Curve and Surface Constructions

This chapter introduces algorithms for the generation of Curves and Surfaces. It discusses some of the most fundamental interpolation and approximation methods in computer aided geometric design (CAGD). The developments emphasizes on Bezier and B-spline techniques because of their intuitive geometric definitions. The chapter focusses on polynomial curve methods, including Lagrange interpolation, point approximation, and Hermite interpolation. Next, a piecewise polynomial scheme, C 2 cubic spline interpolation is presented. For C 2 cubic spline interpolation, the choice of end conditions is important for the shape of the interpolant near the endpoints. The focus then moves to surface methods. The chapter describes interpolation to boundary curve data with Coons patches, interpolation to rectangular data with tensor product surfaces, approximation to large sets of data, and interpolation to point and derivative data. Mirroring the curve presentation, a piecewise polynomial surface scheme, C 2 bicubic spline interpolation, is discussed. The chapter concludes with a discussion on volume deformations.

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