T-Splines: a new schema for C2 spline interpolation

Interactive CAGD systems produce faster and more accurate curve and surface analysis and design by means of geometric or algebraic interpolation algorithms, where users can directly modify and edit the position and shape of curves or surfaces. For this reason, many claim that interpolation schemata are more flexible and intuitive than approximation schemata. Also, in engineering and architectural design problems there are position and slope constraints to fulfill that, with standard approximation schemata, are harder to produce. However, interpolation methods are limited with respect to the analytic properties of the resulting curves and surfaces. Piecewise polynomial interpolation methods only guarantees a C¹ continuity order, which is usually unacceptable. In this work we propose an interpolation method, named T-Splines or tangent splines. Starting with interpolating points and derivatives, a T-Spline is the (approximating) cubic Spline over an ancillary control graph, whose approximation is also a solution of the required interpolation. We present the necessary conditions for finding the ancillary control graph from the interpolation points and derivatives. In addition we give a nonuniform parametrization schema such that a nonuniform B-Spline approximation of the ancillary control points is a C² piecewise polynomial interpolation of the original control points. Some applications are also discussed