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7973Definition: Projectivities, Ranges and Pencils

In projective geometry, a projectivity “$\overline\wedge$” is a bijection (one-to-one-relationship) of collinear points on a straight line $o$ to concurrent straight lines passing through a point $O$ (and vice versa). It is required that $o$ and $O$ are not incident and that each of the collinear points lies on the concurrent line it has a bijection with.

If there is a projectivity of the point $P$ to the straight line $p$, then we write $P\overline\wedge p$ (or $p\overline\wedge P$).

The collinear points in a projectivity are called the range, the corresponding concurrent lines are called the pencil of the projectivity.

Example

The following interactive figure demonstrates a projectivity of the range $P_1,P_2$ to the pencil $p_1,p_2$, in other words, we have $P_1\overline\wedge p_1$ and $P_2\overline\wedge p_2.$

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