Laguerre formula
The Laguerre formula (named after Edmond Laguerre) provides the acute angle ϕ {\displaystyle \phi } between two proper real lines, as follows: ϕ = | 1 2 i Log Cr ( I 1 , I 2 , P 1 , P 2 ) | {\displaystyle \phi =|{\frac {1}{2i}}\operatorname {Log} \operatorname {Cr} (I_{1},I_{2},P_{1},P_{2})|} where: Log {\displaystyle \operatorname {Log} } is the principal value of the complex logarithm Cr {\displaystyle \operatorname {Cr} } is the cross-ratio of four collinear points P 1 {\displaystyle P_{1}} and P 2 {\displaystyle P_{2}} are the points at infinity of the lines I 1 {\displaystyle I_{1}} and I 2 {\displaystyle I_{2}} are the intersections of the absolute conic, having equations x 0 = x 1 2 + x 2 2 + x 3 2 = 0 {\displaystyle x_{0}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=0} , with the line joining P 1 {\displaystyle P_{1}} and P 2 {\displaystyle P_{2}} . The expression between vertical bars is a real number.