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9 Conics and Their Duals 9 Conics and their duals You always admire what you really don’t understand. Blaise Pascal So far we dealt almost exclusively with situations in which only points and lines were involved. Geometry would be quite a pure topic if these were the only objects to be treated. Large parts of classical elementary geometry deal with constructions involving circles. Already the most elementary drawing tools treated by Euclid (the ruler and the compass) contain a tool for gener- ating circles. In a sense so far we dealt with the ruler alone. Unfortunately, circles are not a concept of projective geometry. This can be easily seen by observing that a shape of a circle is not invariant under projective transfor- mations. If you look at a sheet of paper on which a circle is drawn from a skew angle you will see an ellipse. In fact projective transformations of cir- cles include ellipses, hyperbolas and parabolas. They are subsumed under the term conic sections or conic, for short. Conics are the concept of projective geometry that comes closest to the concept of circles in Euclidean geometry. It is the purpose of this section to give a purely projective treatment of conics. Later on we will see how certain specializations give interesting insights in the geometry of circles. 9.1 The equation of a conic Let us start with the unit circle in the euclidean plane: 2 2 2 (x, y) R x + y = 1 . { ∈ | } In this section we will investigate which shapes can arise if we transform this circle projectively. For this we again consider the Euclidean plane embedded 146 9 Conics and their duals at the z = 1 plane of R3 and represented by homogeneous coordinates (any other affine embedding not containing the origin would serve as well and would lead to the same result). Thus the points of the circle correspond to points with homogeneous coordinates (x, y, 1) with x2 + y2 = 1. Taking into account that homogeneous coordinates are only specified up to a scalar multiple, we may rewrite this condition in a more general way as points (x, y, z) with x2 + y2 = z2. Setting z = 1 we obtain the original formula. According to the fact that every term of the expression x2 + y2 = z2 is quadratic a vector (x, y, z) satisfies this expression if (λx, λy, λz); λ = 0 satisfies it. We may rewrite the quadratic equation as " 1 0 0 x (x, y, z) 0 1 0 y = 0. · 0 0 1 · z − Thus the homogeneous coordinate vectors that satisfy this equation are exactly those that represent points of our circle. We now want to transform these points projectively. Applying a projective transformation to the points can be carried out by replacing both occurrences of the vector p = (x, y, z) by a transformed vector M p. Thus we obtain that a projectively transformed unit circle can be express·ed algebraically as the solutions of the equation 1 0 0 x ((x, y, z) M T ) 0 1 0 (M y ) = 0, · · 0 0 1 · · z − where M is a real 3 3 matrix with non-zero determinant. Multiplying the three matrices in the×middle of this expression we are led to an equation a b d x (x, y, z) b c e y = 0, ( ) · d e f · z ∗ for suitable parameters a, b, c, d, e, f. Observe that the matrix in the middle is necessarily symmetric. Expanding the above product expression yields the following quadratic equation a x2 + c y2 + f z2 + 2b xy + 2d xz + 2e yz = 0. · · · · · · We call such an expression of the form pT Ap a quadratic form (regardless of the matrix A being symmetric or not). The set of points that satisfies such an equation will be called a conic. In a sense we are dealing on three different levels when we speak about matrices, quadratic forms and conics. Before we continue we want to clarify this relation. Let A be a 3 3 matrix with entries × in some field K the associated quadratic form is a homogeneous quadratic 3 function A : K K defined by Q → (p) = pT Ap. QA 9.1 The equation of a conic 147 It is important to notice that different matrices may lead to the same quadratic form. This can be seen by expanding a b1 d2 x = (x, y, z) b2 c e1 y QA · · d1 e2 f z to = a x2 + c y2 + f z2 + (b + b ) xy + (d + d ) xz + (e + e ) yz. QA · · · 1 2 · 1 2 · 1 2 · Observe that, the quadratic form associated to a matrix does only depend on the diagonal entries and the sums of matrix entries Aij + Aji. For every potentially non-symmetric matrix A the matrix (A + AT )/2 creates the same quadratic form: = T . QA Q(A+A )/2 We call this process symmetrization. Furthermore it is also clear that for every homogeneous quadratic polynomial f(x, y, z) also a (symmetric) matrix A with A(x, y, z) = f(x, y, z). The Qconics themselves are the third level with which we have to deal. They consist of all points in K for which a quadratic form vanishes. The conic associated to a matrix (oPr quadratic form) is = [p] pT Ap = 0 . CA { ∈ PK | } Since the expression pT Ap = 0 is stable under scalar multiplication of p with a non-zero scalar we will by common abuse of notation as usual work on the level of homogeneous coordinates and omit the brackets [. .]. How much information about the matrix A is still present in the conic in a sense depends on the underlying field. Over R it may for instance happen that a conic (like the solution set of x2 + y2 + z2 = 0) has no non-zero solutions at all. In such a case then there are still complex solutions. We will deal with them later. We come back to the case of a projectively transformed circle and the quadratic form described in equation ( ). In this case the parameters a, . , f are not completely independent. Sylvest∗ ers law of inertia from linear algebra tells us that the signature of the eigenvalues must be the same as in the matrix 1 0 0 0 1 0 0 0 1 − and that this is the only restriction1. 1 Sylvesters law of inertia states that if M is nonsingular then A and M T AM have the same signature of eigenvalues for any symmetric matrix A. Furthermore this is the only restriction on the coefficients of the symmetric matrix M T AM. 148 9 Conics and their duals ! ! ! ∞ ∞ ∞ PSfrag replacements PSfrag replacements PSfrag replacements ellipse parabola hyperbola Fig. 9.1. The possible images of a circle. If we allow general real parameters in equation ( ) we can classify the different cases by the signature of the eigenvalues. Sinc∗e the quadratic forms of the matrices A and A describe identical point sets, we may identify a sig- nature vector with its −negative. Thus we end up with five essentially different cases of signatures. (+, +, ), (+, +, +), , (+, , 0), (+, +, 0), (+, 0, 0). − − Each of these cases describes a geometric situation that cannot be trans- formed projectively into any of the other cases. The case of a circle corresponds to the first entry in the list. The last three cases correspond to degenerate con- ics (the determinant of A is zero) and the second case corresponds to a case like x2 + y2 + z2 = 0 in which we have no real solutions at all (but where still complex solutions exist). Before we clarify the geometric meaning of the other four cases in detail we will have a closer look at the circular case. Projectively all quadratic forms with eigenvalue signature (+, +, ) have to be considered as isomorphic (they are the projective images of a un−it circle). However if we 2 fix a certain embedding of the Euclidean plane into RP (say the standard z = 1 embedding) and by this single out a specific line at infinity, then we may classify them also with respect to Euclidean motions. In fact there is an infinite variety of forms of conics that are in-equivalent under Euclidean transformations. Still there is a very useful (and well known) coarser classifi- 9.2 Polars and tangents 149 cation if we just count the intersection of the conic with the line at infinity. Intersecting the quadratic form of a projectively transformed circle a x2 + c y2 + f z2 + 2b xy + 2d xz + 2e yz = 0. · · · · · · with the line at infinity (i.e. setting z = 0) we a are left with the equation a x2 + c y2 + 2b xy = 0. · · · This homogeneous quadratic equation may lead to zero, one or two solutions up to scalar multiples. The three cases are shown in Figure 9.1. They corre- spond to the well known cases of ellipses, parabola, and hyperbola. Thus one could say that a parabola is a conic that touches the line at infinity, while a hy- perbola has two points at infinity. The three cases can be algebraically distin- guished by considering the discriminant of the equation a x2+c y2+2b xy = 0. This sign of the discriminant turns out to be the sign of· the d·etermin·ant a b det . b c % & If this sign is negative we are in the hyperbolic case. If it is zero we get a parabola and if it is positive we get an ellipse. 9.2 Polars and tangents For the moment we will stick to the case of non-degenerate conics that are projective images of a circle.
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