Scalar- and planar-valued curve fitting using splines under tension

The spline under tension was introduced by Schweikert in an attempt to imitate cubic splines but avoid the spurious critical points they induce. The defining equations are presented here, together with an efficient method for determining the necessary parameters and computing the resultant spline. The standard scalar-valued curve fitting problem is discussed, as well as the fitting of open and closed curves in the plane. The use of these curves and the importance of the tension in the fitting of contour lines are mentioned as application.