# Perspective projection

Perspective projection is a type of drawing that graphically approximates on
a planar (two-dimensional) surface (e.g. paper) the images of three-dimensional
objects so as to approximate actual visual perception. It is sometimes also
called **perspective view** or **perspective drawing** or simply **
perspective**.

All perspectives on a planar surface have some degree of distortion, similar to the distortion created when portraying the earth's surface on a planar map.

# Linear perspective

Linear perspective is the art of representing three-dimensional constructions on a two-dimensional surface. It presupposes a fixed viewpoint and a desire to create an "objective" recording of one's visual experience - two conditions which have been the most dominant in the Western art of the past half-millennium.

Once the observer assumes a single point of view, several conclusions follow logically. The first and most important is this: objects appear to get smaller as their distance from the observer increases. That this is not self-evident in art is apparent from even a casual perusal of the art of other cultures and eras - most frequently objects are drawn or painted a certain size for reasons that have nothing to do with their position in space. In medieval Last Judgement paintings, for example, the relative scale of the various figures is determined only by their sacred significance; the most important are the largest.

Once the diminishment of scale with distance is noted, it is an easy step to understanding why the space between parallel lines must also appear to diminish. A wall retreating from the observer will appear to get progressively shorter, and the top and bottom edges of the wall will thus appear to move closer together.

It was an enormous conceptual leap when artists concluded that these lines, if extended indefinitely, would appear to meet at a single point on the horizon. This idea, long since verified, was likely made from a theoretical analysis of the process of seeing rather than direct observation. Van Eyck, for example, was unable to create a consistent structure for the converging lines in paintings like London's The Arnolfini Portrait because he was not aware of the theoretical breakthrough just then occurring in Italy.

Below are descriptions of the three main varieties of perspective technique. Note that the only difference among these three varieties is the orientation of linear objects being viewed relative to the viewer.

## 1. One-Point Perspective

If the viewpoint is pointing directly into a linear object like a building or a road, one would use one vanishing point, that is the principal focus. All lines perpendicular to the painting plate would vanish in the vanishing point.

More precisely, one-point perspective exists when the painting plate (also known as the picture plane) is parallel to a "Cartesian scene" (see also Cartesian coordinate system) --- a scene which is composed entirely of linear elements that intersect only at right angles. Therefore, all elements are either parallel to the painting plate (either horizontally or vertically) or perpendicular to it. All elements that are parallel to the painting plate are drawn as parallel lines. All elements that are perpendicular to the painting plate converge at a single point on the horizon.

## 2. Two-Point Perspective

If the lines have angles to the painting plate, they would vanish in the other vanishing points. There are lot of vanishing points homologous to different angles. But all vanishing points should be located in the same horizontal line with the focus.

In other words, two-point perspective is derived from one-point perspective by yawing the line of vision so that the line of vision will be at an acute angle away from the focus. Then the lines which used to be horizontal and parallel will now be concurrent, intersecting at the horizon. Interpreted according to projective geometry, the horizontal parallel lines of one-point perspective are actually concurrent, intersecting at the point at infinity [1:0:1]. When the head is turned by a slight angle, these lines no longer intersect at an ideal point, but at an affine point on the horizon, so they are no longer parallel.

More precisely, two-point perspective exists when the painting plate is parallel to a "Cartesian scene" (a scene composed entirely of linear elements intersecting only at right angles) in one axis (usually the z-axis) but not parallel to the other two axes. Note that if the scene being viewed consists solely of a cylinder sitting on a horizontal plane, no difference exists in the image of the cylinder between a one-point and two-point perspective.

### 3. Three-point Perspective

If the lines have angles from the painting plate up or down, one would use the other kind vanishing points. Those vanishing points must located in the same vertical line with the focus. Looking at the object from above or below, the horizontal line with the focus and all other 2nd vanishing points would left the horizon up or down.

Three-point perspective exists when the image plane is viewing a "Cartesian scene" (a scene composed entirely of linear elements intersecting at right angles) and is not parallel to any of the scenes three axes. Elements that are parallel to each of the three axes will converge to three vanishing points respectively.

## Other varieties of perspective

A key point to note is that one-point, two-point, and three-point perspective are dependent on the structure of the scene being viewed. These only exist for strict Cartesian scenes.

Note that by inserting into a Cartesian scene a set of parallel lines that are not parallel to any of the three axes of the scene, a new distinct vanishing point is created.

Therefore, it is possible to have an infinite-point perspective if the scene being viewed is not a Cartesian scene but instead consists of infinite pairs of parallel lines, where each pair is not parallel to any other pair.

Due to the fact that vanishing points exist only when parallel lines are present in the scene, a zero-point perspective is also possible if the viewer is observing a nonlinear scene. One example is a random (ie, not aligned in a three-dimensional Cartesian coordinate system) arrangement of spherical objects. Another would be a scene composed entirely of three-dimensionally curvilinear strings. A third example would be a scene consisting of lines where no two are parallel to each other.

**See also**

2D | Two-dimensional coordinate system |

3D | Three-dimensional coordinate system |

Angle | Definition of an angle |

Axis | Definition of Cartesian axis |

Cartesian geometry | What is Cartesian geometry? |

Coordinate system | Definition of coordinates |

Curve | Definition of a curve |

Distance | Definition of distance |

Euclidean geometry | What is Euclidean geometry? |

Geometry | Definition of geometry |

Length | Definition of length |

Line | Definition of a line |

Origin | Definition of origin in a Cartesian coordinate system |

Perspective projection | Definition of perspective projection |

Planar homography | Definition of planar homography |

Plane | Definition of a plane |

Point | Definition of a point |

Point (kinematics) | Definition of a point (kinematics) |

Projective geometry | What is projective geometry? |

Segment (kinematics) | Definition of a segment (kinematics) |

Vanishing points | Definition of vanishing points and vanishing lines in perspective projection |

Vector | Definition of a vector |