# Definition: Perspectivities

In projective geometry, a perspectivity "$\overline{\overline\wedge}$" is a composition of two projectivities. A perspectivity establishes a bijection (one-to-one-relationship) between two ranges of two distinct straight lines $p_1$ and $p_2$ in relation to a pencil of a point $O,$ called the projective center of the projectivity. We write in this case $$p_1\overline{\overline\wedge}p_2.$$

In its dual formulation, a perspectivity establishes a bijection between two pencils of two distinct points $P_1$ and $P_2$ in relation to a range of a straight line $o,$ called the projective axis. We write in this case $$P_1\overline{\overline\wedge}P_2.$$

### Example

The following interactive figure demonstrates a perspectivity of the ranges $P_1P_2\overline{\overline\wedge}Q_1Q_2$ and the corresponding dual perspectivity of the pencils $p_1\cdot p_2\overline{\overline\wedge}q_1\cdot q_2.$

Perspectivity Dual Definition

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