Chapter 4: Geometry

4.10.3 PROJECTIVE TRANSFORMATIONS
A transformation that maps lines to lines (but does not necessarily preserve parallelism)
is a projective transformation. Any spatial projective transformation can be
expressed by an invertible ¢ matrix in homogeneous coordinates; conversely, any
invertible ¢ matrix defines a projective transformation of space. Projective transformations
(if not affine) are not defined on all of space, but only on the complement
of a plane (the missing plane is “mapped to infinity”).
The following particular case is often useful, especially in computer graphics,
in projecting a scene from space to the plane. Suppose an observer is at the point
´Ü ¼ Ý ¼ Þ ¼ µ of space, looking toward the origin Ç ´¼ ¼ ¼µ. Let È , the
screen, be the plane through Ç and perpendicular to the ray Ç. Place a rectangular
coordinate system on È with origin at Ç so that the positive -axis lies in the halfplane
determined by and the positive Þ-axis of space (that is, the Þ-axis is pointing
“up” as seen from ). Then consider the transformation that associates with a point
´Ü Ý Þµ the triple ´µ, where ´µ are the coordinates of the point, where
the line intersects È (the screen coordinates of as seen from ), and is the
inverse of the signed distance from to along the line Ç (this distance is the
depth of as seen from ). This is a projective transformation, given by the matrix
¾
Ö ¾ Ý ¼ Ö ¾ Ü ¼ ¼ ¼
ÖÜ ¼ Þ ¼ ÖÝ ¼ Þ ¼ Ö ¾ ¼
¼ ¼ ¼ Ö
with
Ô
Ü
¾
¼ · ݾ ¼ and Ö Ô
Ü
¾
¼ · ݾ ¼ · Þ¾ ¼ .
(4.10.5)
Ü ¼ Ý ¼ Þ ¼ Ö ¾
¿
4.11 DIRECTION ANGLES AND DIRECTION
COSINES
Given a vector ´ µ in three-dimensional space, the direction cosines of this vector
are
Ó× « Ô
¾ · ¾ ·
¾
Ó× ¬ Ô
¾ · ¾ ·
¾ (4.11.1)
Ó× ­ Ô
¾ · ¾ · ¾
Here the direction angles «, ¬, ­ are the angles that the vector makes with the positive
Ü-, Ý- and Þ-axes, respectively. In formulae, usually the direction cosines appear,
rather than the direction angles. We have
Ó× ¾ « ·Ó× ¾ ¬ · Ó× ¾ ­ ½ (4.11.2)
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