4.6 Dynamic
Illustrations for Projectivities
Metrical geometry is thus a part of [projective]
geometry, and [projective] geometry is all geometry.
—
Arthur Cayley (1821–1895)
- A one-to-one mapping between
two pencils of points is called a projectivity if the mapping is a
composition of finitely many elementary correspondences or perspectivities. A
projectivity is denoted oror.
When a projectivity exists between two pencils, the pencils are said to be
projectively related. Also, note
that elementary correspondences and perspectivities themselves are
projectivities.
Illustrations of projectivities: Figure 1 , Figure 2
, and Figure 3
.
The points in red may be dragged to different locations to explore each projectivity.
To better see the projectivities in each figure, we describe a path to follow
beginning with one of the points/lines and following the "path of its
projection."
-
In Figure 1, from A follow the path with center P to axis p', from axis p' follow the corresponding path with center O to A".
-
In Figure 2, follow the path a from center P to axis o, from axis o follow the corresponding path from center P' to axis o' which leads to the corresponding path a" with center P".
-
In Figure 3, follow the path a from center P to axis o' , then from axis o' follow the corresponding path through center Q to A".
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