BIT Numerical Mathematics (2006)46:000-000 c  Springer 2006. DOI:10.1007/s10543-000-0000-x GEOMETRIC INTERPOLATION BY PLANAR CUBIC G 1 SPLINES J. KOZAK 1 and M. KRAJNC 2 1 FMF and IMFM, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia. Email: jernej.kozak@fmf.uni-lj.si 2 IMFM, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia. Email: marjetka.krajnc@fmf.uni-lj.si Abstract. In this paper, geometric interpolation by G 1 cubic spline is studied. A wide class of sufficient conditions that admit a G 1 cubic spline interpolant is determined. In particular, convex data as well as data with inflection points are included. The existence requirements are based upon geometric properties of data entirely, and can be easily verified in advance. The algorithm that carries out the verification is added. AMS subject classification (2000): 65D05, 65D07, 65D17. Key words: cubic spline curve, G 1 continuity, geometric interpolation. 1 Introduction. Geometric interpolation schemes, introduced in [1], are becoming more and more important practical tool in the approximation of curves and surfaces. Per- haps the main reason could be found in the fact that such interpolants please the human eye more than usual linear counterparts. This is clearly a consequence of the basic principle of the geometric interpolation: free parameters of a para- metric interpolant are determined by geometric conditions only. An interpolant should pass through a point, should have a prescribed tangent or normal direc- tion, a curvature, etc. But, no additional artificial conditions are imposed on it such as at which parameter values the interpolation conditions should be met. Since no free parameters are used ineffectively, geometric interpolation often re- sults in higher approximation order than one would expect from the functional case. But geometric schemes involve a nonlinear part, and the questions like the existence and the efficient implementation require a more subtle analysis. Most of results obtained are based upon the asymptotic analysis, and only a few papers examine geometric conditions on given data ([7], [5], [4]). For an excellent recent overview of planar Hermite geometric interpolation the reader is referred to [2]. However, results offered by the asymptotic analysis are not always adequate in practical applications. If one is merely looking for an interpolant of a nice shape, suppositions like ”if data points are sampled dense enough” are not very encouraging. Therefore robust algorithms should be based upon conditions that ensure the existence in advance if only possible. But in geometric interpolation,