Approximate spline of \(G^2\)-continuity on a generalized hyperbolic paraboloid
The authors provide a unified method to create a special kind of approximate splines of \(G^2\)-continuity on a generalized hyperbolic paraboloid. The proposed splines have a shape parameter added into the parametric form of the generalized hyperbolic paraboloid. The shape parameter is able to control or adjust a curve segment or a surface patch closing to the control polygon or net. Section 2 begins with some basic definitions and notations, e.g., of the control polygon. The most important part in this section is the parametric equation of generalized hyperbolic paraboloids in which the authors use a shape parameter and weight functions. In Section 3, the authors introduce curve segments over the generalized hyperbolic paraboloid. They start with the definition of the family of functions \(V\). The family together with weight functions which satisfy conditions from Theorem 1 are used to introduce the curve segment. In Theorem 1, the authors prove that the curve segment lies on the generalized hyperbolic paraboloid, interpolates the first and the last control point, and that the tangent direction in the first control point is identified with the first two control points and the tangent direction in the last control point is identified with the last two control points. Next, the authors analyse the curvature of the curve segments at the endpoints and the approximate behaviour of the shape parameter. After these analysis, the authors give a method of joining consecutive segments into a spline curve of \(G^2\)-continuity. At the end of Section 3, the authors give a brief description of properties of the proposed curve segments and spline curves. In Section 4, the authors describe how to improve the approximating effect to the control polygon. For this purpose they modify the parametric equation from Section 2. They introduce further weight functions. The modified curve has got similar geometric properties at the endpoints as the curve from Section 3, but it does not lie on the generalized hyperbolic paraboloid. Section 5 is devoted to tensor product patches and composite surfaces. First, the authors give the tensor product representation of the surface using the proposed basis functions. Next, they give some issues which arise in connecting two surface patches. Finally, some examples of the construction of \(G^2\)-continuous surfaces are given.
scientific article; zbMATH DE number 6155457