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This paper presents a novel approach for constructing a piecewise triangular cubic polynomial surface with C1 continuity around a common corner vertex. A C1 continuity condition between two cubic triangular patches is first derived using mixed directional derivatives. An approach for constructing a surface with C1 continuity around a corner is then developed. Our approach is easy and fast with the virtue of cubic reproduction, local shape controllability, C2 continuous at the corner vertex. Some experimental results are presented to show the applicability and flexibility of the approach.