A perspective projection with center 
and image plane 
not containing 
maps a point 
in three-dimensional Euclidean
space to the intersection of the line 
with 
. If 
is parallel to 
, its image is a point at
infinity. All points on the same line
through 
have the same image, so distance from 
along the line is lost.
Choose a coordinate system with 
and image plane 
. A point 
with 
is mapped to the image point 
,
where
 |
In homogeneous coordinates, the same projection is represented by
 |
These formulas give the idealized geometry of imaging from a single viewpoint (Hartley and Zisserman 2004, Torralba et al. 2024).
A perspective projection maps lines not passing through 
to lines. Images of parallel lines generally meet at a vanishing
point. Consequently, perspective projection does not preserve lengths,
angles, or parallelism, unlike
an affine transformation (Torralba et
al. 2024).
