# Perspective

*with centre $S$*

A mapping of a plane $\pi$ into a plane $\pi_1$ under which each point $M$ in $\pi$ is put into correspondence with the point $M_1$ of intersection of the straight line $SM$ with $\pi_1$ (if $SM$ is not parallel to $\pi_1$, see Fig.).

Figure: p072410a

More generally, let $V$ and $V_1$ be proper subspaces of identical dimension in a projective space $\Omega$ and let $T$ be a subspace of maximal dimension not having points in common with $V$ or $V_1$. Let $U$ be a subspace contained in $V$, let $W$ be the subspace of minimal dimension containing $U$ and $T$ and let $U_1$ be the intersection of $W$ and $V_1$.

The correspondence by which each subspace $U$ contained in $V$ is put into correspondence with the subspace $U_1$ contained in $V_1$ is called a perspective mapping from $V$ into $V_1$ with perspective centre $T$.

A perspective is a collineation. If the subspaces $V$ and $V_1$ intersect, then each point in the subspace $V\cap V_1$ corresponds to itself.

If projective coordinates are introduced into the spaces $V$ and $V_1$, then a perspective correspondence can be specified by a linear mapping.

#### References

[1] | E. Artin, "Geometric algebra" , Interscience (1957) |

[2] | N.A. Glagolev, "Projective geometry" , Moscow (1963) (In Russian) |

#### Comments

Perspective mappings are also called central projections or perspectivities.

#### References

[a1] | M. Berger, "Geometry" , I , Springer (1987) |

[a2] | H. Busemann, P.J. Kelly, "Projective geometry and projective metrics" , Acad. Press (1953) |

[a3] | H.S.M. Coxeter, "Introduction to geometry" , Wiley (1963) |

[a4] | H.S.M. Coxeter, "Projective geometry" , Blaisdell (1964) |

[a5] | H.S.M. Coxeter, "The real projective plane" , McGraw-Hill (1949) |

**How to Cite This Entry:**

Perspective.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Perspective&oldid=31589