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A formula for calculating the angle between straight lines in Euclidean and pseudo-Euclidean spaces. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057290/l0572901.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057290/l0572902.png" /> be the points at infinity on two straight lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057290/l0572903.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057290/l0572904.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057290/l0572905.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057290/l0572906.png" /> be the points of intersection of these lines with the absolute of the space. Then the angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057290/l0572907.png" /> between these lines can be expressed in terms of the [[Cross ratio|cross ratio]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057290/l0572908.png" />:
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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|cross ratio]] $W(G,K,X,Y)$:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057290/l0572909.png" /></td> </tr></table>
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$$\phi=\left|\frac i2\ln W(G,K,X,Y)\right|.$$
  
For two-dimensional pseudo-Euclidean space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057290/l05729010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057290/l05729011.png" /> are the direction vectors of the isotropic lines passing through the point of intersection of the lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057290/l05729012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057290/l05729013.png" />.
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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 .
 
The formula was introduced by E. Laguerre .
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A formula according to which, for all curves on a given surface that touch at some point, the quantity
 
A formula according to which, for all curves on a given surface that touch at some point, the quantity
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057290/l05729014.png" /></td> </tr></table>
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$$\left(3\frac{d\theta}{ds}+2k_2\right)\sin\theta k_1-\left(\frac{d}{ds}k_1\right)\cos\theta$$
  
is invariant, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057290/l05729015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057290/l05729016.png" /> are the curvature and the torsion of the curve, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057290/l05729017.png" /> is the angle between the principal normal of the curve and the normal to the surface, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057290/l05729018.png" /> is the natural parameter on the curve. The formula was obtained by E. Laguerre (1870, see [[#References|[2]]]).
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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 [[#References|[2]]]).
  
 
====References====
 
====References====

Latest revision as of 09:19, 7 August 2014

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)


Comments

References

[a1] M. Berger, "Geometry" , 1–2 , Springer (1987) (Translated from French)
How to Cite This Entry:
Laguerre formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Laguerre_formula&oldid=32750
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article