Duality (projective geometry)

In projective geometry, duality or plane duality is a formalization of the striking symmetry of the roles played by points and lines in the definitions and theorems of projective planes. There are two approaches to the subject of duality, one through language (Template:Section link) and the other a more functional approach through special mappings. These are completely equivalent and either treatment has as its starting point the axiomatic version of the geometries under consideration. In the functional approach there is a map between related geometries that is called a duality. Such a map can be constructed in many ways. The concept of plane duality readily extends to space duality and beyond that to duality in any finite-dimensional projective geometry.

Principle of duality

A projective plane Template:Math may be defined axiomatically as an incidence structure, in terms of a set Template:Math of points, a set Template:Math of lines, and an incidence relation Template:Math that determines which points lie on which lines. These sets can be used to define a plane dual structure.

Interchange the role of "points" and "lines" in

Template:Math

to obtain the dual structure

Template:Math,

where Template:Math is the converse relation of Template:Math. Template:Math is also a projective plane, called the dual plane of Template:Math.

If Template:Math and Template:Math are isomorphic, then Template:Math is called self-dual. The projective planes Template:Math for any field (or, more generally, for every division ring (skewfield) isomorphic to its dual) Template:Math are self-dual. In particular, Desarguesian planes of finite order are always self-dual. However, there are non-Desarguesian planes which are not self-dual, such as the Hall planes and some that are, such as the Hughes planes.

In a projective plane a statement involving points, lines and incidence between them that is obtained from another such statement by interchanging the words "point" and "line" and making whatever grammatical adjustments that are necessary, is called the plane dual statement of the first. The plane dual statement of "Two points are on a unique line" is "Two lines meet at a unique point". Forming the plane dual of a statement is known as dualizing the statement.

If a statement is true in a projective plane Template:Math, then the plane dual of that statement must be true in the dual plane Template:Math. This follows since dualizing each statement in the proof "in Template:Math" gives a corresponding statement of the proof "in Template:Math".

The principle of plane duality says that dualizing any theorem in a self-dual projective plane Template:Math produces another theorem valid in Template:Math.<ref name=Cox25>Template:Harvnb</ref>

The above concepts can be generalized to talk about space duality, where the terms "points" and "planes" are interchanged (and lines remain lines). This leads to the principle of space duality.<ref name=Cox25 />

These principles provide a good reason for preferring to use a "symmetric" term for the incidence relation. Thus instead of saying "a point lies on a line" one should say "a point is incident with a line" since dualizing the latter only involves interchanging point and line ("a line is incident with a point").<ref>Template:Harvnb</ref>

The validity of the principle of plane duality follows from the axiomatic definition of a projective plane. The three axioms of this definition can be written so that they are self-dual statements implying that the dual of a projective plane is also a projective plane. The dual of a true statement in a projective plane is therefore a true statement in the dual projective plane and the implication is that for self-dual planes, the dual of a true statement in that plane is also a true statement in that plane.<ref>Template:Harvnb</ref>

Dual theorems

As the real projective plane, Template:Math, is self-dual there are a number of pairs of well known results that are duals of each other. Some of these are:

• Desargues' theorem ⇔ Converse of Desargues' theorem
• Pascal's theorem ⇔ Brianchon's theorem
• Menelaus' theorem ⇔ Ceva's theorem

Dual configurations

File:Dualquads.svg

Dual configurations

Not only statements, but also systems of points and lines can be dualized.

A set of Template:Math points and Template:Math lines is called an Template:Math configuration if Template:Math of the Template:Math lines pass through each point and Template:Math of the Template:Math points lie on each line. The dual of an Template:Math configuration, is an Template:Math configuration. Thus, the dual of a quadrangle, a (4₃, 6₂) configuration of four points and six lines, is a quadrilateral, a (6₂, 4₃) configuration of six points and four lines.<ref>Template:Harvnb</ref>

The set of all points on a line, called a projective range, has as its dual a pencil of lines, the set of all lines on a point, in two dimensions, or a pencil of hyperplanes in higher dimensions. A line segment on a projective line has as its dual the shape swept out by these lines or hyperplanes, a double wedge.<ref>Template:Citation.</ref>

Duality as a mapping

Plane dualities

A plane duality is a map from a projective plane Template:Math to its dual plane Template:Math (see Template:Section link above) which preserves incidence. That is, a plane duality Template:Math will map points to lines and lines to points (Template:Math and Template:Math) in such a way that if a point Template:Math is on a line Template:Math (denoted by Template:Math) then Template:Math. A plane duality which is an isomorphism is called a correlation.<ref>Template:Harvnb</ref> The existence of a correlation means that the projective plane Template:Math is self-dual.

The projective plane Template:Math in this definition need not be a Desarguesian plane. However, if it is, that is, Template:Math with Template:Math a division ring (skewfield), then a duality, as defined below for general projective spaces, gives a plane duality on Template:Math that satisfies the above definition.

In general projective spaces

A duality Template:Math of a projective space is a permutation of the subspaces of Template:Math (also denoted by Template:Math with Template:Math a field (or more generally a skewfield (division ring)) that reverses inclusion,<ref>Some authors use the term "correlation" for duality, while others, as shall we, use correlation for a certain type of duality.</ref> that is:

Template:Math implies Template:Math for all subspaces Template:Math of Template:Math.<ref>Template:Harvnb Dembowski uses the term "correlation" for duality.</ref>

Consequently, a duality interchanges objects of dimension Template:Math with objects of dimension Template:Math ( = codimension Template:Math). That is, in a projective space of dimension Template:Math, the points (dimension 0) correspond to hyperplanes (codimension 1), the lines joining two points (dimension 1) correspond to the intersection of two hyperplanes (codimension 2), and so on.

Classification of dualities

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The dual Template:Math of a finite-dimensional (right) vector space Template:Math over a skewfield Template:Math can be regarded as a (right) vector space of the same dimension over the opposite skewfield Template:Math. There is thus an inclusion-reversing bijection between the projective spaces Template:Math and Template:Math. If Template:Math and Template:Math are isomorphic then there exists a duality on Template:Math. Conversely, if Template:Math admits a duality for Template:Math, then Template:Math and Template:Math are isomorphic.

Let Template:Pi be a duality of Template:Math for Template:Math. If Template:Pi is composed with the natural isomorphism between Template:Math and Template:Math, the composition Template:Math is an incidence preserving bijection between Template:Math and Template:Math. By the Fundamental theorem of projective geometry Template:Math is induced by a semilinear map Template:Math with associated isomorphism Template:Math, which can be viewed as an antiautomorphism of Template:Math. In the classical literature, Template:Pi would be called a reciprocity in general, and if Template:Math it would be called a correlation (and Template:Math would necessarily be a field). Some authors suppress the role of the natural isomorphism and call Template:Math a duality.<ref>for instance Template:Harvnb</ref> When this is done, a duality may be thought of as a collineation between a pair of specially related projective spaces and called a reciprocity. If this collineation is a projectivity then it is called a correlation.

Let Template:Math denote the linear functional of Template:Math associated with the vector Template:Math in Template:Math. Define the form Template:Math by:

<math>\varphi (v,w) = T_w (v).</math>

Template:Math is a nondegenerate sesquilinear form with companion antiautomorphism Template:Math.

Any duality of Template:Math for Template:Math is induced by a nondegenerate sesquilinear form on the underlying vector space (with a companion antiautomorphism) and conversely.

Homogeneous coordinate formulation

Homogeneous coordinates may be used to give an algebraic description of dualities. To simplify this discussion we shall assume that Template:Math is a field, but everything can be done in the same way when Template:Math is a skewfield as long as attention is paid to the fact that multiplication need not be a commutative operation.

The points of Template:Math can be taken to be the nonzero vectors in the (Template:Math)-dimensional vector space over Template:Math, where we identify two vectors which differ by a scalar factor. Another way to put it is that the points of Template:Math-dimensional projective space are the 1-dimensional vector subspaces, which may be visualized as the lines through the origin in Template:Math.<ref>Dimension is being used here in two different senses. When referring to a projective space, the term is used in the common geometric way where lines are 1-dimensional and planes are 2-dimensional objects. However, when applied to a vector space, dimension means the number of vectors in a basis, and a basis for a vector subspace, thought of as a line, has two vectors in it, while a basis for a vector space, thought of as a plane, has three vectors in it. If the meaning is not clear from the context, the terms projective or geometric are applied to the projective space concept while algebraic or vector are applied to the vector space one. The relation between the two is simply: algebraic dimension = geometric dimension + 1.</ref> Also the Template:Math- (vector) dimensional subspaces of Template:Math represent the (Template:Math)- (geometric) dimensional hyperplanes of projective Template:Math-space over Template:Math, i.e., Template:Math.

A nonzero vector Template:Math in Template:Math also determines an Template:Math - geometric dimensional subspace (hyperplane) Template:Math, by

Template:Math.

When a vector Template:Math is used to define a hyperplane in this way it shall be denoted by Template:Math, while if it is designating a point we will use Template:Math. They are referred to as point coordinates or hyperplane coordinates respectively (in the important two-dimensional case, hyperplane coordinates are called line coordinates). Some authors distinguish how a vector is to be interpreted by writing hyperplane coordinates as horizontal (row) vectors while point coordinates are written as vertical (column) vectors. Thus, if Template:Math is a column vector we would have Template:Math while Template:Math. In terms of the usual dot product, Template:Math. Since Template:Math is a field, the dot product is symmetrical, meaning Template:Math.

A fundamental example

A simple reciprocity (actually a correlation) can be given by Template:Math between points and hyperplanes. This extends to a reciprocity between the line generated by two points and the intersection of two such hyperplanes, and so forth.

Specifically, in the projective plane, Template:Math, with Template:Math a field, we have the correlation given by: points in homogeneous coordinates Template:Math lines with equations Template:Math. In a projective space, Template:Math, a correlation is given by: points in homogeneous coordinates Template:Math planes with equations Template:Math. This correlation would also map a line determined by two points Template:Math and Template:Math to the line which is the intersection of the two planes with equations Template:Math and Template:Math.

The associated sesquilinear form for this correlation is:

Template:Math,

where the companion antiautomorphism Template:Math. This is therefore a bilinear form (note that Template:Math must be a field). This can be written in matrix form (with respect to the standard basis) as:

Template:Math,

where Template:Math is the Template:Math identity matrix, using the convention that Template:Math is a row vector and Template:Math is a column vector.

The correlation is given by:

<math> \pi ( \mathbf{x}_P) = (G \mathbf{x}_P)^{\mathsf{T}} = (\mathbf{x}_P)^{\mathsf{T}} = \mathbf{x}_H.</math>

Geometric interpretation in the real projective plane

This correlation in the case of Template:Math can be described geometrically using the model of the real projective plane which is a "unit sphere with antipodes<ref>the points of a sphere at opposite ends of a diameter are called antipodal points.</ref> identified", or equivalently, the model of lines and planes through the origin of the vector space Template:Math. Associate to any line through the origin the unique plane through the origin which is perpendicular (orthogonal) to the line. When, in the model, these lines are considered to be the points and the planes the lines of the projective plane Template:Math, this association becomes a correlation (actually a polarity) of the projective plane. The sphere model is obtained by intersecting the lines and planes through the origin with a unit sphere centered at the origin. The lines meet the sphere in antipodal points which must then be identified to obtain a point of the projective plane, and the planes meet the sphere in great circles which are thus the lines of the projective plane.

That this association "preserves" incidence is most easily seen from the lines and planes model. A point incident with a line in the projective plane corresponds to a line through the origin lying in a plane through the origin in the model. Applying the association, the plane becomes a line through the origin perpendicular to the plane it is associated with. This image line is perpendicular to every line of the plane which passes through the origin, in particular the original line (point of the projective plane). All lines that are perpendicular to the original line at the origin lie in the unique plane which is orthogonal to the original line, that is, the image plane under the association. Thus, the image line lies in the image plane and the association preserves incidence.

Matrix form

As in the above example, matrices can be used to represent dualities. Let Template:Pi be a duality of Template:Math for Template:Math and let Template:Math be the associated sesquilinear form (with companion antiautomorphism Template:Math) on the underlying (Template:Math)-dimensional vector space Template:Math. Given a basis Template:Math of Template:Math, we may represent this form by:

<math> \varphi(\mathbf{u}, \mathbf{x}) = \mathbf{u}^{\mathsf{T}} G (\mathbf{x}^{\sigma}),</math>

where Template:Math is a nonsingular Template:Math matrix over Template:Math and the vectors are written as column vectors. The notation Template:Math means that the antiautomorphism Template:Math is applied to each coordinate of the vector Template:Math.

Now define the duality in terms of point coordinates by:

<math> \pi ( \mathbf{x}) = (G (\mathbf{x}^{\sigma}))^{\mathsf{T}}.</math>

Polarity

A duality that is an involution (has order two) is called a polarity. It is necessary to distinguish between polarities of general projective spaces and those that arise from the slightly more general definition of plane duality. It is also possible to give more precise statements in the case of a finite geometry, so we shall emphasize the results in finite projective planes.

Polarities of general projective spaces

If Template:Pi is a duality of Template:Math, with Template:Math a skewfield, then a common notation is defined by Template:Math for a subspace Template:Math of Template:Math. Hence, a polarity is a duality for which Template:Math for every subspace Template:Math of Template:Math. It is also common to bypass mentioning the dual space and write, in terms of the associated sesquilinear form:

<math>S^{\bot} = \{\mathbf{u} \text{ in }V \colon \varphi (\mathbf{u},\mathbf{x}) =0 \text{ for all }\mathbf{x} \text{ in } S \}.</math>

A sesquilinear form Template:Math is reflexive if Template:Math implies Template:Math.

A duality is a polarity if and only if the (nondegenerate) sesquilinear form defining it is reflexive.<ref name=Demb42>Template:Harvnb</ref>

Polarities have been classified, a result of Template:Harvtxt that has been reproven several times.<ref name=Demb42 /><ref>Template:Harvnb</ref><ref>Template:Harvnb</ref> Let Template:Math be a (left) vector space over the skewfield Template:Math and Template:Math be a reflexive nondegenerate sesquilinear form on Template:Math with companion anti-automorphism Template:Math. If Template:Math is the sesquilinear form associated with a polarity then either:

1. Template:Math (hence, Template:Math is a field) and Template:Math for all Template:Math in Template:Math, that is, Template:Math is a bilinear form. In this case, the polarity is called orthogonal (or ordinary). If the characteristic of the field Template:Math is two, then to be in this case there must exist a vector Template:Math with Template:Math, and the polarity is called a pseudo polarity.<ref>Template:Harvnb</ref>
2. Template:Math (hence, Template:Math is a field) and Template:Math for all Template:Math in Template:Math. The polarity is called a null polarity (or a symplectic polarity) and can only exist when the projective dimension Template:Math is odd.
3. Template:Math (here Template:Math need not be a field) and Template:Math for all Template:Math in Template:Math. Such a polarity is called a unitary polarity (or a Hermitian polarity).

A point Template:Math of Template:Math is an absolute point (self-conjugate point) with respect to polarity Template:Math if Template:Math. Similarly, a hyperplane Template:Math is an absolute hyperplane (self-conjugate hyperplane) if Template:Math. Expressed in other terms, a point Template:Math is an absolute point of polarity Template:Pi with associated sesquilinear form Template:Math if Template:Math and if Template:Math is written in terms of matrix Template:Math, Template:Math.

The set of absolute points of each type of polarity can be described. We again restrict the discussion to the case that Template:Math is a field.<ref>Template:Harvnb</ref>

1. If Template:Math is a field whose characteristic is not two, the set of absolute points of an orthogonal polarity form a nonsingular quadric (if Template:Math is infinite, this might be empty). If the characteristic is two, the absolute points of a pseudo polarity form a hyperplane.
2. All the points of the space Template:Math are absolute points of a null polarity.
3. The absolute points of a Hermitian polarity form a Hermitian variety, which may be empty if Template:Math is infinite.

When composed with itself, the correlation Template:Math (in any dimension) produces the identity function, so it is a polarity. The set of absolute points of this polarity would be the points whose homogeneous coordinates satisfy the equation:

Template:Math.

Which points are in this point set depends on the field Template:Math. If Template:Math then the set is empty, there are no absolute points (and no absolute hyperplanes). On the other hand, if Template:Math the set of absolute points form a nondegenerate quadric (a conic in two-dimensional space). If Template:Math is a finite field of odd characteristic the absolute points also form a quadric, but if the characteristic is even the absolute points form a hyperplane (this is an example of a pseudo polarity).

Under any duality, the point Template:Math is called the pole of the hyperplane Template:Math, and this hyperplane is called the polar of the point Template:Math. Using this terminology, the absolute points of a polarity are the points that are incident with their polars and the absolute hyperplanes are the hyperplanes that are incident with their poles.

Polarities in finite projective planes

By Wedderburn's theorem every finite skewfield is a field and an automorphism of order two (other than the identity) can only exist in a finite field whose order is a square. These facts help to simplify the general situation for finite Desarguesian planes. We have:<ref name="Dembowski 1968 page=153">Template:Harvnb</ref>

If Template:Pi is a polarity of the finite Desarguesian projective plane Template:Math where Template:Math for some prime Template:Math, then the number of absolute points of Template:Pi is Template:Math if Template:Pi is orthogonal or Template:Math if Template:Pi is unitary. In the orthogonal case, the absolute points lie on a conic if Template:Math is odd or form a line if Template:Math. The unitary case can only occur if Template:Math is a square; the absolute points and absolute lines form a unital.

In the general projective plane case where duality means plane duality, the definitions of polarity, absolute elements, pole and polar remain the same.

Let Template:Math denote a projective plane of order Template:Math. Counting arguments can establish that for a polarity Template:Pi of Template:Math:<ref name="Dembowski 1968 page=153"/>

The number of non-absolute points (lines) incident with a non-absolute line (point) is even.

Furthermore,<ref>Template:Citation</ref>

The polarity Template:Pi has at least Template:Math absolute points and if Template:Math is not a square, exactly Template:Math absolute points. If Template:Pi has exactly Template:Math absolute points then;

1. if Template:Math is odd, the absolute points form an oval whose tangents are the absolute lines; or
2. if Template:Math is even, the absolute points are collinear on a non-absolute line.

An upper bound on the number of absolute points in the case that Template:Math is a square was given by Seib<ref>Template:Citation</ref> and a purely combinatorial argument can establish:<ref>Template:Harvnb</ref>

A polarity Template:Pi in a projective plane of square order Template:Math has at most Template:Math absolute points. Furthermore, if the number of absolute points is Template:Math, then the absolute points and absolute lines form a unital (i.e., every line of the plane meets this set of absolute points in either Template:Math or Template:Math points).<ref>Template:Harvnb</ref>

Poles and polars

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File:Reciprocation.svg

Pole and polar with respect to circle C. P and Q are inverse points, p is the polar of P, P is the pole of p.

Reciprocation in the Euclidean plane

A method that can be used to construct a polarity of the real projective plane has, as its starting point, a construction of a partial duality in the Euclidean plane.

In the Euclidean plane, fix a circle Template:Math with center Template:Math and radius Template:Math. For each point Template:Math other than Template:Math define an image point Template:Math so that Template:Math. The mapping defined by Template:Math is called inversion with respect to circle Template:Math. The line Template:Math through Template:Math which is perpendicular to the line Template:Math is called the polar<ref name=comment1>Although no duality has yet been defined these terms are being used in anticipation of the existence of one.</ref> of the point Template:Math with respect to circle Template:Math.

Let Template:Math be a line not passing through Template:Math. Drop a perpendicular from Template:Math to Template:Math, meeting Template:Math at the point Template:Math (this is the point of Template:Math that is closest to Template:Math). The image Template:Math of Template:Math under inversion with respect to Template:Math is called the pole<ref name=comment1 /> of Template:Math. If a point Template:Math is on a line Template:Math (not passing through Template:Math) then the pole of Template:Math lies on the polar of Template:Math and vice versa. The incidence preserving process, in which points and lines are transformed into their polars and poles with respect to Template:Math is called reciprocation.<ref>Template:Harvnb</ref>

In order to turn this process into a correlation, the Euclidean plane (which is not a projective plane) needs to be expanded to the extended euclidean plane by adding a line at infinity and points at infinity which lie on this line. In this expanded plane, we define the polar of the point Template:Math to be the line at infinity (and Template:Math is the pole of the line at infinity), and the poles of the lines through Template:Math are the points of infinity where, if a line has slope Template:Math its pole is the infinite point associated to the parallel class of lines with slope Template:Math. The pole of the Template:Math-axis is the point of infinity of the vertical lines and the pole of the Template:Math-axis is the point of infinity of the horizontal lines.

The construction of a correlation based on inversion in a circle given above can be generalized by using inversion in a conic section (in the extended real plane). The correlations constructed in this manner are of order two, that is, polarities.

Algebraic formulation

File:ProjectiveDuality.PNG

Three pairs of dual points and lines: one red pair, one yellow pair, and one blue pair.

We shall describe this polarity algebraically by following the above construction in the case that Template:Math is the unit circle (i.e., Template:Math) centered at the origin.

An affine point Template:Math, other than the origin, with Cartesian coordinates Template:Math has as its inverse in the unit circle the point Template:Math with coordinates,

<math>\left ( \frac{a}{a^2 + b^2}, \frac{b}{a^2 + b^2} \right).</math>

The line passing through Template:Math that is perpendicular to the line Template:Overbar has equation Template:Math.

Switching to homogeneous coordinates using the embedding Template:Math, the extension to the real projective plane is obtained by permitting the last coordinate to be 0. Recalling that point coordinates are written as column vectors and line coordinates as row vectors, we may express this polarity by:

<math> \pi : \mathbb{R}P^2 \rightarrow \mathbb{R}P^2 </math>

such that

<math> \pi \left ( (x,y,z)^{\mathsf{T}} \right ) = (x, y, -z).</math>

Or, using the alternate notation, Template:Math. The matrix of the associated sesquilinear form (with respect to the standard basis) is:

<math> G = \left (\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{matrix} \right ). </math>

The absolute points of this polarity are given by the solutions of:

<math> 0 = P^{\mathsf{T}} G P = x^2 + y^2 - z^2, </math>

where Template:Math^TTemplate:Math. Note that restricted to the Euclidean plane (that is, set Template:Math) this is just the unit circle, the circle of inversion.

Synthetic approach

File:Conicpolediag.svg

Diagonal triangle Template:Math of quadrangle Template:Math on conic. Polars of diagonal points are colored the same as the points.

The theory of poles and polars of a conic in a projective plane can be developed without the use of coordinates and other metric concepts.

Let Template:Math be a conic in Template:Math where Template:Math is a field not of characteristic two, and let Template:Math be a point of this plane not on Template:Math. Two distinct secant lines to the conic, say Template:Overline and Template:Overline determine four points on the conic (Template:Math) that form a quadrangle. The point Template:Math is a vertex of the diagonal triangle of this quadrangle. The polar of Template:Math with respect to Template:Math is the side of the diagonal triangle opposite Template:Math.<ref>Template:Harvnb</ref>

The theory of projective harmonic conjugates of points on a line can also be used to define this relationship. Using the same notation as above;

If a variable line through the point Template:Math is a secant of the conic Template:Math, the harmonic conjugates of Template:Math with respect to the two points of Template:Math on the secant all lie on the polar of Template:Math.<ref>Template:Harvnb</ref>

Properties

There are several properties that polarities in a projective plane have.<ref>Template:Harvnb</ref>

Given a polarity Template:Pi, a point Template:Math lies on line Template:Math, the polar of point Template:Math if and only if Template:Math lies on Template:Math, the polar of Template:Math.

Points Template:Math and Template:Math that are in this relation are called conjugate points with respect to Template:Pi. Absolute points are called self-conjugate in keeping with this definition since they are incident with their own polars. Conjugate lines are defined dually.

The line joining two self-conjugate points cannot be a self-conjugate line.

A line cannot contain more than two self-conjugate points.

A polarity induces an involution of conjugate points on any line that is not self-conjugate.

A triangle in which each vertex is the pole of the opposite side is called a self-polar triangle.

A correlation that maps the three vertices of a triangle to their opposite sides respectively is a polarity and this triangle is self-polar with respect to this polarity.

History

The principle of duality is due to Joseph Diaz Gergonne (1771−1859) a champion of the then emerging field of Analytic geometry and founder and editor of the first journal devoted entirely to mathematics, Annales de mathématiques pures et appliquées. Gergonne and Charles Julien Brianchon (1785−1864) developed the concept of plane duality. Gergonne coined the terms "duality" and "polar" (but "pole" is due to F.-J. Servois) and adopted the style of writing dual statements side by side in his journal.

Jean-Victor Poncelet (1788−1867) author of the first text on projective geometry, Traité des propriétés projectives des figures, was a synthetic geometer who systematically developed the theory of poles and polars with respect to a conic. Poncelet maintained that the principle of duality was a consequence of the theory of poles and polars.

Julius Plücker (1801−1868) is credited with extending the concept of duality to three and higher dimensional projective spaces.

Poncelet and Gergonne started out as earnest but friendly rivals presenting their different points of view and techniques in papers appearing in Annales de Gergonne. Antagonism grew over the issue of priority in claiming the principle of duality as their own. A young Plücker was caught up in this feud when a paper he had submitted to Gergonne was so heavily edited by the time it was published that Poncelet was misled into believing that Plücker had plagiarized him. The vitriolic attack by Poncelet was countered by Plücker with the support of Gergonne and ultimately the onus was placed on Gergonne.<ref>Template:Harvnb</ref> Of this feud, Pierre Samuel<ref>Template:Harvnb</ref> has quipped that since both men were in the French army and Poncelet was a general while Gergonne a mere captain, Poncelet's view prevailed, at least among their French contemporaries.

See also

• Dual curve

References

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Further reading

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• F. Bachmann, 1959. Aufbau der Geometrie aus dem Spiegelungsbegriff, Springer, Berlin.
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• Coxeter, H. S. M., 1995. The Real Projective Plane, 3rd ed. Springer Verlag.
• Coxeter, H. S. M., 2003. Projective Geometry, 2nd ed. Springer Verlag. Template:ISBN.
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• Greenberg, M. J., 2007. Euclidean and non-Euclidean geometries, 4th ed. Freeman.
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• Hartshorne, Robin, 2000. Geometry: Euclid and Beyond. Springer.
• Hilbert, D. and Cohn-Vossen, S., 1999. Geometry and the imagination, 2nd ed. Chelsea.
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External links

{{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web |_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:DualityPrinciple%7CDualityPrinciple.html}} |title = Duality Principle |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}