We present a class of non-polynomial parametric splines that interpolate the given control points and show that some curve types in this class have a set of highly desirable properties that were not previously demonstrated for interpolating curves before. In particular, the formulation of this class guarantees that the resulting curves have C² continuity everywhere and local support, such that only four control points define each curve segment between consecutive control points. These properties are achieved directly due to the mathematical formulation used for defining this class, without the need for a global numerical optimization step. We also provide four example spline types within this class. These examples show how guaranteed self-intersection-free curve segments can be achieved, regardless of the placement of control points, which has been a limitation of prior interpolating curve formulations. In addition, they present how perfect circular arcs and linear segments can be formed by splines within this class, which also have been challenging for prior methods of interpolating curves.