In a course in Geometry we where asked to geometrically construct the sum of two points $x$ and $y$ on a projective line by help of "the theorem on complete quadrilaterals". This theorem (as stated in our lecture notes) says that if $A,B,C,D$ are four points in general position in a real projective plane and if $P=AB\cap CD$ and $Q=AD\cap BC$, $l=PQ$, $X=l\cap BD$, $Y=l\cap AC$. Then the pair of points $\{P,Q\}$ on $l$ separates the pair $\{X,Y\}$ harmonically, meaning that the cross ratio $Cr(P,X,Q,Y)=-1$

Projective planes are assumed to satisfy desargues theorem in my course.

I have the following idea, which I first present and then I will add my concerns.

If $x=y$, then $Cr(0,x,2x,\infty )=-1$

If $y>x$ then $Cr(x, \frac{x+y}{2},y, \infty )=-1$

We use this to construct $\frac{x+y}{2}$. Let $P=x$ and $Q=y$, we let $r_1$, $r_2$ be arbitrary lines intersecting in $x$. Furthermore let $A$ and $C$ be the intersection points of $r_1$ and $r_2$ with the line $l_{\infty }$ at infinity respectively. Now draw the lines $AQ=h_1$, $CQ=h_2$. We let $D=r_2\cap h_1$ and $B=r_1\cap h_2$. Now, we are in the context of the above mentioned theorem. The line $BD$ intersects $l$ in the point $s$ such that $Cr(x, s,y, \infty )=-1$, hence $s=\frac{x+y}{2}$.

My concerns are the following:

  1. I have problems calculating cross ratios, using Berger I would believe $Cr(x, y, \frac{x+y}{2}, \infty )=\frac{\frac{x+y}{2}-x}{\frac{x+y}{2}-y}=-1$, and in the same way I get $Cr(0,2x,x,\infty )=-1$, this seems completely unreasonable however since $2x$ is not between $0$ and $x$ which makes it impossible for these calculations to be correct, please help me clearly write down the standard expression for these crossratios, because I am somehow misunderstanding page 125 in berger $Cr[a,b,c, \infty ]=\frac{\overline{ca } }{\overline{cb}}$.

  2. Independent on which one of the calculations of crossratio is correct, I run into problems in making the argument when the unknown quantity is on position three, i.e "when I want to construct the point $Q$".

I realized that for constructing 2x, I just map the x-axis or the y-axis to $l$ by the projective transformation taking 0 to 0 and 1 to x and infinity to infinity then the image of 2 is my desired point, hence I think that this together with previous calculations solves the problem, if $Cr(x, \frac{x+y}{2},y, \infty )=-1$ that is.

Any comments, hints or general opinions about crossratios and "geometric construction" are welcome.

  • $\begingroup$ Does en.wikipedia.org/wiki/Projective_range help? $\endgroup$ – Andrew D. Hwang Mar 21 '15 at 18:22
  • 1
    $\begingroup$ Thank you for the added details. :) I don't have access to Berger's book, but your formulas do look compatible with en.wikipedia.org/wiki/Cross-ratio. Two thoughts: 1. When you compute cross-ratios, are you taking into account a fixed orientation of the line? 2. Your calculations of $Cr(x, y, \frac{x+y}{2}, \infty)$ and $Cr(0, 2x, x, \infty)$ don't appear to be incompatible: In each, the third argument is between the first and second...? $\endgroup$ – Andrew D. Hwang Mar 21 '15 at 22:25
  • $\begingroup$ It is true that these two is not incompatible, they are derieved from the same formula and completely symmetric, however they should not equal minus one, since this order does not describe harmonic conjugates from the geometric point of view. And also, it is the other expression that I would need for my solution to work, however I can not derive it from the formulas for cross ratio eventhough it is intuitevely clear that I should have $Cr(0,x,2x,\infty )=-1$ and $Cr(x,\frac{x+y}{2},y, \infty )=-1$ $\endgroup$ – harajm Mar 22 '15 at 0:22

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