The problem of fitting reduced data \(Q_m\) is discussed here. Reduced data form the ordered sequence of interpolation points \(q_i=\gamma (t_i)\) in arbitrary Euclidean space. Here the corresponding unknown knots \(\mathcal {T}\) are replaced with \(\hat{\mathcal {T}}\) compensated by the so-called exponential parameterization determined by reduced data \(Q_m\) and a single parameter \(\lambda \in [0,1]\) . In sequel, a modified complete spline is used to interpolate \(Q_m\) with the aid of exponential parameterization. The main theoretical contribution of this work is to prove a linear convergence order in \(\gamma \) estimation by fitting \(Q_m\) (getting denser) with modified complete spline based on exponential parameterization for \(\lambda \in [0,1)\) . The latter holds for sufficiently smooth, regular curves sampled more-or-less uniformly. The asymptotics established here is subsequently verified numerically in affirmative as sharp. The respective tests are conducted on 2D and 3D curves.