projection plane are parallel to each other. Thus, lines that are parallel in three-dimensional space remain parallel in the two-dimensional projected image. Parallel projection also corresponds to a
perspective projection with an infinite
focal length (the distance from a camera's
lens and
focal point), or "
zoom".
Images drawn in parallel projection rely upon the technique of axonometry ("to measure along axes"), as described in Pohlke's theorem. In general, the resulting image is oblique (the rays are not perpendicular to the image plane); but in special cases the result is orthographic (the rays are perpendicular to the image plane). Axonometry should not be confused with axonometric projection, as in English literature the latter usually refers only to a specific class of pictorials (see below).
Orthographic projection
The orthographic projection is derived from the principles of descriptive geometry and is a two-dimensional representation of a three-dimensional object. It is a parallel projection (the lines of projection are parallel both in reality and in the projection plane). It is the projection type of choice for working drawings.
If the normal of the viewing plane (the camera direction)
Images drawn in parallel projection rely upon the technique of axonometry ("to measure along axes"), as described in Pohlke's theorem. In general, the resulting image is oblique (the rays are not perpendicular to the image plane); but in special cases the result is orthographic (the rays are perpendicular to the image plane). Axonometry should not be confused with axonometric projection, as in English literature the latter usually refers only to a specific class of pictorials (see below).
The orthographic projection is derived from the principles of descriptive geometry and is a two-dimensional representation of a three-dimensional object. It is a parallel projection (the lines of projection are parallel both in reality and in the projection plane). It is the projection type of choice for working drawings.
If the normal of the viewing plane (the camera direction) is parallel to one of the primary axes (which is the x, y, or z axis), the mathematical transformation is as follows;
To project the 3D point
Oblique projection
In
oblique projections the parallel projection rays are not perpendicular to the viewing plane as with orthographic projection, but strike the projection plane at an angle other than ninety degrees. In both orthographic and oblique projection, parallel lines in space appear parallel on the projected image. Because of its simplicity,
oblique projection is used exclusively for pictorial purposes rather than for formal, working drawings. In an oblique pictorial
drawing, the displayed angles among the axes as well as the foreshortening factors (scale) are arbitrary. The distortion created thereby is usually attenuated by aligning one plane of the imaged object to be parallel with the plane of projection thereby creating a true shape, full-size image of the chosen plane. Special types of oblique projections are:
Cavalier projection (45°)
In cavalier projection (sometimes cavalier perspective or high view point) a point of the object is represented by three coordinates, x, y and z. On the drawing, it is represented by only two coordinates, x″ and y″. On the flat drawing, two axes, x and z on the figure, are perpendicular and the length on these axes are drawn with a 1:1 scale; it is thus similar to the dimetric projections, although it is not an axonometric projection, as the third axis, here y, is drawn in diagonal, making an arbitrary angle with the x″ axis, usually 30 or 45°. The length of the third axis is not scaled.
Cabinet projection
The term cabinet projection (sometimes cabinet perspective) stems from its use in illustrations by the furniture industry.[citation needed] Like cavalier perspective, one face of the projected object is parallel to the viewing plane, and the third axis is projected as going off in an angle (typically 30° or 45° or arctan(2) = 63.4°). Unlike cavalier projection, where the third axis keeps its length, with cabinet projection the length of the receding lines is cut in half.
Military projection
A variant of oblique projection is called military projection. In this case the horizontal sections are isometrically drawn so that the floor plans are not distorted and the verticals are drawn at an angle. The military projection is given by rotation in the xy-plane and a vertical translation an amount z.[4]
Limitations of parallel projectIn cavalier projection (sometimes cavalier perspective or high view point) a point of the object is represented by three coordinates, x, y and z. On the drawing, it is represented by only two coordinates, x″ and y″. On the flat drawing, two axes, x and z on the figure, are perpendicular and the length on these axes are drawn with a 1:1 scale; it is thus similar to the dimetric projections, although it is not an axonometric projection, as the third axis, here y, is drawn in diagonal, making an arbitrary angle with the x″ axis, usually 30 or 45°. The length of the third axis is not scaled.
Cabinet projection
The term cabinet projection (sometimes cabinet perspective)
The term cabinet projection (sometimes cabinet perspective) stems from its use in illustrations by the furniture industry.[citation needed] Like cavalier perspective, one face of the projected object is parallel to the viewing plane, and the third axis is projected as going off in an angle (typically 30° or 45° or arctan(2) = 63.4°). Unlike cavalier projection, where the third axis keeps its length, with cabinet projection the length of the receding lines is cut in half.
Military projectionA variant of oblique projection is called military projection. In this case the horizontal sections are isometrically drawn so that the floor plans are not distorted and the verticals are drawn at an angle. The military projection is given by rotation in the xy-plane and a vertical translation an amount z.[4]
Limitations of parallel projection
op art, as well as "impossible object" drawings.
M. C. Escher's
Waterfall (1961), while not strictly using parallel projection, is a well-known example, in which a channel of water seems to travel unaided along a downward path, only to then paradoxically fall once again as it returns to its source. The water thus appears to disobey the
law of conservation of energy. An extreme example is depicted in the film
Inception, where by a
forced perspective trick an immobile stairway changes its connectivity.
Perspective projection or perspective transformation is a linear projection where three dimensional objects are projected on a picture plane. This has the effect that distant objects appear smaller than nearer objects.
It also means that lines which are parallel in nature (that is, meet at the point at infinity) appear to intersect in the projected image, for example if railways are pictured with perspective projection, they appear to converge towards a single point, called the vanishing point. Photographic lenses and the human eye work in the same way, therefore perspective projection looks most realistic.[5] Perspective projection is usually categorized into one-point, two-point and three-point perspective, depending on the orientation of the projection plane towards the axes of the depicted object.[6]
Graphical projection methods rely on the duality between lines and points, whereby two straight lines determine a point while two points determine a straight line. The orthogonal projection of the eye point onto the picture plane is called the principal vanishing point (P.P. in the scheme on the left, from the Italian term punto principale, coined during the renaissance).[7]
Two relevant points of a line are:
- its intersection with the picture plane, and
- its vanishing point, found at the intersection between the parallel line from the eye point and the picture plane.
The principal vanishing point is the vanishing point of all horizontal lines perpendicular to the picture plane. The vanishing points of all horizontal lines lie on the horizon line. If, as is often the case, the picture plane is vertical, all vertical lines are drawn vertically, and have no finite vanishing point on the picture plane. Various graphical methods can be easily envisaged for projecting geometrical scenes. For example, lines traced from the eye point at 45° to the picture plane intersect the latter along a circle whose radius is the distance of the eye point from the plane, thus tracing that circle aids the construction of all the vanishing points of 45° lines; in particular, the intersection of that circle with the horizon line consists of two distance points. They are useful for drawing chessboard floors which, in turn, serve for locating the base of objects on the scene. In the perspective of a geometric solid on the right, after choosing the principal vanishing point —which determines the horizon line— the 45° vanishing point on the left side of the drawing completes the characterization of the (equally distant) point of view. Two lines are drawn from the orthogonal projection of each vertex, one at 45° and one at 90° to the picture plane. After intersecting the ground line, those lines go toward the distance point (for 45°) or the principal point (for 90°). Their new intersection locates the projection of the map. Natura
It also means that lines which are parallel in nature (that is, meet at the point at infinity) appear to intersect in the projected image, for example if railways are pictured with perspective projection, they appear to converge towards a single point, called the vanishing point. Photographic lenses and the human eye work in the same way, therefore perspective projection looks most realistic.[5] Perspective projection is usually categorized into one-point, two-point and three-point perspective, depending on the orientation of the projection plane towards the axes of the depicted object.[6]
Graphical projection methods rely on the duality between lines and points, whereby two straight lines determine a point while two points determine a straight line. The orthogonal projection of the eye point onto the picture plane is called the principal vanishing point (P.P. in the scheme on the left, from the Italian term punto principale, coined during the renaissance).[7]
Two relevant points of a line are:
The principal vanishing point is the vanishing point of all horizontal lines perpendicular to the picture plane. The vanishing points of all horizontal lines lie on the horizon line. If, as is often the case, the picture plane is vertical, all vertical lines are drawn vertically, and have no finite vanishing point on the picture plane. Various graphical methods can be easily envisaged for projecting geometrical scenes. For example, lines traced from the eye point at 45° to the picture plane intersect the latter along a circle whose radius is the distance of the eye point from the plane, thus tracing that circle aids the construction of all the vanishing points of 45° lines; in particular, the intersection of that circle with the horizon line consists of two distance points. They are useful for drawing chessboard floors which, in turn, serve for locating the base of objects on the scene. In the perspective of a geometric solid on the right, after choosing the principal vanishing point —which determines the horizon line— the 45° vanishing point on the left side of the drawing completes the characterization of the (equally distant) point of view. Two lines are drawn from the orthogonal projection of each vertex, one at 45° and one at 90° to the picture plane. After intersecting the ground line, those lines go toward the distance point (for 45°) or the principal point (for 90°). Their new intersection locates the projection of the map. Natural heights are measured above the ground line and then projected in the same way until they meet the vertical from the map.
While orthographic projection ignores perspective to allow accurate measurements, perspective projection shows distant objects as smaller to provide additional realism.
While orthographic projection ignores perspective to allow accurate measurements, perspective projection shows distant objects as smaller to provide additional realism.
The perspective projection requires a more involved definition as compared to orthographic projections. A conceptual aid to understanding the mechanics of this projection is to imagine the 2D projection as though the object(s) are being viewed through a camera viewfinder. The camera's position, orientation, and field of view control the behavior of the projection transformation. The following variables are defined to describe this transformation:
-
Most conventions use positive z values (the plane being in front of the pinhole), however negative z values are physically more correct, but the image will be inverted both horizontally and vertically.
Which results in:
-
b
x
,
y
{\displaystyle \mathbf {b} _{x,y}}
- the 2D projection of
a
.
{\displaystyle \mathbf {a} .}

When
c
x
,
y
,
z
=
⟨
0
,
0
,
0
⟩
,
{\displaystyle \mathbf {c} _{x,y,z}=\langle 0,0,0\rangle ,}
and
θ
x
,
y
,
z
=
⟨
0
,
0
,
0
⟩
,
{\displaystyle \mathbf {\theta } _{x,y,z}=\langle 0,0,0\rangle ,}
the 3D vector When
c
x
,
y
,
z
=
⟨
0
,
0
,
0
⟩
,
{\displaystyle \mathbf {c} _{x,y,z}=\langle 0,0,0\rangle ,}
and
θ
x
,
y
,
z
=
⟨
0
,
0
,
0
⟩
,
{\displaystyle \mathbf {\theta } _{x,y,z}=\langle 0,0,0\rangle ,}
the 3D vector
⟨
1
,
2
,
0
⟩
{\displaystyle \langle 1,2,0\rangle }
is projected to the 2D vector
⟨
1
,
2
⟩
{\displaystyle \langle 1,2\rangle }
.
Otherwise, to compute
b
x
,
y
{\displaystyle \mathbf {b} _{x,y}}
we first define a vector
d
x
,
y
,
z
{\displaystyle \mathbf {d} _{x,y,z}}
b
x
,
y
{\displaystyle \mathbf {b} _{x,y}}
we first define a vector
d
x
,
y
,
z
{\displaystyle \mathbf {d} _{x,y,z}}
as the position of point A with respect to a coordinate system defined by the camera, with origin in C and rotated by
θ
{\displaystyle \mathbf {\theta } }
with respect to the initial coordinate system. This is achieved by subtracting
c
{\displaystyle \mathbf {c} }
from
a
{\displaystyle \mathbf {a} }
and then applying a rotation by
−
θ
{\displaystyle -\mathbf {\theta } }
to the result. This transformation is often called a camera transform, and can be expressed as follows, expressing the rotation in terms of rotations about the x, y, and z axes (these calculations assume that the axes are ordered as a left-handed system of axes):
[9]
[10]
This representation corresponds to rotating by three Euler angles (more properly, Tait–Bryan angles), using the xyz convention, which can be interpreted either as "rotate about the extrinsic axes (axes of the scene) in the order z, y, x (reading right-to-left)" or "rotate about the intrinsic axes (axes of the camera) in the order x, y, z (reading left-to-right)". Note that if the camera is not rotated (
θ
x
,
y
,
z
=
⟨
0
,
0
,
0
⟩
{\displaystyle \mathbf {\theta } _{x,y,z}=\langle 0,0,0\rangle }
), then the matrices drop out (as identities), and this reduces to simply a shift:
d
=
a
−
c
.
{\displaystyle \mathbf {d} =\mathbf {a} -\mathbf {c} .}
Alternatively, without using matrices (let us replace
a
x
−
c
x
{\displaystyle a_{x}-c_{x}}
with
x
{\displaystyle \mathbf {x} }
and so on, and abbreviate
cos
(
θ
α
)
{\displaystyle \cos \left(\theta _{\alpha }\right)}
to
c
α
{\displaystyle c_{\alpha }}
and
sin
(
θ
α
)
{\displaystyle \sin \left(\theta _{\alpha }\right)}
to
s
α
{\displaystyle s_{\alpha }}
):
-
d
x
=
c
y
(
s
z
y
+
c
z
x
)
−
s
y
z
d
y
=
s
x
(
c
y
z
+
s
y
(
s
z
y
+
c
z
x
)
)
+
c
x
(
c
z
y
−
s
z
x
)
d
z
=
c
x
(
c
y
z
+
s
y
(
s
z
y
+
c
z
x
)
)
−
s
x
(
c
z
y
−
s
z
x
)
{\displaystyle {\begin{aligned}\mathbf {d} _{x}&=c_{y}(s_{z}\mathbf {y} +c_{z}\mathbf {x} )-s_{y}\mathbf {z} \\\mathbf {d} _{y}&=s_{x}(c_{y}\mathbf {z} +s_{y}(s_{z}\mathbf {y} +c_{z}\mathbf {x} ))+c_{x}(c_{z}\mathbf {y} -s_{z}\mathbf {x} )\\\mathbf {d} _{z}&=c_{x}(c_{y}\mathbf {z} +s_{y}(s_{z}\mathbf {y} +c_{z}\mathbf {x} ))-s_{x}(c_{z}\mathbf {y} -s_{z}\mathbf {x} )\end{aligned}}}

This transformed point can then be projected onto the 2D plane using the formula (here, x/y is used as the projection plane; literature also may use x/z):[11]
-
a
x
−
c
x
{\displaystyle a_{x}-c_{x}}
with
x
{\displaystyle \mathbf {x} }
and so on, and abbreviate
cos
(
θ
α
)
{\displaystyle \cos \left(\theta _{\alpha }\right)}
to
c
α
{\displaystyle c_{\alpha }}
and
sin
(
θ
α
)
{\displaystyle \sin \left(\theta _{\alpha }\right)}
to
s
α
{\displaystyle s_{\alpha }}
):
This transformed point can then be projected onto the 2D plane using the formula (here, x/y is used as the projection plane; literature also may use x/z):[11]
-
b
x
=
e
z
d
z
d
x
+
e
x
,
b
y
=
e
z
d
z
d
y
+
e
y
.
{\displaystyle {\begin{aligned}\mathbf {b} _{x}&={\frac {\mathbf {e} _{z}}{\mathbf {d} _{z}}}\mathbf {d} _{x}+\mathbf {e} _{x},\\[5pt]\mathbf {b} _{y}&={\frac {\mathbf {e} _{z}}{\mathbf {d} _{z}}}\mathbf {d} _{y}+\mathbf {e} _{y}.\end{aligned}}}
![{\displaystyle {\begin{aligned}\mathbf {b} _{x}&={\frac {\mathbf {e} _{z}}{\mathbf {d} _{z}}}\mathbf {d} _{x}+\mathbf {e} _{x},\\[5pt]\mathbf {b} _{y}&={\frac {\mathbf {e} _{z}}{\mathbf {d} _{z}}}\mathbf {d} _{y}+\mathbf {e} _{y}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f002d3d4ed5e51f66a9e80bad596258adb82ed25)
Or, in matrix form using homogeneous coordinates, the system
-
[
f
x
f
y
f
w
]
=
[
1
0
e
x
e
z
0
1
e
y
e
z
Or, in matrix form using homogeneous coordinates, the system
-
[
f
x
f
y
f
w
]
=
[
1
0
e
x
e
z
0
1
e
y
e
z
0
0
1
e
z
]
[
d
in conjunction with an argument using similar triangles, leads to division by the homogeneous coordinate, giving
-
b
x
=
f
x
/
f
w
b
y
=
f
y
/
f
w
{\displaystyle {\begin{aligned}\mathbf {b} _{x}&=\mathbf {f} _{x}/\mathbf {f} _{w}\\\mathbf {b} _{y}&=\mathbf {f} _{y}/\mathbf {f} _{w}\end{aligned}}}

The distance of the viewer from the display surface,
e
z
{\displaystyle \mathbf {e} _{z}}
, directly relates to the field of view, where
α
=
2
⋅
arctan
(
1
/
e
z
)
{\displaystyle \alpha =2\cdot \arctan(1/\mathbf {e} _{z})}
is the viewed angle. (Note: This assumes that you map the points (-1,-1) and (1,1) to the corners of your viewing surface)
The above equations can also be rewritten as:
-
b
x
=
(
e
z
{\displaystyle \mathbf {e} _{z}}
, directly relates to the field of view, where
α
=
2
⋅
arctan
(
1
/
e
z
)
{\displaystyle \alpha =2\cdot \arctan(1/\mathbf {e} _{z})}
is the viewed angle. (Note: This assumes that you map the points (-1,-1) and (1,1) to the corners of your viewing surface)
The above equations can also be rewritten as:
-
b
x
=
(
d
x
s
x
)
/
(
d
z
r
x
)
r
z
,
b
y
=
(
d
s
x
,
y
{\displaystyle \mathbf {s} _{x,y}}
is the display size,
r
x
,
y
{\displaystyle \mathbf {r} _{x,y}}
is the recording surface size (CCD or film),
r
z
{\displaystyle \mathbf {r} _{z}}
is the distance from the recording surface to the entrance pupil (camera center), and
d
z
{\displaystyle \mathbf {d} _{z}}
is the distance, from the 3D point being projected, to the entrance pupil.
Subsequent clipping and scaling operations may be necessary to map the 2D plane onto any particular display media.
Weak perspective projection
A "weak" perspective projection uses the same principles of an orthographic projection, but requires the scaling factor to be specified, thus ensuring that closer objects appear bigger in the projection, and vice versa. It can be seen as a hybrid between an orthographic and a perspective projection, and described either as a perspective projection with individual point depths
Z
i
{\displaystyle Z_{i}}
replaced by an average constant depth
Z
ave
{\displaystyle Z_{\text{ave}}}
,[12] or simply as an orthographic projection plus a scaling.[13]
The weak-perspective model thus approximates perspective projection while using a simpler model, similar to the pure (unscaled) orthographic perspective.
It is a reasonable approximation when the depth of the object along the line of sight is small compared to the distance from the camera, and the field of view is small. With these conditions, it can be assumed that all points on a 3D object are at the same distance
Z
ave
{\displaystyle Z_{\text{ave}}}
from the camera without significant errors in the projection (compared to the full perspective model).
Equation
-
P
x
=
X
Z
ave
P
y
=
Y
Subsequent clipping and scaling operations may be necessary to map the 2D plane onto any particular display media.
A "weak" perspective projection uses the same principles of an orthographic projection, but requires the scaling factor to be specified, thus ensuring that closer objects appear bigger in the projection, and vice versa. It can be seen as a hybrid between an orthographic and a perspective projection, and described either as a perspective projection with individual point depths
Z
i
{\displaystyle Z_{i}}
replaced by an average constant depth
Z
ave
{\displaystyle Z_{\text{ave}}}
,[12] or simply as an orthographic projection plus a scaling.[13]
The weak-perspective model thus approximates perspective projection while using a simpler model, similar to the pure (unscaled) orthographic perspective.
It is a reasonable approximation when the depth of the object along the line of sight is small compared to the distance from the camera, and the field of view is small. With these conditions, it can be ass
The weak-perspective model thus approximates perspective projection while using a simpler model, similar to the pure (unscaled) orthographic perspective.
It is a reasonable approximation when the depth of the object along the line of sight is small compared to the distance from the camera, and the field of view is small. With these conditions, it can be assumed that all points on a 3D object are at the same distance
Z
ave
{\displaystyle Z_{\text{ave}}}
from the camera without significant errors in the projection (compared to the full perspective model).
Equation
assuming focal length
f
=
1
f=1
.
Diagram
To determine which screen x-coordinate corresponds to a point at
A
x
,
A
z
{\displaystyle A_{x},A_{z}}
multiply the point coordinates by:
-
B
x
=
A
x
To determine which screen x-coordinate corresponds to a point at
To determine which screen x-coordinate corresponds to a point at
A
x
,
A
z
{\displaystyle A_{x},A_{z}}
multiply the point coordinates by:
where
-
B
x
{\displaystyle B_{x}}
is the screen x coordinate
-
A
x
{\displaystyle A_{x}}
See also