The ''n''-vector representation (also called geodetic normal or ellipsoid normal vector) is a three-parameter
non-singular
In the mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In ca ...
representation well-suited for replacing
geodetic coordinates
Geodetic coordinates are a type of curvilinear orthogonal coordinate system used in geodesy based on a ''reference ellipsoid''.
They include geodetic latitude (north/south) , ''longitude'' (east/west) , and ellipsoidal height (also known as geod ...
(
latitude
In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pol ...
and
longitude
Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek lette ...
) for
horizontal position representation
A position representation is the parameters used to express a position relative to a reference. When representing positions relative to the Earth, it is often most convenient to represent ''vertical position'' (height or depth) separately, and to ...
in mathematical calculations and computer algorithms.
Geometrically, the ''n''-vector for a given position on an
ellipsoid is the outward-pointing
unit vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction v ...
that is
normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
in that position to the ellipsoid. For representing horizontal positions on Earth, the ellipsoid is a
reference ellipsoid and the vector is
decomposed in an
Earth-centered Earth-fixed coordinate system. It behaves smoothly at all Earth positions, and it holds the mathematical
one-to-one property.
More in general, the concept can be applied to representing positions on the boundary of a strictly
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
bounded subset of ''k''-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
, provided that that boundary is a
differentiable manifold. In this general case, the ''n''-vector consists of ''k'' parameters.
General properties
A
normal vector to a strictly
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
surface can be used to uniquely define a surface position. ''n''-vector is an outward-pointing normal vector with
unit length
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (al ...
used as a position representation.
For most applications the surface is the
reference ellipsoid of the Earth, and thus ''n''-vector is used to represent a horizontal position. Hence, the angle between ''n''-vector and the equatorial plane corresponds to
geodetic latitude
Geodetic coordinates are a type of curvilinear orthogonal coordinate system used in geodesy based on a '' reference ellipsoid''.
They include geodetic latitude (north/south) , ''longitude'' (east/west) , and ellipsoidal height (also known as ...
, as shown in the figure.
A surface position has two
degrees of freedom, and thus two parameters are sufficient to represent any position on the surface. On the reference ellipsoid,
latitude
In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pol ...
and
longitude
Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek lette ...
are common parameters for this purpose, but like all ''two-parameter representations'', they have
singularities. This is similar to
orientation
Orientation may refer to:
Positioning in physical space
* Map orientation, the relationship between directions on a map and compass directions
* Orientation (housing), the position of a building with respect to the sun, a concept in building de ...
, which has three degrees of freedom, but all ''three-parameter representations'' have singularities.
In both cases the singularities are avoided by adding an extra parameter, i.e. to use ''n''-vector (three parameters) to
represent horizontal position and a unit
quaternion (four parameters) to
represent orientation.
''n''-vector is a
one-to-one representation, meaning that any surface position corresponds to one unique ''n''-vector, and any ''n''-vector corresponds to one unique surface position.
As a
Euclidean 3D vector, standard 3D
vector algebra In mathematics, vector algebra may mean:
* Linear algebra, specifically the basic algebraic operations of vector addition and scalar multiplication; see vector space.
* The algebraic operations in vector calculus, namely the specific additional stru ...
can be used for the position calculations, and this makes ''n''-vector well-suited for most horizontal position calculations.
Converting latitude/longitude to ''n''-vector
Based on the definition of the
ECEF
The Earth-centered, Earth-fixed coordinate system (acronym ECEF), also known as the geocentric coordinate system, is a cartesian spatial reference system that represents locations in the vicinity of the Earth (including its surface, interior, ...
coordinate system, called ''e'', it is clear that going from latitude/longitude to ''n''-vector, is achieved by:
: