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Summary: We present a geometric approach to piecewise quadric $C^1$-interpolants constructed algebraically by {\it W. Dahmen} [Mathematical methods in computer aided geometric design, Pap. Int. Conf., Oslo/Norw. 1988, 181-193 (1989; Zbl 0682.41001)]. These piecewise quadrics interpolate the vertices of a triangular net with prescribed normals. In Dahmen’s construction certain free parameters were set to arbitrarily chosen constants. Our approach provides a geometric interpretation of these constants. It renders the Powell-Sabin interpolant as a special case and provides another class of quadric splines by dualization. Furthermore, we show how to avoid the global dependencies of Dahmen’s and {\it B. Guo}’s transversal system [Modeling arbitrary smooth objects with algebraic surfaces, Technical Report CORNELLCS//TR91-1226, Cornell University, Computer Science Department (1991)].