Other instances include:
*''[Point on plane closest to origin](/source/Point_on_plane_closest_to_origin)'', for the perpendicular distance from the origin to a plane in three-dimensional space
*''[Nearest distance between skew lines](/source/Nearest_distance_between_skew_lines)'', for the perpendicular distance between two non-parallel lines in three-dimensional space<ref>{{cite journal |last1=Clarke |first1=L. E. |title=2212. The shortest distance between two skew lines |journal=The Mathematical Gazette |date=May 1951 |volume=35 |issue=312 |pages=120–121 |doi=10.2307/3609345}}</ref><ref>{{cite journal |last1=Johar |first1=Syafiq |title=A New Proof for Distance Between Skew Lines |journal=The College Mathematics Journal |date=21 January 2026 |pages=1–8 |doi=10.1080/07468342.2025.2603159}}</ref>

[Perpendicular regression](/source/Perpendicular_regression) fits a line to data points by minimizing the sum of squared perpendicular distances from the data points to the line.
Other [geometric curve fitting](/source/geometric_curve_fitting) methods using perpendicular distance to measure the quality of a fit exist, as in [total least squares](/source/total_least_squares).

The concept of perpendicular distance may be generalized to
* '''orthogonal distance''', between more abstract non-geometric [orthogonal](/source/orthogonal_(mathematics)) objects, as in [linear algebra](/source/Projection_(linear_algebra)) (e.g., [principal components analysis](/source/principal_components_analysis));
* '''normal distance''', involving a [surface normal](/source/surface_normal), between an arbitrary point and its [foot](/source/Foot_of_a_normal) on the surface. It can be used for [surface fitting](/source/surface_fitting) and for defining [offset surface](/source/offset_surface)s.

==See also==
*[Distance between sets](/source/Distance_between_sets)
*[Hypercycle (geometry)](/source/Hypercycle_(geometry))
*[Moment of inertia](/source/Moment_of_inertia)
*[Signed distance](/source/Signed_distance)

==References==
{{reflist}}

Category:Orthogonality
Category:Distance

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