perspectivity
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In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
and in its applications to
drawing Drawing is a form of visual art in which an artist uses instruments to mark paper or other two-dimensional surface. Drawing instruments include graphite pencils, pen and ink, various kinds of paints, inked brushes, colored pencils, crayons, ...
, a perspectivity is the formation of an image in a
picture plane In painting, photography, graphical perspective and descriptive geometry, a picture plane is an image plane located between the "eye point" (or '' oculus'') and the object being viewed and is usually coextensive to the material surface of the w ...
of a scene viewed from a fixed point.


Graphics

The science of
graphical perspective Linear or point-projection perspective (from la, perspicere 'to see through') is one of two types of graphical projection perspective in the graphic arts; the other is parallel projection. Linear perspective is an approximate representation, ...
uses perspectivities to make realistic images in proper proportion. According to
Kirsti Andersen Kirsti Andersen (born December 9, 1941, Copenhagen), published under the name Kirsti Pedersen, is a Danish historian of mathematics. She is an Associate Professor of the History of Science at Aarhus University, where she had her Candidate exami ...
, the first author to describe perspectivity was
Leon Alberti Leon Battista Alberti (; 14 February 1404 – 25 April 1472) was an Italian Renaissance humanist author, artist, architect, poet, Catholic priest, priest, linguistics, linguist, philosopher, and cryptography, cryptographer; he epitomised the natu ...
in his ''De Pictura'' (1435). 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". 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''


Projective geometry

In
projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, pro ...
the points of a line are called a
projective range In mathematics, a projective range is a set of points in projective geometry considered in a unified fashion. A projective range may be a projective line or a conic. A projective range is the dual of a pencil of lines on a given point. For instanc ...
, and the set of lines in a plane on a point is called a
pencil A pencil () is a writing or drawing implement with a solid pigment core in a protective casing that reduces the risk of core breakage, and keeps it from marking the user's hand. Pencils create marks by physical abrasion, leaving a trail ...
. Given two
lines Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Arts ...
\ell and m in a
projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that do ...
and a point ''P'' of that plane on neither line, the
bijective mapping In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
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''). 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 \ Y. 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 In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, ...
'' are
synonym A synonym is a word, morpheme, or phrase that means exactly or nearly the same as another word, morpheme, or phrase in a given language. For example, in the English language, the words ''begin'', ''start'', ''commence'', and ''initiate'' are all ...
s). There are several results concerning projectivities and perspectivities which hold in any pappian projective plane: 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 . This correspondence ''f''''P'' is also called a perspectivity. The central perspectivity described above is the case with and .


Perspective collineations

Let ''S''2 and ''T''2 be two distinct projective planes in a projective 3-space ''R''3. With ''O'' and ''O''* being points of ''R''3 in neither plane, use the construction of the last section to project ''S''2 onto ''T''2 by the perspectivity with center ''O'' followed by the projection of ''T''2 back onto ''S''2 with the perspectivity with center ''O''*. This composition is a
bijective map In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
of the points of ''S''2 onto itself which preserves
collinear In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned ...
points and is called a ''perspective collineation'' (''central collineation'' in more modern terminology). Let φ be a perspective collineation of ''S''2. Each point of the line of intersection of ''S''2 and ''T''2 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 ''S''2. ''P'' is also fixed by φ and every line of ''S''2 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 ''S''2 not passing through ''P'' is the central perspectivity in ''S''2 with center ''P'' between that line and the line which is its image under φ.


See also

* Perspective projection *
Desargues's theorem In projective geometry, Desargues's theorem, named after Girard Desargues, states: :Two triangles are in perspective ''axially'' if and only if they are in perspective ''centrally''. Denote the three vertices of one triangle by and , and tho ...


Notes


References

* * * * * {{citation, first=John Wesley, last=Young, title=Projective Geometry, year=1930, publisher=Mathematical Association of America, series=The Carus Mathematical Monographs (#4)


External links

* Christopher Coope
Perspectivities and Projectivities
* James C. Morehead Jr. (1911
Perspective and Projective Geometries: A Comparison
from
Rice University William Marsh Rice University (Rice University) is a Private university, private research university in Houston, Houston, Texas. It is on a 300-acre campus near the Houston Museum District and adjacent to the Texas Medical Center. Rice is ranke ...
. * John Taylo
Projective Geometry
from
University of Brighton The University of Brighton is a public university based on four campuses in Brighton and Eastbourne on the south coast of England. Its roots can be traced back to 1858 when the Brighton School of Art was opened in the Royal Pavilion. It achieve ...
. Projective geometry Perspective projection Technical drawing Functions and mappings Composition in visual art