The Laguerre formula (named after
Edmond Laguerre
Edmond Nicolas Laguerre (9 April 1834, Bar-le-Duc – 14 August 1886, Bar-le-Duc) was a French mathematician and a member of the Académie des sciences (1885). His main works were in the areas of geometry and complex analysis. He also investigate ...
) provides the acute angle
between two proper real lines,
as follows:
:
where:
*
is the principal value of the
complex logarithm
*
is the
cross-ratio
In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points ''A'', ''B'', ''C'' and ''D'' on a line, th ...
of four collinear points
*
and
are the
points at infinity
In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line.
In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. Ad ...
of the lines
*
and
are the intersections of the
absolute conic, having equations
, with the line joining
and
.
The expression between vertical bars is a real number.
Laguerre formula can be useful in
computer vision, since the absolute conic has an image on the retinal plane which is invariant under camera displacements, and the cross ratio of four collinear points is the same for their images on the retinal plane.
Derivation
It may be assumed that the lines go through the origin. Any
isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' me ...
leaves the absolute conic invariant, this allows to take as the first line the ''x'' axis and the second line lying in the plane ''z''=0. The
homogeneous coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. ...
of the above four points are
:
respectively. Their nonhomogeneous coordinates on the infinity line of the plane ''z''=0 are
,
, 0,
. (Exchanging
and
changes the cross ratio into its inverse, so the formula for
gives the same result.) Now from the
formula of the cross ratio we have
References
{{Reflist
* O. Faugeras. Three-dimensional computer vision. MIT Press, Cambridge, London, 1999.
Equations
Geometry in computer vision