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Projective Line by Paris Pamfilos Projective Line by Paris Pamfilos Exclusive of the abstract sciences, the largest and worthiest portion of our knowledge consists of aphorisms: and the greatest and best of men is but an aphorism. S. Coleridge, Aids to Reflection XXVII Contents 1 Projective line2 2 Homogeneous coordinates2 3 Projective base3 4 Relation between euclidean and projective base4 5 Change of projective bases5 6 Line projectivities or homographies6 7 The fundamental theorem for line projectivities6 8 Homographic relations7 9 Homographic relations, the group properties8 10 Line perspectivities8 11 Circle and tangents homography9 12 Cross ratio or anaharmonic ratio 10 13 Projectivities defined by cross ratios 11 14 Cross ratio in euclidean coordinates 11 1 Projective line 2 1 Projective line The standard model of the “projective line” consists of the classes »x1; x2¼ of non-zero vectors of R2 modulo non-zero multiplicative constants. For every non-zero vector 2 a = ¹a1; a2º of R , the symbol »a¼ = »a1; a2¼ denotes a “point” of the projective line and 0 a = ¹a1; a2º is called a “representative” of the point. Two representatives fa; a g define the same point if and only if a0 = ka , with a non-zero real number k. The most important examples of projective lines are the usual lines of the euclidean plane to which we add an additional point, called point at infinity. This additional point makes the line closed and is called “projectification” of the line (or “one-point-compactifica- tion”). The projectification is illustrated by figure 1. In this we consider the points of (0,1) α (t,1) t=x1/x2 O(0,0) Figure 1: Projective line represented by (a line + a point at infinity) line α : y = 1; parallel to the x−axis. Each line represented by »x1; x2¼; with non-zero x2; intersects α at the point ¹t;1º with t = x1/x2: The line parallel to α which is the x−axis ∗ is represented by »x1;0¼ considered as an additional point of α . The set A of all lines of the plane through the fixed point A; generalizes this basic model of projective line and is called the “pencil” of lines through A: Figure 2 shows that a pencil A∗ is like a circle. The * t A = A A Figure 2: A pencil of lines is like a circle idea is to pass a circle κ through A and associate to each line through A its intersection with κ: The only line that has no associate is the tangent tA at A . The remedy for this is ∗ to associate to tA the point A . This makes the projective line via its “pencil model” A a “closed” set and “homeomorphic” to a circle. 2 Homogeneous coordinates The pairs f¹x1; x2ºg defining line »x1; x2¼ , considered as a “point” of the projective line are called “homogeneous projective coordinates. They are defined modulo a non-zero multiplica- tive constant, since ¹kx1; kx2º defines the same point. Also they cover all the points of the 0 0 projective line except a single one. Obviously another system ¹x1; x2º of coordinates of 3 Projective base 3 2 R (with the same origin) is related to ¹x1; x2º by an invertible matrix: 0 0 a b x1 x1 x1 a b x1 A = defines a map 0 = A i.e. 0 = : (1) c d x2 x2 x2 c d x2 0 0 0 And for the corresponding quotients t = x1/x2;t = x1/x2 we get the so called “homo- graphic relation” (see file Homographic relation) at + b t0 = : (2) ct + d Notice that a non-zero multiple A0 = k A of the matrix defines the same transformation between the homogeneous coordinates. 3 Projective base Three distinguished points fA; B;Cg on a projective line determine a so-called “projective base” and through it a corresponding “homogeneous coordinates system”. The mechanism can be explained in terms of bases of the vector space R2: α A C B a D b Figure 3: Projective base defined through three collinear points The points fA = »a¼; B = »b¼;C = »c¼g of the projective line determine three lines repre- sented by corresponding vectors fa; b; cg. The vectors are selected so that c = a + b: This condition uniquely determines fa; b; cg from the data fA = »a¼; B = »b¼;C = »c¼g, up to a multiplicative constant. The homogeneous coordinate system results by using the base fa; bg of R2 and writ- ing vor every point D on this line its coordinates with respect to this base D = »d¼; where d = d1a + d2b: Thus, to D we correspond ¹d1; d2º; and it is clear that a non-zero multiple k¹d1; d2º defines the same point D on the projective line, therefore we often write D = d1 A + d2B; (3) using the symbols for the points fA; Bg rather than the symbols fa; bg for the vectors representing them. Obviously in this coordinate system the points fA; B;Cg have correspondingly the co- ordinates f¹1;0º;¹0;1º;¹1;1ºg. Point C is called “coordinator” or “unit” of the projective base fA; B;Cg. It is used to calibrate a basis defined by the other two points. The calibra- tion is done by projecting some vector c; parallel to the other lines determined by fA; Bg and using the base of R2 resulting from these projections fa; bg (See Figure 3). Having this calibration, the point P of the line can be written as a linear combination P = xA + yB (4) 4 Relation between euclidean and projective base 4 Passing from this equation to the vector equation, there is an ambiguity, which we elim- inate using the “unit” C of the base. In fact, we could select two other vectors a0 = µa; and b0 = νb representing the same points fA; Bg and write for the same point P : P = »x0a0 + y0b0¼ = x0 A + y0B = xA + yB = »xa + yb¼: (5) Then, we would obtain two different pairs f¹x0; y0º;¹x; yºg for the same point and this is not good, except if it happens to be ¹x0; y0º = k¹x; yº with a k , 0: But this follows from the requirement to have C = A + B: (6) Thus if we select the vectors fa; bg to represent fA; Bg and then two other fa0; b0g to repre- sent the same base points, the requirement C = A + B implies the vectorial relation c = λ¹a + bº = λ0¹a0 + b0º = λ0¹µa + νbº ) λ = λ0µ = λ0ν ) µ = ν; (7) and we are salvaged. Thus, you have a certain freedom to select two vectors a; b 2 R2 to represent your base points fA; Bg, but you must always respect the rule a + b = kc with a k , 0; so that if you select two others fa0; b0g, this rule will imply that fa0 = ra; b0 = rbg with r , 0: 4 Relation between euclidean and projective base In this section we consider a line " in the euclidean plane, defined by two position vec- tors fa; bg and also a point on this line x = xaa + xb b; where the variables fxa; xb g satisfy xa + xb = 1 (See Figure 4). Often the running point x on the line determined by the po- A=[a] C=[c] X=[x] B=[b] ε b x b bx a c a' b' xa a O Figure 4: Euclidean base fa; bg versus projective base fA; B;Cg sition vectors fa; bg is described parametrically by x = a + ¹b − aºt ) x = ¹1 − tºa + tb ) xa = ¹1 − tº; xb = t; (8) which for the euclidean signed ratio of the segments defined by x imply xa t xb = = − : (9) xb t − 1 xa We consider also the points fA = »a¼; B = »b¼; X = »x¼;g defined by these vectors on the same “extended” projective line. We want to find the expression of the projective ho- mogeneous coordinates of X relative to the projective base fA; B;Cg, where C = »c¼ for 5 Change of projective bases 5 an arbitrary other vector different from fa; bg. We can represent c in the vectorial base a; b through 0 0 c = caa + cb b = a + b : Thus if 0 0 0 0 0 0 x = xaa + xb b = xacaa + xbcb b = xaa + xb b; 0 0 0 0 the projective coordinates of X are fλxa; λxb g, so that X = ¹λxaºA + ¹λxbºB and from the previous equalities we obtain 0 0 xa = xaca; xb = xbcb: Thus, if we know the euclidean (cartesian) coordinates fxa; xb g of x relative to the vecto- 0 0 rial base a; b; then the projective coordinates fxa; xb g of X = »x¼ relative to the projective base fA; B;Cg are 0 0 0 xa 0 xb X = xa A + xb B with xa = λ ; xb = λ ; and λ , 0: (10) ca cb Thus, taking into account the equation9, we obtain the relation between the ratio of the projective coordinates and the euclidean signed ratio of distances xa/xb: 0 xa cb xa cb xb 0 = · = − · : (11) xb ca xb ca xa 5 Change of projective bases Having two projective bases fA; B;Cg, fA0; B0;C0g of the line α; we can express the same point P of the line as linear combination of the base points: P = xA + yB = x0 A0 + yB0; where »a¼ = A;»a0¼ = A0;»b¼ = B;»b0¼ = B: (12) Using a bit of linear algebra and the matrix K for the basis-change in R2 we obtain 0 0 0 k11 k12 a = k11a + k21b; λx = k11 x + k12 y ; with K = ; 0 ) 0 0 (13) k21 k22 b = k12a + k22b; λy = k21 x + k22 y : Thus, we obtain the rule, that by homogeneous bases change the coordinates transform by a matrix and corresponding linear equations: x x0 x0 x λ = K ) = µK−1 : (14) y y0 y0 y Taking the quotients of coordinates in equation 13 we see that they are related by a simple function 0 0 x 0 x k11t + k12 t = and t = 0 ) t = 0 : (15) y y k21t + k22 Such relations between two variables are called “homographic”, so that we can state the following trivial theorem.
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