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6 Determinants 6 Determinants One person’s constant is another person’s variable. Susan Gerhart While the previous chapters had their focus on the exploration of the logic and structural properties of projective planes this chapter will focus on the question: “what is the easiest way to calculate with geometry?”. Many textbooks on geometry introduce coordinates (or homogeneous coordinates) and base all analytic computations on calculations on this level. An analytic proof of a geometric theorem is carried out in the parameter space. For a different parameterization of the theorem the proof may look entirely different. In this chapter we will see, that this way of thinking is very often not the most economic one. The reason for this is that the coordinates of a geometric object are in a way a basis dependent artifact and carry not only information on the geometric object but also on the relation of this object with respect to the basis of the coordinate system. For instance if a point is represented by its homogeneous coordinates (x, y, z) we have encoded its relative position to a frame of reference. From the perspective of projective geometry the perhaps most important fact that one can say about the point is simply the fact that it is a point. All other properties are not projectively invariant. Similarly if we consider three points p1 =(x1,y1,z1), p2 =(x2,y2,z2), p3 =(x3,y3,z3), the statement that these points are collinear reads: x y z + x y z + x y z x y z + x y z + x y z =0, 1 2 3 2 3 1 3 1 2 − 1 3 2 2 1 3 3 2 1 a3 3 determinant. Again from a structural point of view this expression is by far× too complicated. It would be much better to encode the collinearity directly into a short algebraic expression and deal with this. The simplest way to do this is to change the role of primary and derived algebraic objects. 100 6 Determinants If we consider the determinants themselves as “first class citizens” the state- ment of collinearity simply reads det(p1,p2,p3) = 0 where the determinant is considered as an unbreakable unit rather than just a shorthand for the above expanded formula. In this chapter we will explore the roles determinants play within projective geometry. 6.1 A “determinantal” point of view Before we start with the treatment of determinants on a more general level we will review and emphasize the role of determinants in topics we have treated so far. One of our first encounters of determinants was when we expressed the collinearity of points in homogeneous coordinates. Three points p1,p2,p3 are 2 collinear in RP if and only if det(p1,p2,p3) = 0. One can interpret this fact either geometrically (if p1,p2,p3 are collinear, then the corresponding vectors of homogenous coordinates are coplanar) or algebraically (if p1,p2,p3 are collinear, then the system of linear equations p1,l = p2,l = p3,l =0 has a non-trivial solution l = (0, 0, 0)). Dually we can# say$ that# the$ determinant# $ % of three lines l1,l2,l3 vanishes if and only if these lines have a point in common (this point may be at infinity). A second instance where determinants played a less obvious role occurred when we calculated the join of two points p and q by the cross-product p q. We will give an algebraic interpretation of this. If (x, y, z) is any point on× the line p q then it satisfies ∨ p1 p2 p3 det q1 q2 q3 =0. x y z If we develop the determinant by the last row, we can rewrite this as p p p p p p det 2 3 x det 1 3 y + det 1 2 z =0. q2 q3 · − q1 q3 · q1 q2 · % & % & % & Or expressed as a scalar product p p p p p p (det 2 3 , det 1 3 , det 1 2 ), (x, y, z) =0. q2 q3 − q1 q3 q1 q2 ' % & % & % & ( We can geometrically reinterpret this equation by saying that p p p p p p (det 2 3 , det 1 3 , det 1 2 ). q2 q3 − q1 q3 q1 q2 % & % & % & must be the homogeneous coordinates of the line l through p and q since every vector (x, y, z) on this line satisfies l, (x, y, z) = 0. This vector is nothing else than the cross-product p q. # $ × 6.2 A few useful formulas 101 A third situation in which determinants played a fundamental role was in the definition of cross ratios. Cross ratios were defined as the product of two determinants divided by the product of two other determinants. We will later on see that all three circumstances described here will have nice and interesting generalizations: In projective d-space coplanarity will be expressed as the vanishing of a • (d + 1) (d + 1) determinant. In projective× d-space joins and meets will be nicely expressed as vectors • of sub-determinants. Projective invariants can be expressed as certain rational functions of de- • terminants. 6.2 A few useful formulas We will now see how we can translate geometric constructions into expressions that only involve determinants and base points of the construction. Since from now on we will have to deal with many determinants at the same time we first introduce a useful abbreviation. For three points p, q, r RP2 we set: ∈ p1 p2 p3 [p, q, r] := det q q q . 1 2 3 r1 r2 r3 Similarly, we set for two points in RP1 p p [p, q] := det 1 2 . q q % 1 2& We call an expression of the form [...]abracket. Here are a few fundamental and useful properties of 3 3 determinants: × Alternating sign-changes: [p, q, r] = [q, r, p] = [r, p, q]= [p, r, q]= [r, q, p]= [q, p, r]. − − − Linearity (in every row and column): [λ p + µ p , q, r]=λ [p , q, r]+µ [p , q, r]. · 1 · 2 · 1 · 2 Pl¨uckers Formula: [p, q, r]= p, q r . # × $ The last formula can be considered as a shorthand for our developments on cross products, scalar products and determinants in the previous section. 102 6 Determinants c2 c1 l1 q l2 q Fig. 6.1. Two applications of Pl¨uckers µ. 6.3 Pl¨ucker’s µ We now introduce a very useful trick with which one can derive formulas for geometric objects that should simultaneously satisfy several constraints. The trick was frequently used by Pl¨ucker and is sometimes called Pl¨ucker’s µ. Imagine you have an equation f: Rd R whose zero-set describes a geo- metric object. For instance, think of a line→ equation (x, y, z) a x+b y +c z or a circle equation in the plane (x, y) (x a)2 +(y b))→2 r·2. Often· one· is interested in objects that share intersection)→ − points wit− h a− given reference object and in addition pass trough a third object. If the linear combination λ f(p)+µ g(p) again describes an object of the same type then one can apply· Pl¨uckers· µ. All objects described by p λ f(p)+µ g(p) )→ · · will pass through the common zeros of f and g. This is obvious since whenever f(p) = 0 and g(p) = 0 any linear combination is also 0. If one in addition wants to have the object pass through a specific point q then the linear combination p g(q) f(p) f(q) g(p) )→ · − · is the desired equation. To see this simply plugin the point q into the equation. Then one gets g(q) f(q) f(q) g(q) = 0. With this trick· we can− very· easily describe the homogeneous of a line " that passes through the intersection of two other lines l1 and l2 and through a third point q by l ,q l l ,q l . # 2 $ 1 −#1 $ 1 Testing whether this line passes through q yields: l ,q l l ,q l ,q = l ,q l ,q l ,q l ,q =0 ## 2 $ 1 −#1 $ 1 $ # 2 $# 1 $ − # 1 $# 2 $ which is obviously true. Later on we will make frequent use of this trick whenever we need a fast and elegant way to calculate a specific geometric 6.3 Pl¨ucker’s µ 103 [a, b, d] c [a, b, c] b [c, d, b][e, f, a] [c, d, a][e, f, b] = 0 · − · − b d b e d c c a a f Fig. 6.2. Conditions for lines meeting in a point. object. We will now use Pl¨uckers µ to calculate intersections of lines spanned by points. What is the intersection the two lines spanned by the point pairs (a, b) and (c, d). On the one hand the point has to be on the line a b thus it must be of the form λ a + µ b it has also to be on c d hence it∨ must be of the form ψ c + φ d.· Identifying· these two expressions∨ and resolving for λ, µ, ψ and φ would· be· one possibility to solve the problem. But we can directly read of the right parameters by using (a dual version of) Pl¨uckers µ. The property that encodes that a point p is on the line c d is simply [c, d, p] = 0. Thus we immediately obtain that the point: ∨ [c, d, b] a [c, d, a] b · − · must be the desired intersection.
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