Unicursal curve

A plane curve $ \Gamma $ which may be traversed such that the points of self-intersection are visited only twice. For a curve to be unicursal it is necessary and sufficient that there are at most two points through which there pass an odd number of paths. If $ \Gamma $ is a plane algebraic curve of order $ n $ having the maximum number $ \delta $ of double points (including improper and imaginary ones), then $ \delta = ( n - 1) ( n - 2)/2 $( where a point of multiplicity $ k $ is counted as $ k ( k - 1)/2 $ double points).

Every integral $ \int R ( x, y) dx $, where $ y $ is the function of $ x $ defined by the equation $ F ( x, y) = 0 $ giving an algebraic unicursal curve and $ R ( x, y) $ is a rational function, can be reduced to an integral of a rational function and can be expressed in terms of elementary functions.

Comments

In algebraic geometry, a unicursal curve $ U $ is a rational curve, i.e. a curve that admits a parametric representation $ x = \phi ( t) $, $ y = \psi ( t) $ with $ \phi $ and $ \psi $ rational functions. Such a curve is an algebraic curve of effective genus $ 0 $. For every irreducible curve $ \Gamma $ there exists a birationally equivalent non-singular curve $ \widetilde \Gamma $. This $ \widetilde \Gamma $ is unique up to isomorphism. The genus of $ \widetilde \Gamma $ is called the effective genus of $ \Gamma $. The unicursal curves are the irreducible algebraic curves of effective genus zero. This (more or less) agrees with the general geometric definition above, in that the parametrization provides a "traversion" .

References

[a1]	R.J. Walker, "Algebraic curves" , Dover, reprint (1950) pp. 149–151 MR0033083 Zbl 0039.37701
[a2]	P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) pp. 178; 674; 179; 349; 525; 532; 535; 632; 743 MR0507725 Zbl 0408.14001