View source for Perspective

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''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.).

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p072410a.gif" />

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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Artin,   "Geometric algebra" , Interscience  (1957)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.A. Glagolev,   "Projective geometry" , Moscow  (1963)  (In Russian)</TD></TR></table>



====Comments====
Perspective mappings are also called central projections or perspectivities.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,   "Geometry" , '''I''' , Springer  (1987)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Busemann,   P.J. Kelly,   "Projective geometry and projective metrics" , Acad. Press  (1953)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H.S.M. Coxeter,   "Introduction to geometry" , Wiley  (1963)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H.S.M. Coxeter,   "Projective geometry" , Blaisdell  (1964)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  H.S.M. Coxeter,   "The real projective plane" , McGraw-Hill  (1949)</TD></TR></table>