In
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, the perpendicular distance between two objects is the
distance
Distance is a numerical or occasionally qualitative measurement of how far apart objects, points, people, or ideas are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two co ...
from one to the other, measured along a
line that is
perpendicular
In geometry, two geometric objects are perpendicular if they intersect at right angles, i.e. at an angle of 90 degrees or π/2 radians. The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', � ...
to one or both.
The
distance from a point to a line is the distance to the nearest
point on that line. That is the point at which a
segment from it to the given point is perpendicular to the line.
Likewise, the distance from a point to a
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
is measured by a line segment that is perpendicular to a
tangent line
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...
to the curve at the nearest point on the curve.
The
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.
Other instances include:
*''
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'', for the perpendicular distance between two non-parallel lines in three-dimensional space
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 methods using perpendicular distance to measure the quality of a fit exist, as in
total least squares.
The concept of perpendicular distance may be generalized to
* orthogonal distance, between more abstract non-geometric
orthogonal
In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...
objects, as in
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrix (mathemat ...
(e.g.,
principal components analysis
Principal component analysis (PCA) is a Linear map, linear dimensionality reduction technique with applications in exploratory data analysis, visualization and Data Preprocessing, data preprocessing.
The data is linear map, linearly transformed ...
);
* normal distance, involving a
surface normal
In geometry, a normal is an object (e.g. a line, ray, or vector) that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the infinite straight line perpendicular to the tangent line to the ...
, between an arbitrary point and its
foot
The foot (: feet) is an anatomical structure found in many vertebrates. It is the terminal portion of a limb which bears weight and allows locomotion. In many animals with feet, the foot is an organ at the terminal part of the leg made up o ...
on the surface. It can be used for
surface fitting and for defining
offset surfaces.
See also
*
Distance between sets
*
Hypercycle (geometry)
*
Moment of inertia
*
Signed distance
References
{{reflist
Orthogonality
Distance