Projective Geometry

The projective geometry is the field of mathematics which models the intuitive concepts of Perspective and of horizon . She studies the properties of the unchanged figures by projection.

Historical considerations

The projective geometry finds its origins in the work of Pappus (ive after Jesus-Christ) which introduces the Anharmonic ratio and refers to a work of Apollonius de Perga.

She was then studied in XVII^e century by mathematicians like Pascal or Desargues, before falling into the lapse of memory. It is Poncelet in its treated geometrical properties of the figures which gives him its true rise in the beginnings of the XIX° century, on the basis of considerations of pure geometry. Indeed the geometry refines would not have allowed this discovery since it prohibited the intersection of the parallel straight lines, essential concept in projective geometry.

Consequently the pure geometry strongly will prevail during all the XIX° century until analytical methods are finally discovered by August Ferdinand Möbius and Julius Plücker. But it is Felix Klein which, at the end of XIX^e century, clarifies the bond between projective geometry and Euclidean geometry.

It is as at the same time as took place a major conceptual evolution; previously the geometry was the science of the figures, the geometricians of the turning of the century concentrated on the transformations of the aforesaid figures, the internal laws of composition of the various transformations, the structure of certain groups of transformations (questions of commutation, associativeness, the reverse transformation, etc), the invariants of such or such family of transformations, the minimal axioms allowing these properties of transformations. She today is largely used by the systems of vision per computer and of made graphic (OpenGL).

Elementary outline

For those which wish only one elementary outline of what is the Projective Geometry compared to the ordinary Euclidean Geometry one can say that the Projective Geometry is the science of the figures which are traced only with the rule alone whereas the Euclidean geometry is to some extent the science of the figures which are traced with the rule and the compass.

The Projective Geometry is unaware of the parallel straight lines, the perpendicular lines, the isométries, the circles, the right-angled triangles, isosceles, equilateral, etc In its definition it comprises less axioms than the Euclidean geometry and consequently it is more general.

Finally it is remarkable by the fact that it is possible to pose certain conventions of language (for example to call parallels two lines which are cut on a selected line of the plan) which makes it possible by the projective geometry to find the results of the geometry closely connected. (See Ci below)

Projective space

See detailed article: projective Space.

A projective space is defined in mathematics like the whole of the vectorial right of a vector space; one can imagine the eye of an observer placed on the origin of a vector space, and each element of projective space corresponds to a direction of its glance.

A projective space is dissociated from a vector space by its homogeneity : one can distinguish in his center no particular point like the origin from a vector space. In that it approaches a Espace refines.

Vectorial definition

That is to say $E\; \backslash ,\; \backslash !$ a vectorial K-space (K is a body, in general $\backslash \; R\; \backslash ,\; \backslash !$ or $\backslash \; mathbb\; \{C\}\; \backslash ,\; \backslash !$), nonreduced to $\backslash \; \{0\; \backslash \}$. One defines on $E\; -\; \backslash \; \{0\; \backslash \}\; \backslash ,\; \backslash !$ the Relation of following equivalence:

$x\; \backslash \; sim\; there\; \backslash \; Leftrightarrow\; \backslash \; exists\; \backslash \; lambda\; \backslash \; in\; K^*,\; x=\; \backslash \; lambda\; there\; \backslash ,\; \backslash !$.

Then one calls projective space on $E\; \backslash ,\; \backslash !$ the unit quotient of $E\; -\; \backslash \; \{0\; \backslash \}\; \backslash ,\; \backslash !$ by the relation of equivalence $\backslash \; sim\; \backslash ,\; \backslash !$: $P\; (E)\; =\; (E\; -\; \backslash \; \{0\; \backslash \})/\backslash \; sim\; \backslash ,\; \backslash !$.

For each element $x\; \backslash \; neq\; 0\; \backslash ,\; \backslash !$ of $E\; \backslash ,\; \backslash !$ one will note $\backslash \; pi\; (X)\; \backslash \; in\; P\; (E)\; \backslash ,\; \backslash !$ its class of equivalence: $\backslash \; pi\; (X)\; =\; \backslash \; \{\backslash \; lambda\; X,\; \backslash \; lambda\; \backslash \; in\; K\; \backslash \}\; \backslash ,\; \backslash !$. One thus has: $\backslash \; pi\; (X)\; =\; \backslash \; pi\; (there)\; \backslash ,\; \backslash !$ if and only if $x\; \backslash ,\; \backslash !$ and $y\; \backslash ,\; \backslash !$ is colinéaires.

The application $\backslash \; pi:\; E\; \backslash \; rightarrow\; P\; (E)\; \backslash ,\; \backslash !$ is called canonical projection .

More simply projective space $P\; (E)\; \backslash ,\; \backslash !$ is the vectorial line whole of $E\; \backslash ,\; \backslash !$; the element $\backslash \; pi\; (X)\; \backslash ,\; \backslash !$ of projective space is the vectorial line of $E\; \backslash ,\; \backslash !$ whose directing vector is $x\; \backslash ,\; \backslash !$.

If $E\; \backslash ,\; \backslash !$ is of finished size $n\; \backslash ,\; \backslash !$ then one says that $P\; (E)\; \backslash ,\; \backslash !$ is of finished size and one notes $n-1=dim\; \backslash ,\; P\; (E)\; \backslash ,\; \backslash !$ the dimension of projective space. In particular:

• If n=1 then $P\; (E)\; \backslash ,\; \backslash !$ is a singleton (null dimension);
• If n=2 then $E\; \backslash ,\; \backslash !$ is a vectorial plan and $P\; (E)\; \backslash ,\; \backslash !$ is called projective Droite.
• If n=3 then $P\; (E)\; \backslash ,\; \backslash !$ is called projective Plan; it is the framework more running to make geometry.

If space $E\; \backslash ,\; \backslash !$ is the vector space of dimension $n\; \backslash ,\; \backslash !$ “typical”, i.e. $K^n\; \backslash ,\; \backslash !$ then one has a particular notation for projective space: $P^\; \{n-1\}\; (K)\; \backslash ,\; \backslash !$ instead of $P\; (K^n)\; \backslash ,\; \backslash !$.

Definition closely connected

This very formal definition of a projective space should not make forget that this concept was born from the central Projection and is, above all, a geometrical concept. To take the example of the projective space of $\backslash \; mathbb\; \{R\}\; ^3$, one can observe the drawing opposite where the points $m$, $n$ and $r$ belong to the plan $(P\text{'})$. It is necessary to imagine an observer placed in $O$. This observer sees all the points of the right-hand side $(OM)$ in $m$, those of the right-hand side $(GOLD)$ in $r$ and those of the right-hand side $(ONE)$ in $n$. the lines $(d)$ of the plan $(P)$ are not seen like points of $(P\text{'})$. There is thus bijection between the vectorial lines of $\backslash \; mathbb\; \{R\}\; ^3$ not parallels with $(P)$ and the points of the plan $(P\text{'})$.

The projective space of $\backslash \; mathbb\; \{R\}\; ^3$ is thus in bijection with the plan closely connected $(P\text{'})$ to which one adds the vectorial line whole of $(P)$. A projective Plan $\backslash \; tilde\; \{P\text{'}\}$ thus consists of a plan refines $(P\text{'})$ which contains the whole of the clean points of $\backslash \; tilde\; \{P\text{'}\}$ with which one associates all the vectorial lines (or directions) of $(P\text{'})$. Each point of the second unit is called not unsuitable $\backslash \; tilde\; \{P\text{'}\}$ or not ad infinitum .

This concept allows, for example, to speak, in a plan, of intersection between two unspecified lines: the lines will be secant in a clean point of $(P\text{'})$ or in an unsuitable point if the lines are parallel.

This concept spreads with any projective space $\backslash \; P$ tilde of dimension $n$: it is a space closely connected $(P)$ of dimension $n$ with which one associates the whole of the directions of $(P)$.

In particular, if $(P)$ = $K$, the associated projective line is the unit $\backslash \; tilde\; \{K\}\; =\; K\; \backslash \; cup\; \{\backslash \; infty\}\; \backslash ,\; \backslash !$ where $\backslash \; infty$ is a point external with $K\; \backslash ,\; \backslash !$, prolonging the algebraic operations in the following way:

• for all $x$ of $K$, $x/\backslash \; infty\; =0$
• for all $x$ of $K^*$, $x/0\; =\; \backslash \; infty$

This double relation, on the one hand with a quotienté vector space, on the other hand with a space refines supplemented makes the richness of the study of the projective geometry. In the same way, this double aspect will be important to preserve when it is a question of giving coordinates to the points of projective space.

Definition of space closely connected

Conversely, by the introduction of the unsuitable elements (an unsuitable plan, in fact), Desargues showed that space refines falls under projective space. The definition of space closely connected is extremely simple, since it consists of the elimination of these unsuitable elements.

See also: Arguesian Geometry

Location

Homogeneous coordinates

See detailed article: homogeneous Coordinates.

In a projective space of dimension $n$, therefore associated with a vector space of dimension $n\; +\; 1$, each point $m$ of $P\; (E)$ is associated with a family of vectors of $E$ all colinéaires. If $E$ is provided with a canonical base, one calls coordinated homogeneous of the point $m$, the coordinates of an unspecified vector $x$ such as $\backslash \; pi\; (X)\; =\; m\; \backslash ,$. A point thus has a family of coordinates all proportional between them. In other words, if $(x\_1,\; x\_2,\; .....,\; x\_\; \{n+1\})\; \backslash ,$ is a homogeneous frame of reference of $m$, it is the same of $(kx\_1,\; kx\_2,\; .....,\; kx\_\; \{n+1\})\; \backslash ,$ for any element $k$ not no one of $K$.

Among all these coordinates, it often happens that one privileges some to find a space closely connected of dimension $n$. Among all the representatives of $m$, one privileges that whose last coordinate, for example, is worth $1$. In other words, one projected space in the hyperplane of equation $x\_\; \{n+1\}\; =\; 1\; \backslash ,$. If $(x\_1,\; x\_2\dots ,\; x\_\; \{n+1\})\; \backslash ,$ is a frame of reference of $m$, one privileges the frame of reference $(\{x\_1\; \backslash \; over\; x\_\; \{n+1\}\},\; \{x\_2\; \backslash \; over\; x\_\; \{n+1\}\},\dots ,\; \{x\_n\; \backslash \; over\; x\_\; \{n+1\}\},\; 1)\; \backslash ,$. That is worth obviously only if $m$ is a clean point of $P\; (E)$.

The unsuitable points are represented by homogeneous frames of reference whose last coordinate is null.

One then notices well there the correspondence between

• the point clean of $P\; (E)$ and the points of a space closely connected of dimension $n$
• the unsuitable points of $P\; (E)$ and the directions of a vector space of dimension $n$

To arbitrarily choose to put a coordinate at $1$ in the homogeneous coordinates makes it possible to define different charts.

Locate of a projective space

See the detailed article: projective Reference mark.

A vector space of dimension N is located by a base of N independent vectors. A space refines dimension N is located using N + 1 nondependant points. A projective space of dimension N is located using n+2 points. One could think that n+1 points would be sufficient by taking for example $(\backslash \; pi\; (e\_1),\; \backslash \; pi\; (e\_2),\dots ,\; \backslash \; pi\; (e\_\; \{n+1\}))\backslash ,$ where $(e\_i)\; \_\; \{I\; \backslash \; in\; \backslash \; \{1;\; n+1\; \backslash \}\}$ form a base of the vector space of dimension n+1 associated with projective space. The punctual coordinates m in this reference mark would be then $(x\_1,\dots ,\; x\_\; \{n+1\})\; \backslash ,$ where $(x\_1,\dots ,\; x\_\; \{n+1\})\; \backslash ,$ is the coordinates of X such as $\backslash \; pi\; (X)\; =\; m\; \backslash ,$ but it would be necessary that these coordinates are independent of the representative chosen for the vectors of the base: $\backslash \; pi\; (e\_1)\; \backslash ,$, for example, has another representative who is $2e\_1\; \backslash ,$. And in the base $(2e\_1,\; e\_2,\dots ,\; e\_\; \{n+1\})\; \backslash ,$ X does not have the same frame of reference $(x\_1/2,\; x\_2,\dots ,\; x\_\; \{n+1)\}\backslash ,$.

It is thus necessary to prevent this ambiguity and to limit the choice of other representatives of the basic vectors to vectors colinéaires for the precedents but of the same coefficient of colinearity. It is enough for that to define one n+2ième point corresponding to $\backslash \; pi\; (e\_1\; +\; e\_2\; +\dots \; +\; e\_\; \{n+1\})\; \backslash ,$. Thus, if one chooses other representatives of $\backslash \; pi\; (e\_1)\dots \; \backslash \; pi\; (e\_\; \{n+1\})\; \backslash ,$ with different coefficients of colinearity, the vector $k\_1e\_1\; +\dots \; +\; k\_\; \{n+1\}\; e\_\; \{n+1\}\; \backslash ,$ will not be any more one representative of $\backslash \; pi\; (e\_1\; +\; e\_2\; +\dots \; +\; e\_\; \{n+1\})\; \backslash ,$.

Projective subspace

See the detailed article: projective Subspace.

As there exist vectorial subspaces of vector space as well as subspaces closely connected of space refines, there exist the same projective subspaces of projective space. They are consisted of projected vectorial subspaces of the associated vector space. One will thus speak projective line in a projective plan, of projective plan in a projective space. The rule of dimensions and the existence of points ad infinitum make it possible to simplify the rules of incidence.

Birapport on a projective line

See detailed article: Anharmonic ratio.

If has, B, C and D are 4 projective line points (has, B and C distinct) D, it exists a single isomorphism of D on $\backslash \; K$ tilde, $f\_\; \{has,\; B,\; C\}$ such as

• $f\_\; \{has,\; B,\; C\}\; (A)\; =\; \backslash \; infty$
• $f\_\; \{has,\; B,\; C\}\; (b)\; =\; 0\; \backslash ,$
• $f\_\; \{has,\; B,\; C\}\; (c)\; =\; 1\; \backslash ,$

One calls birapport of has, B, C, D, noted the value of $f\_\; \{has,\; B,\; C\}\; (d)$.

If has, B, C and D are 4 clean points distinct from D, one finds the old definition of the birapport or anharmonic ratio:

$\backslash \; frac\; \{\backslash \; overline\; \{Ca\}\; \backslash ,/\backslash ,\; \backslash \; overline\; \{cb\}\}\; \{\backslash \; overline\; \{da\}\; \backslash ,/\backslash ,\; \backslash \; overline\; \{dB\}\}$

Projective transformation or homography

See the article: Projective application.

The projective transformations or homographies are transformations studied into projective geometry. They are obtained like made up of a finished number of central projections. They describe what arrives at the positions observed of various objects when the eye of the observer changes place. The projective transformations do not preserve by always the distances nor the angles but preserves the properties of incidence and the birapport - two important properties in projective geometry. One finds transformations projective on lines, in plans and space.

fundamental Property : In finished dimension, a projective transformation is entirely determined by the image of a reference mark of projective space.

Analytical definition of an homography

$\backslash \; mathcal\; P\_1$ and $\backslash \; mathcal\; P\_2$ are 2 projective spaces respectively associated with the vector spaces $\backslash \; quad\; E\_1$ and $\backslash \; quad\; E\_2$. One indicates by $\backslash \; quad\; \backslash \; pi\_1$ and $\backslash \; quad\; \backslash \; pi\_2$ the canonical projections of $\backslash \; quad\; E\_1$ (resp. $\backslash \; quad\; E\_2$) on $\backslash \; mathcal\; P\_1$ (resp.$\backslash \; mathcal\; P\_2$).

One can then carry out a “passage to the quotient” of the linear applications injective of $\backslash \; quad\; E\_1$ in $\backslash \; quad\; E\_2$. Such a linear application $\backslash \; quad\; \backslash \; varphi$ being given one can define an application $\backslash \; quad\; h$ of $\backslash \; mathcal\; P\_1$ in $\backslash \; mathcal\; P\_2$ transforming the point $\backslash \; quad\; M$ into $h\; (M)\; =\; \backslash \; pi\_2\; \backslash \; circ\; \backslash \; varphi\; (m),\; \backslash \; quad\; m$ naming a representative of $\backslash \; quad\; M$. Naturally so that this definition is coherent, we must check that it does not depend on the representative chosen, which is immediate considering the linearity of $\backslash \; varphi$ and the definition of $\backslash \; quad\; \backslash \; pi\_2$.

the application $\backslash \; quad\; h$ is the homography associated with $\backslash \; quad\; \backslash \; varphi$. It is in a more concise way defined by the equality: $\backslash \; pi\_2\; \backslash \; circ\; \backslash \; varphi\; =\; H\; \backslash \; circ\; \backslash \; pi\_1$.

One can also speak more generally about projective application, by not requiring the injectivity of the linear application $\backslash \; quad\; \backslash \; varphi$ initial; the same process of passage to the quotient will provide an application only defined on part of $\backslash \; mathcal\; P\_1$: $\backslash \; mathcal\; P\_1-\; \backslash \; pi\_1^\; \{-\; 1\}\; (\backslash \; ker\; (\backslash \; quad\; \backslash \; varphi))$, and with values in $\backslash \; mathcal\; P\_2$. One will not speak then about homography.

There exists an infinity of linear applications associated with an homography but these linear applications form a vectorial line of $\backslash \; mathcal\; L\; (E\_1,\; E\_2)$ since $\backslash \; quad\; h\_1=h\_2$ involves $\backslash \; pi\_2\; \backslash \; circ\; \backslash \; varphi\; \_1=\; \backslash \; pi\_2\; \backslash \; circ\; \backslash \; varphi\; \_2$.

In finished dimensions p, N, if one has a homogeneous frame of reference, an homography could be defined by a class of nonnull matrices of format (n+1) * (p+1) all multiples of the one of them. With being one of these matrices and X matrix-columns of homogeneous coordinates of $\backslash \; quad\; M$, AX will be matrix column of homogeneous coordinates of $\backslash \; quad\; H\; (M)$ (all this being thus defined except for a factor).

; Example and discussion (plane geometry).

We take for $\backslash \; quad\; E\_1$ and $\backslash \; quad\; E\_2$ space $\backslash \; mathbb\; R^3$. $\backslash \; mathcal\; P\_1\; =\; \backslash \; mathcal\; P\_2$ is the projective plan $\backslash \; mathcal\; P$. Let us consider an homography $\backslash \; quad\; h$ defined by the matrix 3*3 has that we suppose diagonalisable . One can thus calculate the homogeneous coordinates of transformed of any point.

the 3 clean directions are independent and define 3 points invariants by $\backslash \; quad\; h$ of $\backslash \; mathcal\; P$. These 3 points respectively have like matrix-column of homogeneous coordinates $X\_1,\; X\_2,\; X\_3$ (clean vectors of the matrix, except for a factor not no one).

Inversement the knowledge of does these 3 points invariants determine the homography, i.e. has , except for a factor? For that it would be necessary to be able to calculate the eigenvalues of has (with a factor of proportionality close always). However there is obviously no means for that while not knowing that the clean directions.

On the other hand if one gives oneself for example transformed point of homogeneous coordinates $X\_1+X\_2+X\_3$ into the point of homogeneous coordinates $Y$, one will have while indicating by $\backslash \; lambda\_1,\; \backslash \; lambda\_2,\; \backslash \; lambda\_3$ the eigenvalues of a: $\backslash \; lambda\_1X\_1+\; \backslash \; lambda\_2X\_2+\; \backslash \; lambda\_3X\_3=kY,\; \backslash \; quad\; K$ unspecified not no one, which makes it possible well to calculate by solving the system the eigenvalues except for a proportionality factor.

the 4 points (the 3 points invariants plus the 4th definite one above) define a projective reference mark (see higher) and the knowledge of the transformation of this projective reference mark entirely determines the homography.

; Example of homography

the transformations by polar reciprocal.

Topology

See the detailed article: Topology in projective geometry.

If E is a vector space on $\backslash \; mathbb\; R$ or $\backslash \; mathbb\; C$ of finished size, one can define on E a Topologie resulting from the distance induced by the standard $||X||\; =\; \backslash \; sqrt\; \{x\_1^2\; +\; x\_2^2\; +\dots .\; +\; x\_n^2\}$ in the real case and $||X||\; =\; \backslash \; sqrt\; \{x\_1\; \backslash \; overline\; \{x\_1\}\; +\; x\_2\; \backslash \; overline\; \{x\_2\}\; +\dots .\; +\; x\_n\; \backslash \; overline\; \{x\_n\}\}$ in the complex case.

This topology makes it possible to define on space quotient $P\; (E)\; =E-\; \{0\}/\backslash \; sim$ a topology, said Topology quotient. If $p:\; E\; \backslash \; setminus\; \backslash \; \{0\; \backslash \}\; \backslash \; rightarrow\; P\; (E)$ indicate the application of passage to the quotient, one will say that a part $A\; \backslash \; subset\; P\; (E)\; \backslash ,$ open is if its reciprocal image $p^\; \{-\; 1\}\; (A)\; \backslash ,$ is open in $E\; \backslash \; setminus\; \backslash \; \{0\; \backslash \}\; \backslash ,$. It is checked that one defines a topological Espace well thus

It is shown that $P\; (E)\; \backslash ,$ is compact.

One will thus provide projective space P (E) with this topology. It makes it possible to speak of Homéomorphisme and to notice, for example, that the real projective line is homeomorphic with a circle, the projective line complexes being homeomorphic with a sphere (see the article Sphère of Riemann for a homeomorphism explicit).

Duality

See the detailed article: Duality (projective geometry).

If E is a vectorial K-space of finished size N, its dual E* is also a vectorial K-space of dimension N. One can thus associate with projective space P (E), his dual P (E*). A line of P (E*) will correspond to a beam of hyperplanes in P (E). The passage to dual makes it possible to reverse a great number of geometrical properties.

For what is used the projective geometry?

1. the projective geometry made it possible to largely simplify theorems of plane geometry like the Théorème of Pappus or the Théorème of Desargues.

2. Provided with a topology, projective space is the starting point of the study of the differential Géométrie.
3. Lastly, with the development of the representation in 2D of objects in 3D, the projective geometry showed the power of the drawing tools computer-assisted which were installed.
4. If the projective space, compared with usual space, i.e. space refines, can seem to be a more complicated object, it is undeniable that for many situations, projective space is the good framework to work. To give an example, if $C$ and $C\text{'}$ are two plane curves (complex) of respective degree $d$ and $d\text{'}$ then, if one sees these curves like subvarieties of the plan refines, the Théorème of Bezout says that the number of points of intersection between $C$ and $C\text{'}$ is always lower or equal to $dd\text{'}$. On the other hand, if one sees these curves like subvarieties of the projective plan, then the theorem says that the number of points of intersection (taken into account multiplicity) is equal to $dd\text{'}$. There are many other situations where the theorems are stated in a more beautiful form in projective geometry.
5. projective space is similar to a closing of space closely connected.
6. In addition to the utility aspects, one can insist on the mental gymnastics and the esthetic feeling of perfection which get certain theorems and some Axiomes of projective plans.

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