Perspectivity

In geometry and in its applications to drawing, a perspectivity is the formation of an image in a picture plane of a scene viewed from a fixed point.

Graphics

The science of graphical perspective uses perspectivities to make realistic images in proper proportion. According to Kirsti Andersen, the first author to describe perspectivity was Leon Alberti in his De Pictura (1435).^[1] In English, Brook Taylor presented his Linear Perspective in 1715, where he explained "Perspective is the Art of drawing on a Plane the Appearances of any Figures, by the Rules of Geometry".^[2] In a second book, New Principles of Linear Perspective (1719), Taylor wrote

When Lines drawn according to a certain Law from the several Parts of any Figure, cut a Plane, and by that Cutting or Intersection describe a figure on that Plane, that Figure so described is called the Projection of the other Figure. The Lines producing that Projection, taken all together, are called the System of Rays. And when those Rays all pass thro’ one and same Point, they are called the Cone of Rays. And when that Point is consider’d as the Eye of a Spectator, that System of Rays is called the Optic Cone^[3]

Projective Geometry

A perspectivity:

ABCD \doublebarwedge A'B'C'D',

In projective geometry the points of a line are called a projective range, and the set of lines in a plane on a point is called a pencil.

Given two lines $\ell$ and $m$ in a plane and a point P of that plane on neither line, the bijective mapping between the points of the range of $\ell$ and the range of $m$ determined by the lines of the pencil on P is called a perspectivity (or more precisely, a central perspectivity with center P).^[4] A special symbol has been used to show that points X and Y are related by a perspectivity; $X \doublebarwedge Y .$ In this notation, to show that the center of perspectivity is P, write $X \ \overset {P}{\doublebarwedge} \ Y.$ Using the language of functions, a central perspectivity with center P is a function $f_P \colon [\ell] \mapsto [m]$ (where the square brackets indicate the projective range of the line) defined by $f_P (X) = Y \text{ whenever } P \in XY$ .^[5] This map is an involution, that is, $f_P (f_P (X)) = X \text{ for all }X \in [\ell]$ .

The existence of a perspectivity means that corresponding points are in perspective. The dual concept, axial perspectivity, is the correspondence between the lines of two pencils determined by a projective range.

Projectivity

The composition of two perspectivities is, in general, not a perspectivity. A perspectivity or a composition of two or more perspectivities is called a projectivity (projective transformation, projective collineation and homography are synonyms).

There are several results concerning projectivities and perspectivities which hold in any pappian projective plane:^[6]

Theorem: Any projectivity between two distinct projective ranges can be written as the composition of no more than two perspectivities.

Theorem: Any projectivity from a projective range to itself can be written as the composition of three perspectivities.

Theorem: A projectivity between two distinct projective ranges which fixes a point is a perspectivity.

Higher-dimensional perspectivities

The bijective correspondence between points on two lines in a plane determined by a point of that plane not on either line has higher-dimensional analogues which will also be called perspectivities.

Let S_m and T_m be two distinct m-dimensional projective spaces contained in an n-dimensional projective space R_n. Let P_n−m−1 be an (n − m − 1)-dimensional subspace of R_n with no points in common with either S_m or T_m. For each point X of S_m, the space L spanned by X and P_n-m-1 meets T_m in a point Y = f_P(X). This correspondence f_P is also called a perspectivity.^[7] The central perspectivity described above is the case with n = 2 and m = 1.

Perspective collineations

Let S₂ and T₂ be two distinct projective planes in a projective 3-space R₃. With O and O* being points of R₃ in neither plane, use the construction of the last section to project S₂ onto T₂ by the perspectivity with center O followed by the projection of T₂ back onto S₂ with the perspectivity with center O*. This composition is a bijective map of the points of S₂ onto itself which preserves collinear points and is called a perspective collineation (central collineation in more modern terminology).^[8] Let φ be a perspective collineation of S₂. Each point of the line of intersection of S₂ and T₂ will be fixed by φ and this line is called the axis of φ. Let point P be the intersection of line OO* with the plane S₂. P is also fixed by φ and every line of S₂ that passes through P is stabilized by φ (fixed, but not necessarily pointwise fixed). P is called the center of φ. The restriction of φ to any line of S₂ not passing through P is the central perspectivity in S₂ with center P between that line and the line which is its image under φ.

Notes

↑ Kirsti Andersen (2007) The Geometry of an Art, page 1,Springer ISBN 978-0-387-25961-1
↑ Andersen 1992, p. 75
↑ Andersen 1992, p. 163
↑ Coxeter 1969, p. 242
↑ Pedoe 1988, p. 281
↑ Fishback 1969, pp. 65-66
↑ Pedoe 1988, pp. 282–3
↑ Young 1930, p. 116

References

Andersen, Kirsti (1992), Brook Taylor's Work on Linear Perspective, Springer, ISBN 0-387-97486-5
Coxeter, Harold Scott MacDonald (1969), Introduction to Geometry (2nd ed.), New York: John Wiley & Sons, ISBN 978-0-471-50458-0, MR 123930
Fishback, W.T. (1969), Projective and Euclidean Geometry, John Wiley & Sons
Pedoe, Dan (1988), Geometry/A Comprehensive Course, Dover, ISBN 0-486-65812-0
Young, John Wesley (1930), Projective Geometry, The Carus Mathematical Monographs (#4), Mathematical Association of America

External links

Christopher Cooper Perspectivities and Projectivities from Macquarie University.
James C. Morehead Jr. (1911) Perspective and Projective Geometries: A Comparison from Rice University.
John Taylor Projective Geometry from University of Brighton.

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