Planar projections are the subset of
3D graphical projections constructed by
linearly mapping points in
three-dimensional space
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informa ...
to points on a
two-dimensional
In mathematics, a plane is a Euclidean ( flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as ...
projection plane. The projected point on the
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
is chosen such that it is
collinear
In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned o ...
with the corresponding three-dimensional point and the centre of projection. The lines connecting these points are commonly referred to as projectors.
The centre of projection can be thought of as the location of the observer, while the plane of projection is the surface on which the two dimensional projected
image of the scene is recorded or from which it is viewed (e.g., photographic negative, photographic print, computer monitor). When the centre of projection is at a finite distance from the projection plane, a
perspective projection is obtained. When the centre of projection is at infinity, all the projectors are parallel, and the corresponding subset of planar projections are referred to as parallel projections.
Mathematical formulation
Mathematically, planar projections are
linear transformations
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
acting on a point in three-dimensional space
to give a point
on the projection plane. These transformations consist of various
compositions of the five transformations:
orthographic projection,
rotation,
shear
Shear may refer to:
Textile production
*Animal shearing, the collection of wool from various species
**Sheep shearing
*The removal of nap during wool cloth production
Science and technology Engineering
*Shear strength (soil), the shear strength ...
,
translation
Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
and
perspective.
Map uses
It is also used in maps to show the planet Earth and other planets or objects in space. This is good for maps of close-up areas.
Graphical projections
{{geometry-stub