Orthographic projection (geometry)
Encyclopedia
In Euclidean geometry
, an orthographic projection is an orthogonal projection. In particular, in 3D it is an affine
, parallel
projection
of an object onto a perpendicular plane
.
A simple orthographic projection
onto the plane
z = 0 can be defined by the following matrix:
For each point v = (vx, vy, vz), the transformed point would be
Often, it is more useful to use homogeneous coordinates
, since translation
cannot be accomplished with a 3-by-3 matrix. The transformation above can be represented for homogeneous coordinates as
For each homogeneous vector v = (vx, vy, vz, 1), the transformed vector would be
In computer graphics
, one of the most common matrices used for orthographic projection
can be defined by a 6-tuple, (left, right, bottom, top, near, far), which defines the clipping
planes. These planes form a box with the minimum corner at (left, bottom, near) and the maximum corner at (right, top, far).
The box is translated so that its center is at the origin, then it is scaled to the unit cube which is defined by having a minimum corner at (-1,-1,-1) and a maximum corner at (1,1,1).
The orthographic transform can be given by the following matrix:
which can be given as a translation
followed by a scaling
of the form
The inversion of the Projection Matrix, which can be used as the Unprojection Matrix is defined:
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
, an orthographic projection is an orthogonal projection. In particular, in 3D it is an affine
Affine transformation
In geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...
, parallel
Parallel (geometry)
Parallelism is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. The assumed existence and properties of parallel lines are the basis of Euclid's parallel postulate. Two lines in a plane that do not...
projection
Projection (linear algebra)
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that P2 = P. It leaves its image unchanged....
of an object onto a perpendicular plane
Plane (mathematics)
In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...
.
A simple orthographic projection
Projection (linear algebra)
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that P2 = P. It leaves its image unchanged....
onto the plane
Plane (mathematics)
In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...
z = 0 can be defined by the following matrix:
For each point v = (vx, vy, vz), the transformed point would be
Often, it is more useful to use homogeneous coordinates
Homogeneous coordinates
In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry much as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points,...
, since translation
Translation (geometry)
In Euclidean geometry, a translation moves every point a constant distance in a specified direction. A translation can be described as a rigid motion, other rigid motions include rotations and reflections. A translation can also be interpreted as the addition of a constant vector to every point, or...
cannot be accomplished with a 3-by-3 matrix. The transformation above can be represented for homogeneous coordinates as
For each homogeneous vector v = (vx, vy, vz, 1), the transformed vector would be
In computer graphics
Computer graphics
Computer graphics are graphics created using computers and, more generally, the representation and manipulation of image data by a computer with help from specialized software and hardware....
, one of the most common matrices used for orthographic projection
Projection (linear algebra)
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that P2 = P. It leaves its image unchanged....
can be defined by a 6-tuple, (left, right, bottom, top, near, far), which defines the clipping
Clipping (computer graphics)
Any procedure which identifies that portion of a picture which is either inside or outside a picture is referred to as a clipping algorithm or clipping.The region against which an object is to be clipped is called clipping window.-Examples:...
planes. These planes form a box with the minimum corner at (left, bottom, near) and the maximum corner at (right, top, far).
The box is translated so that its center is at the origin, then it is scaled to the unit cube which is defined by having a minimum corner at (-1,-1,-1) and a maximum corner at (1,1,1).
The orthographic transform can be given by the following matrix:
which can be given as a translation
Translation (geometry)
In Euclidean geometry, a translation moves every point a constant distance in a specified direction. A translation can be described as a rigid motion, other rigid motions include rotations and reflections. A translation can also be interpreted as the addition of a constant vector to every point, or...
followed by a scaling
Scaling (geometry)
In Euclidean geometry, uniform scaling is a linear transformation that enlarges or shrinks objects by a scale factor that is the same in all directions. The result of uniform scaling is similar to the original...
of the form
The inversion of the Projection Matrix, which can be used as the Unprojection Matrix is defined: