دانلود رایگان مقاله منحنی گویای موزون ماتریس و سطوح

Abstract

Rational curves and surfaces are powerful tools for shape representation and geometric modeling. However, the real weights are generally difficult to choose except for a few special cases such as representing conics. This paper presents an extension of rational curves and surfaces by replacing the real weights with matrices. The matrix weighted rational curves and surfaces have the same structures as the traditional rational curves and surfaces but the matrix weights can be defined in geometric ways. In particular, the weight matrices for the extended rational Bézier, NURBS or the generalized subdivision curves and surfaces are computed using the normal vectors specified at the control points. Similar to the effects of control points, the specified normals can be used to control the curve or the surface's shape efficiently. It is also shown that matrix weighted NURBS curves and surfaces can pass through their control points, thus curve or surface reconstruction by the extended NURBS model no longer needs solving any large system but just choosing control points and control normals from the input data.

7. Conclusion

We have generalized conventional rational curves and surfaces to matrix weighted rational curves and surfaces. The new models of rational curves and surfaces maintain the same structures and many similar properties as the traditional ones but the matrix weights permit clear shape parameters for geometric modeling. In addition to control points, control normals have been used to control the shapes of extended rational Bézier or extended NURBS curves and surfaces as well as several extended rational subdivision surfaces efficiently. Due to their nice properties such as almost-interpolation of control points, matrix weighted NURBS curves and surfaces are also suitable for curve or surface reconstruction from sampled points even with no need of solving large systems. As matrix weighted rational Bézier or NURBS curves and surfaces can be transformed from and to the traditional rational representations of curves and surfaces, it is convenient to plug the proposed models into Bézier or NURBS based modeling systems. Several techniques need to be developed further in the future which can make the matrix weighted rational models even more useful for geometric modeling or surface reconstruction: (1) Define the weight matrices in some other geometric ways such that the obtained curves or surfaces can have other desired features; (2) Reconstruct curves and surfaces by adaptively choosing the control points and control normals from the input data; (3) Improve the smoothness of matrix weighted rational subdivision surfaces by applying geometric driven or non-stationary subdivision rules on the weight matrices.