Duality (projective geometry)

The projective duality, creates by Jean-Victor Poncelet (1788 - 1867), founding father of the projective Géométrie, although much less taught than the duality in linear algebra, is probably the most beautiful concept of Dualité which one meets in mathematics.

It is a question of formalizing the very simple observation that there is an analogy - a duality, precisely - between the fact that by two distinct points passes a line and only one , and the fact that two distinct lines cut in a point and only one (with the proviso of precisely placing itself in projective geometry, so that two parallel straight lines meet in a point ad infinitum).

Duality in a projective plan

Definition

Contrary to the traditional plane geometry where the lines are whole of points, it is to better consider in projective geometry that the projective plan $P$ consists of a whole of points $\backslash \; \backslash \; mathcal\; P\; (P)$, of a whole of right-hand sides $\backslash \; mathcal\; D\; (P)$, and of a relation indicating which points are on which line (or which lines pass by which point). For understanding well that it is this relation which is important and not the nature of the points and the right-hand sides, the Hilbert mathematician said: “It is always necessary to be able to say " table" , " chaise" and " glass of bier of bière" in the place of " point" , " droite" and " plan" ”!

We consider initially that the projective plan $P$ is definite in an axiomatic way; it is noted whereas one obtains another projective plan by considering the $P*$ object of which " points" are the lines of $P$ and the " droites" are the points of $P$, a line of $P*$ (which is a point $M$ of $P$) passing by a " point" of $P*$ (which is a line $D$ of $P$) when $D$ passes by $M$.

To simplify, instead of working on 2 different levels, $P$ and $P*$, one can be satisfied to work on only one projective level $P$.

Correlation or duality, déf: a correlation is a transformation of the points of the plan into right-hand sides and lines of the plan in points and which respects the incidence.

Polarity, déf: a duality is an involutive correlation, i.e. its square is the identical transformation.

Examples

Any configuration of points and right-hand sides in $P$ corresponds then in $P*$ a dual configuration obtained by exchanging the points and the lines, and the same, to any theorem in $P$, corresponds a dual theorem. Here some examples:

Note: If one agrees to identify a line with the whole of his points, it is necessary, so that the duality is perfect, to identify a point with the whole of the right-hand sides which pass by this point, in other words, to identify a beam of right-hand sides with its pole!

Duality and birapport

Dualities, correlations and polarities

Let us consider the homographies $P$ on $P*$; in fact bijections $f$ of $\backslash \; mathcal\; P\; (P)$ on $\backslash \; mathcal\; P\; (P^*)\; =\; \backslash \; mathcal\; D\; (P)$ transform a line of $P$ into a " droite" of $P*$; one can thus prolong them in a bijection, always noted $f$, of $\backslash \; mathcal\; P\; (P)\; \backslash \; cup\; \backslash \; mathcal\; D\; (P)$, which transforms a point into a line and reciprocally, and which checks: $M\; \backslash \; in\; D\; \backslash \; yew\; F\; (D)\; \backslash \; in\; F\; (M)$.

Such applications are called dualities or correlations ; when they are involutive ($f=f^\; \{-\; 1\}$), they are called polarities or formerly " transformations by polar réciproques". In this last case, the image of a point is called the polar of this point, and the image of a line, its pole .

According to the fundamental Theorem of the projective geometry, in the real case any duality comes from an homography (in the general case, of an semi-homography).

Theorem of incidence and reciprocity

There are two important theorems which rise from the definitions.

Theorem of incidence: If the point has is incidental with the right-hand side D, then the dual point of D is incidental with the dual right-hand side of A.

Polar theorem of reciprocity: If the point has is on polar point B, then B is on the polar one of A. This theorem is more powerful than the precedent.

Relations with the duality in linear algebra

It is known that there exists a bijection between the points of $P$ and the vectorial lines of a vector space $E$ of dimension 3, and a bijection between the lines of $P$ and the vectorial plans of $E$ (a point pertaining on a line if the vectorial line is included in the vectorial plan).

Orthogonality between $E$ and its dual $E*$, together of the linear forms on $E$, which with any vectorial subspace of $E$ associate a vectorial subspace of $E*$ induces a bijection between the vectorial plans of $E$ and the vectorial lines of $E*$, and between the vectorial lines of $E$ and the vectorial plans of $E*$, which reverses inclusions.

There thus exists a canonical bijection between the points and right-hand sides of $P*$ and the lines and vectorial plans of $E*$ which respects the incidences: if a projective plan $P$ is associated with a vector space $E$, the dual plan $P*$ is well associated with the dual vector space $E*$.

Dual of an homography

An homography $f$ of the projective plan in him even is a bijection in the whole of the points of $P$, which induces a bijection $f*$ in the whole of the right-hand sides of $P$, which is the whole of the " points" of $P*$: $f*$ is the dual homography of $f$ (notice that $f*\; (D)=f\; (D)$!) ; it is checked that if $f$ comes from an automorphism $\backslash \; vec\; f$ of $\backslash \; vec\; E$, then $f*$ comes from the automorphism of $\backslash \; vec\; dual\; E^*$ of $\backslash \; vec\; f$, more often called automorphism transposed of $f$.

Use of the coordinates

Let us bring back the projective plan $P$ to a projective reference mark $R=\; (A\_1,\; A\_2,\; A\_3,\; \backslash \; Omega)\; \backslash ,$, which is associated with a base $B=\; (e\_1,\; e\_2,\; e\_3)\; \backslash ,$ of the vector space $E$; let us consider the isomorphism $\backslash \; varphi$ between $E$ and its dual which transforms $B$ into the dual base $B^*=\; (e\_1^*,\; e\_2^*,\; e\_3^*)\; \backslash ,$, which induces a duality between $P$ and $P*$; at a point $M$ of $P$ a vector defined in a multiplicative constant close to coordinates $is\; associated\; (has,\; B,\; c)$ in $B$ (homogeneous coordinates of $M$ in $R$), with which by $\backslash \; varphi$ the linear form $ax+by+cz$ is associated whose core is the plan of equation $ax+by+cz=0$;

this equation is the homogeneous equation of the right-hand side $D$ image of $M$ by the duality; it is checked that conversely, the image of $D$ is $M$, with the result that this duality is a polarity, defined by:

point of homogeneous coordinates $(has,\; B,\; c)$ <--> right of homogeneous equation: $ax+by+cz=0$

the dual reference mark $R=\; (A\_1^*,\; A\_2^*,\; A\_3^*,\; \backslash \; Omega^*)\; \backslash ,$ of $R$, associated with $B*$ is formed the respective lines of equation: $x=0,\; y=0,\; z=0,\; x+y+z=0$, therefore $A\_1^*=\; (A\_2A\_3),\; A\_2^*=\; (A\_3A\_1),\; A\_3^*=\; (A\_1A\_2)\; \backslash ,$. Let us notice that a point and its polar line have same homogeneous coordinates, one in $R$, the other in $R*$.

Duality associated with a bilinear form, polarity associated with a quadratic form or conical

That is to say $f$ a duality of $P$ towards $P*$ coming from an isomorphism $\backslash \; vec\; f$ of $E$ towards $E*$. It is associated with this last a bilinear form not degenerated on $E$, defined by $\backslash \; varphi\; (X,\; there)\; =\; \backslash \; vec\; F\; (there)\; (X)$ (noted by the hook of duality ) and this correspondence is bijective; the duality $f$ known as is associated with the bilinear form $\backslash \; varphi$ (defined except for a multiplicative constant). Let us note that the matrix of isomorphism $\backslash \; vec\; f$ in a base $B$ and the dual base $B*$ is that of the bilinear form $\backslash \; varphi$dans $B$.

The duality $f$ is a polarity so for all points $M$ and $N$: $M\; \backslash \; in\; F\; (NR)\; \backslash \; yew\; NR\; \backslash \; in\; F\; (M)$, which is translated on the bilinear form $\backslash \; varphi$ by: for all vectors $x$ and $y$: $\backslash \; varphi\; (X,\; there)\; =0\; \backslash \; yew\; \backslash \; varphi\; (there,\; X)\; =0$; it is shown that this last condition is equivalent so that $\backslash \; varphi$ is symmetrical or antisymmetric (if the body is of characteristic different from 2).

All quadratic Forme on $E$ generates a symmetrical bilinear form, which generates of it a polarity in $P$, which known as is associated with $q$. The isotropic cone of $q$ (defined by $q\; (X)\; =\; \backslash \; varphi\; (X,\; X)\; =0$) is a cone of the second degree of $E$ which generates conical projective a $(C)$ in $P$. One then says by abuse that the polarity $f$ associated with $q$ is the polarity compared to (C) . Let us notice that one has then: $M\; \backslash \; in\; (C)\; \backslash \; yew\; M\; \backslash \; in\; F\; (M)$.

Polarity compared to a circle in the Euclidean plan

Let us consider a circle (C) of center O of radius has of an Euclidean plan $\backslash \; breve\; P$ brought back to a reference mark orthonormé $(O,\; e\_1,\; e\_2)\; \backslash ,$; $P$ the is supplemented projective one of $\backslash \; breve\; P$ and $E$ its vectorial envelope, brought back to $(O,\; e\_1,\; e\_2,\; e\_3)\; \backslash ,$.

The Cartesian equation of the circle is $x^2+y^2=a^2$; the polarity $f$ compared to $(C)$ is thus associated with the quadratic form $X^2+Y^2-a^2Z^2$ of $E$ and the isomorphism of $E$ on $E*$ is that which sends $(O,\; e\_1,\; e\_2,\; e\_3)\; \backslash ,$ on $(O,\; e\_1^*,\; e\_2^*,\; -\; ae\_3^*)$

From the point of view of the plan refines $\backslash \; breve\; P$, the polarity $f$ has a very simple definition: to the point of coordinates $(U,\; v)$ corresponds the line of equation $ux+vy=a^2$ and the image of a point ad infinitum is the line passing by $O$ and perpendicular to the direction of the point.

Duality between curves

Configuration of Pappus, detailed example of duality

To illustrate an unspecified duality geometrically, it is necessary to define the process by which one transforms a point into right-hand side. A simple example of duality is given below: one takes quadrangle (4 points) ACZF, one transforms it into quadrilateral (4 lines) aczf, and to supplement the figure a little the straight lines AC, CZ, ZF of the starting figure were plotted, as well as the points of intersection a*c, c*z and z*f of the figure of arrival.

Continuing the drawing of the same example, one can appear the duality of a configuration of Pappus, to see Théorème of Pappus. The starting configuration is formed of the 9 points: AEC DBF XYZ, the configuration of arrival is given by the 9 lines aec dbf xyz. In the starting configuration one took care to supplement the figure by the 9 lines uniting the points, they are the right-hand sides jnp qhk and mgr; in the same way in the configuration of arrival the intersection of the right-hand sides give rise to 9 points JNP QHK Mgr.

For more details, to also see Duality (projective geometry) /Exemples concrete of dual and polar constructions

Duality in a projective space of finished size

It is the generalization of what we have just seen in the plan; in dimension $n$, not only the duality exchanges the points and the hyperplanes, but more generally the subspaces of dimension $k$ with those of dimension $n-1-k$.

For example, in dimension 3, the points are exchanged with the plans, and the lines with themselves. The dual theorem of: " by two points distincts" passes a line and only one becomes: " two distinct plans are cut in a droite". A tetrahedron of tops $ABCD$ becomes by duality a tetrahedron of faces $ABCD$; in the first case, the points $A$ and $B$ determine an edge (that which passes by $A$ and $B$, and in the second also (the intersection of $A$ and $B$).

More precisely dual the $E*$ of a projective space $E$ of dimension $n$ is the space whose subspaces of dimension $k$ are those of dimension $n-1-k$ of $E$, and a duality on $E$ is a bijection of the whole of the projective subspaces of $E$ in itself which reverses inclusions and transforms a subspace of dimension $k$ into one of dimension $n-1-k$; in the real case, a duality comes from an homography of $E$ on $E*$ (of an semi-homography in the general case).

All that was seen in the plane case generalizes here, in particular the concept of polarity compared to conical which becomes here that of polarity compared to hyperquadrique (not degenerated).

References

• Jean Frenkel, Geometry for the pupil professor, Hermann, 1973
• Jean-Claude Sidler, Geometry projective, Interéditions, 1993
• Alain Bigard, Geometry, Masson, 1998
• H.S.M. Coxeter, Projective geometry, Springer, 1998 (3rd edition); it is in Canadian English very easy to include/understand, and the pages on the duality and the polarity are clear, although very abstract.
• Yves Ladegaillerie, Geometry, Ellipses, 2003

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