Murata Umehara [9] gave a representation formula for complete flat fronts with non-empty singular set, and proved the four vertex type theorem: Let [xi] : [S.sup.1] [right arrow] [S.sup.2] be a

regular curve without inflection points, and [alpha] = a(t)dt a 1-form on [S.sup.1] = R/2[pi]Z such that [mathematical expression not reproducible] holds.

We can express a new frame different from the Frenet frame for a

regular curve. Let [alpha]: I [subset] R [right arrow] [S.sup.2.sub.1] be a regular unit speed curve lying fully on [S.sup.2.sub.1].

If the position vector of a

regular curve a is composed by the Frenet frame vectors of another

regular curve [beta], then the curve a is called a Smarandache curve [2].

In the theory of curves in Euclidean space, one of the important and interesting problems is the characterization of a

regular curve. In the solution of the problem, the curvature [kappa] and the torsion [tau] of a

regular curve have an effective role.

Recall that if a

regular curve has constant Frenet-Serret curvature ratios (i.e., [tau]/k and [sigma]/[tau] are constants), it is called a ccr-curve [4,5].

12 This line is a

regular curve in the general plans, and only takes this more complex form in the larger scale stage plan.

Let [alpha](s), s being the arclength parameter, be a non-null

regular curve in semi-Euclidean space [E.sup.2n+1.sub.v].