Unicursal curve

A plane curve 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 is a plane algebraic curve of order having the maximum number of double points (including improper and imaginary ones), then (where a point of multiplicity is counted as double points).

Every integral , where is the function of defined by the equation giving an algebraic unicursal curve and 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 is a rational curve, i.e. a curve that admits a parametric representation , with and rational functions. Such a curve is an algebraic curve of effective genus . For every irreducible curve there exists a birationally equivalent non-singular curve . This is unique up to isomorphism. The genus of is called the effective genus of . 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