Laguerre formula

A formula for calculating the angle between straight lines in Euclidean and pseudo-Euclidean spaces. Let $X$ and $Y$ be the points at infinity on two straight lines $a$ and $b$ and let $G$ and $K$ be the points of intersection of these lines with the absolute of the space. Then the angle $\phi$ between these lines can be expressed in terms of the cross ratio $W(G,K,X,Y)$:

$$\phi=\left|\frac i2\ln W(G,K,X,Y)\right|.$$

For two-dimensional pseudo-Euclidean space, $G$ and $K$ are the direction vectors of the isotropic lines passing through the point of intersection of the lines $a$ and $b$.

The formula was introduced by E. Laguerre .

A formula according to which, for all curves on a given surface that touch at some point, the quantity

$$\left(3\frac{d\theta}{ds}+2k_2\right)\sin\theta k_1-\left(\frac{d}{ds}k_1\right)\cos\theta$$

is invariant, where $k_1$ and $k_2$ are the curvature and the torsion of the curve, $\theta$ is the angle between the principal normal of the curve and the normal to the surface, and $s$ is the natural parameter on the curve. The formula was obtained by E. Laguerre (1870, see [2]).

References

 [1] E. Laguerre, "Sur la théorie des foyers" Nouv. Ann. Math. , 12 (1853) pp. 57–66 [2] E. Laguerre, "Oeuvres" , 2 , Chelsea, reprint (1972) [3] B.A. Rozenfel'd, "Non-Euclidean geometry" , Moscow (1955) (In Russian)