The standard model of the projective plane (often denoted by PR2 or P2), consists of the classes X= [x] = [x1, x2, x3] of points of R3 modulo non-zero multiplicative constants. For every non-zero vector a of R3, A=[a] denotes a point of the projective plane and a is called a representative of the point. Two representatives a, a' defining the same point if and only if a' = ka, with a non-zero real number k.
The most important shapes of the projective plane are its lines consisting of all points X=[x], whose representatives x lie on a plane a1x1+a2x2+a3x3 = 0 of R3, the plane passing through the origin. Since every plane is defined by a tripple of coefficients [a]=[a1,a2,a3] modulo a non-zero multiplicative constant, we see that the set of lines of the projective plane is itself a projective plane. This is called the dual projective plane and denoted by P*2.
The most important invertible transformations of the plane onto itself are the projective transformations or projectivities or homographies (I use occasionally all these terms). These are defined by classes [A] of 3x3 real invertible matrices A and their action on representatives of points: [A][x] = [Ax]. The transformation is well defined and the composition of them reduces to matrix multiplication of the corresponding matrices.
Shapes of particular importance of the projective plane are the algebraic curves defined by equations Sp = {[x] | p(x)=0}, where p(x) = p(x1,x2,x3) is a homogeneous polynomial of three variables i.e. a polynomial satisfying p(kx) = krp(x). r being the degree of the polynomial. The homogeneity of the polynomial implies that p(x)=0 is a condition on the class [x] and not on its representative. Special cases of such curves are the lines, for which the degree r=1 and the conics for which the degree r=2.
Thus, (projective) conics are defined by equations of the form
The matrix representation explains the two's of the previous equation. A conic having the above symmetric matrix non singular is called proper, otherwise is called degenerate. Proper conics correspond to irreducible polynomials of degree two, whereas degenerate correspond to reducible polynomials i.e. products of two linear factors representing two lines or one double line (if the factors are identical).
Obviously every pair of distinct lines has an intersection point. This is radically different from euclidean geometry which has parallel i.e. non-intersecting lines. The other obvious property is that two distinct points A([a]) and B([b]) define a unique line. By the elementary properties of determinants this line is given by the equation:
Analogous questions for higher degree curves are non trivial and need some theory to get to the answer. In particular the equation of a conic involving six coefficients, but determined modulo a non-zero multiplicative constant, one would expect that five points suffice to define a conic through them. This is indeed so as explained in the file ConicsThroughFivePts.html . Equivalence between algebraic curves is the analogon of isometries between shapes of euclidean geometry. Two curves {p(x)=0} and {q(x)=0} are called equivalent if and only if there is a projectivity [A], such that for every [x]: q(x) = kp(Ax), k being a non-zero constant. Identifying a line {a1x1+a2x2+a3x3=0} with the class of a row vector (a1,a2,a3), equivalence between two lines [a] and [b] amounts to a matrix equation:
Similarly equivalence between two conics amounts to a matrix equation of the form:
A being an invertible matrix and At denoting the transposed matrix. From elementary linear algebra considerations follows easily that every two distinct lines are equivalent. Similarly it can be shown that two distinct proper conics are equivalent.
The standard coordinates (x1,x2,x3) used above can be generalized through the idea of projective base examined in ProjectiveBase.html . One can use some of these, more general coordinate systems, to define the algebraic curves through equations, and show subsequently the independence from the particular base used.
One can use also projective bases to show that four points {A,B,C,D} and the prescription of their images {A',B',C',D'} completely determines (under certain restrictions) a projectivity. This is examined in Projectivity.html .
There is a model of projective plane defined by extending the familiar euclidean plane, therefore called the projectification of the euclidean plane. This extension is discussed in Projectification.html .
One final note on the complexification of projective plane, which is the embedding of PR2 into PC2, C denoting the complex numbers. All definitions transfer verbatim to the complex projective plane PC2. The only difference is the use of the word complex number wherever appears the word real number. Due to the fundamental theorem of algebra, there is a difference with the real projective plane in questions of intersections. For example a line and a conic have always common points in the complex projective plane, whereas in the real plane this is not the case. Similarly two conics have always common points in the complex plane.
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