10 Conics and perspectivities Some people have got a mental horizon of radius zero and call it their point of view. Attributed to David Hilbert In the last chapter we treated conics more or less as isolated objects. We defined points on them and lines tangent to them. Now we want to investigate various geometric and algebraic properties of conics. In particular, we will see how we can treat conics on the level of bracket algebra. 10.1 Conic through five points We start by calculating a conic through a given set of points. For this consider the quadratic equation that defines a conic. a x2 + b y2 + c z2 + d xy + e xz + f yz = 0. · · · · · · This equation has six parameters a, . , f 1. Multiplying all of them simultane- ously by the same non-zero scalar leads to the same conic. Thus the parameter vector (a, . , f) behaves like a vector of homogeneous coordinates. Counting degrees of freedom shows that in general it will take five points to uniquely determine a conic. To find the parameters for a conic through five points pi = (xi, yi, zi); i = 1, . , 5 we simply have to solve the following linear system of equations: 1 Compared to Section 9.1 we have relabeled the parameters and put the factor of 2 of the mixed terms into the parameters 164 10 Conics and perspectivities 2 2 2 a x1 y1 z1 x1y1 x1z1 y1z1 0 2 2 2 b x2 y2 z2 x2y2 x2z2 y2z2 0 2 2 2 c x3 y3 z3 x3y3 x3z3 y3z3 = 0 2 2 2 · d x4 y4 z4 x4y4 x4z4 y4z4 0 2 2 2 e x y z x5y5 x5z5 y5z5 0 5 5 5 f If this system has a full rank of 5 then there is an (up to scalar multiple) unique solution (a, . , f) that defines the corresponding conic. If more than 3 points are simultaneously collinear of if two points coincide the rank of the system may be lower than 5. This corresponds to the situation that there are more than one conic passing through the given set of points. This method of determining the parameter vector (a, . , f) is mathematically elegant how- ever it is computationally expensive. We first have to calculate the squared parameters and then have to solve a 5 times 6 system of equalities. There is also another way to calculate such a conic more or less directly. This way will also give us additional structural insight into the geometry and underlying algebra of a conic. In preparation we have to understand how to calculate a degenerate conic that consists of two lines with homogeneous coor- dinates g and h. A conic must be represented by a quadratic form pT AP p = 0 that vanishes if p is on either of the lines. The (non-symmetrized) matrix of such a quadratic form is simply given by A = ghT . This can be easily seen since the quadratic form pT Ap = pT (ghT )p = (pT g)(hT p) = p, g p, h ! "! " vanish if one of the two scalar products on right side vanish. This in turn corresponds geometrically to the situation in which p is on g or on h. Assume that the line g is spanned by two points labeled 1 and 2 and that line h is spanned by two points labeled 3 and 4. Then we have g = 1 2 and h = 3 4. The quadratic form becomes × × p, 1 2 p, 3 4 = 0. ! × "! × " We may as well express this term as the product of two determinants [p, 1, 2][p, 3, 4] = 0. Each factor describes a linear condition on the point p. The product calculates the conjunction between the two expressions. Now, assume that we want to describe the set of conics that passes through for points 1, . , 4 in general position. Clearly, there are many conics that satisfy this condition. The corresponding system of linear equations con- sists of four equations in six variables. Hence the solution space will be two- dimensional. One of these two degrees of freedom goes into the homogeneity of 10.1 Conic through five points 165 1 4 PSfrag replacements PSfrag replacements 1 2 4 3 2 3 Fig. 10.1. Bundles of conics though four points. Three degenerate special cases. the conic parameters. Therefore we have a bundle of geometric solutions with one degree of freedom. Figure 10.1 (left) illustrates such a bundle of conics. Among these conics there are three degenerate conics, each of them passing through a pair of lines spanned by the four points. In Figure 10.1 (right) these pairs of lines are marked by identical colors. They correspond to the following four quadratic forms: [p, 1, 2][p, 3, 4] = 0, [p, 1, 3][p, 2, 4] = 0, [p, 1, 4][p, 2, 3] = 0. A linear combination of two of these forms (say the last two) λ[p, 1, 3][p, 2, 4] + µ[p, 1, 4][p, 2, 3] = 0 generates again a quadratic form. The set of points p satisfying this equation forms again a conic. This conic passes through all four points 1, . , 4 since both summands vanish on these points. If λ and µ run trough all possible values we obtain all the conics in the bundle through the four points. Applying the technique of Plu¨cker’s µ (compare Section 6.3) we can adjust these values 1 4 PSfrag replacements q 3 2 Fig. 10.2. Constructing a conic through five points. 166 10 Conics and perspectivities such that the resulting conic passes through another given point q. For this we have to simply choose λ = [q, 1, 4][q, 2, 3]; µ = [q, 1, 3][q, 2, 4]. − The resulting conic equation can be written as [q, 1, 4][q, 2, 3][p, 1, 3][p, 2, 4] [q, 1, 3][q, 2, 4][p, 1, 4][p, 2, 3] = 0. − Observe that this equation is a multi-homogeneous bracket polynomial that is quadratic in each of the six involved points. Figure 10.2 illustrates the situa- tion. We can also interpret it as a bracket condition encoding the (projectively invariant) property that six points 1, . , 4, p, q are on a conic (compare Sec- tion 6.4 and Section 7.3). We will come back to this interpretation in the next Section. Before this we will give the procedure for calculating calculate the sym- metric matrix for the conic through the five points 1, 2, 3, 4, q. We give it as a kind of simple computer program: 1: g := 1 3; 1 × 2: g := 2 4; 2 × 3: h := 1 3; 1 × 4: h := 2 4; 2 × T 5: G := g1g2 ; T 6: H := h1h2 ; 7: M := qT HqG qT GqH; − 8: A := M + M T ; The matrix A assigned in the last line of the program contains the sym- metrized matrix. 10.2 Conics and cross ratios Let us come back to the equation [q, 1, 4][q, 2, 3][p, 1, 3][p, 2, 4] [q, 1, 3][q, 2, 4][p, 1, 4][p, 2, 3] = 0, ( ) − ∗ which characterizes whether six points are on a conic. First observe that this equation is highly symmetric. For each bracket in one term its complement (the bracket consisting of the other three letters) is in the other term. The symmetry becomes a bit more transparent if we rewrite the equation with new points labels: 10.2 Conics and cross ratios 167 2 3 4 1 PSfrag replacements p q Fig. 10.3. Four points on a conic seen from other points of a conic. [A, B, C][A, Y, Z][X, B, Z][X, Y, C] [A, B, Z][A, Y, C][X, B, C][X, Y, Z] = 0. − There is another important observation that we can make by rewriting equation ( ). We assume that the conic is non-degenerate and that none of the determ∗inants vanishes. In this case we can rewrite ( ) to the form ∗ [q, 1, 4][q, 2, 3] [p, 1, 4][p, 2, 3] = . [q, 1, 3][q, 2, 4] [p, 1, 3][p, 2, 4] Both sides of the equation represent cross ratios. The left side is a cross ratio of the lines p1, p2, p3, p4 the right side of the equation is a cross ratio of the lines q1, q2, q3, q4. We abbreviate [q, 1, 4][q, 2, 3] (1, 2; 3, 4) := . q [q, 1, 3][q, 2, 4] This is the cross ratio of 1, 2, 3, 4 as “seen from” point q. Thus equation ( ) may be restated as ∗ (1, 2; 3, 4)q = (1, 2; 3, 4)p. Point p and point q see the points 1, 2, 3, 4 under the same cross ratio. The situation is shown in Figure 10.3 We summarize this in a theorem: Theorem 10.1. Let 1, 2, 3, 4, p be five points on a conic such that p is distinct from the other four points. Then the cross ratio (1, 2; 3, 4)p is independent of the special choice of p. We will later on see that this theorem is very closely related to the so called exterior angle theorem for circles which states that in a circle a fixed secant is seen from an arbitrary point on the circle under the same angle (modulo π). 168 10 Conics and perspectivities The last theorem enables us to speak of the cross ratio of four points on a fixed conic as long as no more than two of the points (1, 2, 3, 4) coincide and we can speak of a cross ratio at all.