We consider here a natural spline interpolation based on reduced data and the so-called exponential parameterization (depending on parameter lambda is an element of [0, 1]). In particular, the latter is studied in the context of the trajectory approximation in arbitrary euclidean space. The term reduced data refers to an ordered collection of interpolation points without provision of the corresponding knots. The numerical verification of the intrinsic asymptotics alpha(lambda) in gamma approximation by natural spline gamma(N)(3) is conducted here for regular and sufficiently smooth curve gamma sampled more-or-less uniformly. We select in this paper the substitutes for the missing knots according to the exponential parameterization. The outcomes of the numerical tests manifest sharp linear convergence orders alpha(lambda) = 1, for all lambda is an element of [0, 1). In addition, the latter results in unexpected left-hand side dis-continuity at lambda = 1, since as shown again here a sharp quadratic order alpha(1) = 2 prevails. Remarkably, the case of alpha(1)=2 (derived for reduced data) coincides with the well-known asymptotics established for a natural spline to fit non-reduced data determined by the sequence of interpolation points supplemented with the corresponding knots (see e.g. [1]).