4.6 Dynamic
Illustrations for Perspectivity
The only way to learn mathematics is to do
mathematics.
—
Paul Halmos (1916–2006)
- A one-to-one mapping between
a pencil of points and a pencil of lines is called an elementary correspondence
if each point of the pencil of points is incident with the corresponding
line of the pencil of lines. The elementary correspondence is denoted or.
- A one-to-one mapping between
two pencils of points is called a perspectivity if the lines
incident with the corresponding points of the two pencils are concurrent.
The point where the lines intersect is called the center of the
perspectivity. The perspectivity is denoted where O is the center of perspectivity.
- A one-to-one mapping between
two pencils of lines is called a perspectivity if the points of
intersection of the corresponding lines of the two pencils are collinear.
The line containing the points of intersection is called the
axis of
the perspectivity. The perspectivity is denoted where o is the axis of perspectivity.
Note
that a perspectivity is a composition of two elementary correspondences between
either two pencils of points or two pencils of lines.
Illustrations of perspectivities: ,
,
and an elementary correspondence
.
The points in red may be dragged to different locations to
explore the perspectivities and elementary correspondence.
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