A

We say that four points {A,B,C,D} are

The fourth point D, corresponding to vector d = k

[1] Every projective base introduces a coordinate system in which we write X = uA + vB + wC, {u,v,w} being defined modulo a non-zero multiplicative constant, and resulting from the corresponding representetion of [x] = X, through the basis vectors: x = u(k

[2] In particular, points {A,B,C,D} have correspondingly the coordinates {(1,0,0), (0,1,0), (0,0,1), (1,1,1)}.

If a point X can be expressed as a combination in two ways X = pA + qB and X = p'A' + q'B', this means that it belongs to both lines determined by the pairs {A,B} and {A',B'} hence it coincides with the intersection of these two lines.

In the standard model of projective plane, described above, projective lines consist of sets of euclidean lines all of which are contained in some euclidean plane passing through the origin. Such a plane is described by an equation of the form

px + qy + rz = 0.

Given {A=[a], B=[b]} the corresponding coefficients (p,q,r) are the coordinates of the vector product axb.

Inversely, given the equation of the plane and selecting two vectors {a,b} satisfying this equation (lying in the plane), all other vectors of that plane are combinations c = ua + vb of these two vectors.

The structure of projective lines independent from the surrounding plane is studied in ProjectiveLine.html .

Since P is on line CD, we have P = cC - dD = cC - d(A+B+C) = (c-d)C - dA - dB. Since P is on line AB it must be (c-d) = 0, hence P = A+B (coordinates defined modulo multiplicative constant!).

Thinking analogously for U and V, it must be U = C + B and V = C + A. Hence U-V = A - B = Q'. But Q' being both linear combination of {U,V} and {A,B}, must coincide with the intersection point of the corresponding lines i.e. Q' = Q.

More general the harmonic conjugate of a point P = sA+tB is Q = sA-tB. To see this consider again a point D on line CP, this time having coordinates D=C+dP=C+d(sA+tB), implying D-sdA = C+dtB. In this equation the left side represents a point on line DA and the right side represents a point on line CB hence both sides represent the intersection point U of these lines: U=D-sdA=C+dtB. Analogously D-tdB=C+dsA is the representation of V = D-tdB=C+dsA. From these we have U-V=(D-sdA)-(D-tdB)=tdB-sdA. Again U-V is a point on line UV and (tdB-sdA) is a point on line AB, hence coincides with Q. Thus, Q=d(tB-sA), which is equivalent with Q = sA-tB.

Passing though to the

Having this definition for the cross-ratio, points {C,D} are called harmonic conjugate with respect to {A,B}, when their cross-ratio (x/y):(u/v) = -1, implying x/y = -u/v. Thus we find again the result of (3), that the harmonic conjugate of C=xA+yB with respect to (A,B) is D=xA-yB.

[1] Assume ABC, A'B'C' point-perspective i.e. lines AA', BB', CC' have a common point D. Then D can be written: D = uA + u'A' = vB + v'B' = wC + w'C'. Taken in pairs these equations are equivalent to:

(i) uA - vB = v'B' - u'A', (ii) wC-uA = u'A' - w'C', and (iii) vB - wC = w'C'-v'B'. Equation (i) expresses the intersection point C* of lines AB and A'B'. Analogously (ii) and (iii) express the intersection points B* and A* of lines AC, A'C' and BC, B'C'.

The three points satisfying C* + A* + B* = (uA-vB) + (vB-wC) + (wC-uA) = 0 are on a line.

[2] For the inverse, assume {A*, B*, C*} are collinear, and taking appropriate projective base, A* = B*+C*. We may assume that C* = uA - vB = v'B' - u'A' and B* = wC - u

See Desargues.html for a standard geometric proof.

By what we saw in the foregoing applications, taking the projective base {A,B,C,D}, points {A',B',C'} are represented correspondingly as {B+C, C+A, A+B}, and the harmonic conjugate to them {A''=B-C,B''=C-A,C''=A-B}. By adding the expressions we find that A''+B''+C''=0, proving the collinearity of the three points.

The applications indicate that incidence relations are very easily expressed and proved in the framework of projective geometry and with the aid of projective bases and their related coordinate system.

Projectification.html

ProjectiveCoordinates.html

ProjectiveLine.html

ProjectivePlane.html

TrilinearPolar.html

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