In mathematics, analytic geometry (also called
Cartesian geometry) describes any point in three-dimensional space by means of
three coordinates. Three coordinate axes are given, usually each perpendicular
to the other two at the origin, the point at which they cross. They are usually
labeled x, y, and z. Relative to these axes, the position of any point in
three-dimensional space is given by an ordered triple of real numbers, each
number giving the distance of that point from the origin measured along the given
axis, which is equal to the distance of that point from the plane determined by
the other two axes.
Other popular methods of describing the location of
a point in three-dimensional space include cylindrical coordinates and
spherical coordinates, though there is an infinite number of possible methods.
See Euclidean space.
Another mathematical way of viewing
three-dimensional space is found in linear algebra, where the idea of
independence is crucial. Space has three dimensions because the length of a box
is independent of its width or breadth. In the technical language of linear
algebra, space is three-dimensional because every point in space can be
described by a linear combination of three independent vectors. In this view,
space-time is four-dimensional because the location of a point in time is
independent of its location in space.
Three-dimensional space has a number of properties
that distinguish it from spaces of other dimension numbers. For example, at
least three dimensions are required to tie a knot in a piece of string. Many of
the laws of physics, such as the various inverse square laws, depend on
dimension three.
The understanding of three-dimensional space in
humans is thought to be learned during infancy using unconscious inference, and
is closely related to hand-eye coordination. The visual ability to perceive the
world in three dimensions is called depth perception. With the space the topologists locally model all other
3-manifolds.
In physics, our three-dimensional space is viewed as
embedded in four-dimensional spacetime, called Minkowski space (see special
relativity). The idea behind space-time is that time is hyperbolic-orthogonal
to each of the three spatial dimensions.
Orthogonality
In the familiar 3-dimensional space that we live in,
there are three pairs of cardinal directions: north/south (latitude), east/west
(longitude) and up/down (altitude). These pairs of directions are mutually
orthogonal: They are at right angles to each other. Movement along one axis
does not change the coordinate value of the other two axes. In mathematical
terms, they lie on three coordinate axes, usually labelled x, y, and z. The
z-buffer in computer graphics refers to this z-axis, representing depth in the
2-dimensional imagery displayed on the computer screen.
Coordinate systems
In mathematics, analytic geometry (also called
Cartesian geometry) describes every point in three-dimensional space by means
of three coordinates. Three coordinate axes are given, each perpendicular to
the other two at the origin, the point at which they cross. They are usually labeled
x, y, and z. Relative to these axes, the position of any point in
three-dimensional space is given by an ordered triple of real numbers, each
number giving the distance of that point from the origin measured along the
given axis, which is equal to the distance of that point from the plane
determined by the other two 2 axes.
Other popular methods of describing the location of
a point in three-dimensional space include cylindrical coordinates and
spherical coordinates, though there is an infinite number of possible methods.
See Euclidean space.