A class of splines with constrained length

Sometimes one needs to approximate a curve by means of splines that preserve the length of the given curve. This is the case, for instance, of the trajectory of an inclined oil-well, where the length of the path described by the trajectory between any two of its points can be measured by engineers. In a previous paper, using quadratic splines, the first author developed such a model for the petroleum industry and solved the corresponding problem of approximation. But the method employed there does not seem to be appropriate to deal with polynomials of higher degrees that appear when other parameters such as the continuity of the curvature need to be preserved in the models. In this work we introduce another method of focusing on the problem that is independent of the degree of the polynomials and that is simpler. A somewhat surprising result is that despite the quadratic splines, we lose uniqueness of solution in the general case.