An algorithm to improve parameterizations of rational Bézier surfaces using rational bilinear reparameterization
Highlights
► Only bilinear surfaces have uniform iso-parametric curves. ► Only rectangles have both uniform and orthogonal iso-parametric curves. ► We give the bilinear reparameterizations of rational Bezier surfaces. ► We present an optimization algorithm to improve surface parameterizations. ► Our optimization algorithm is based on the bilinear reparameterization.
Introduction
Freeform surfaces play an increasingly important role in contemporary Computer Aided Design (CAD). While the manufacturers are mainly concerned with the final geometric shapes, most algorithms [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13] for surface rendering (e.g. texture mapping), tessellation and blending applications are highly dependent on the surface parameterization. To generate a target geometric shape, the control points and their weights of the rational Bézier surface are adjusted by the designer. Such modification may destroy some desirable properties of the surface parameterization such as the uniformity and orthogonality of iso-parametric curves (see Fig. 1), and affect the subsequent surface manipulations such as surface tessellation and surface rendering (see Fig. 2). As a result, either the designer is forced to make a conservative modification, or some reparameterization technique has to be introduced to improve the surface parameterization, which is the aim of our paper.
In the past twenty years, how to achieve optimal parameterization of Bézier curves has been studied extensively in the literature such as [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24]. Farouki [15] identified arc-length parameterization as the optimal parameterization of Bézier curves. By minimizing an integral which measures the deviation from arc-length parameterization, the optimal representation is obtained by solving a quadratic equation. Jüttler [16] presented a simplified approach to Farouki’s result by using a back substitution in the integral. Costantini et al. [14] obtained closer approximations to the arc-length parameterization by applying composite reparameterizations to Bézier curves.
To our knowledge, little attention has been paid to the Bézier surface reparameterization. The results of rendering and tessellation applications for Bézier surfaces largely depend on the parameterization quality. Moreover, a parameterization with uniform and orthogonal iso-parametric curves will lead to more robust and stable computations for derivative based algorithms such as surface intersection, curvature computation, and so on [1], [6], [9], [11]. To perform texture mappings on a Bézier surface, the parametric coordinate of the surface is usually reused as the texture coordinate. If the iso-parametric curves are far from being uniform and orthogonal, there will be large distortion of the texture image on the surface (see Fig. 2(b)). To tessellate a Bézier surface, most existing algorithms [4], [7], [6] map a triangulation of the parameter domain onto the surface (see Fig. 2(c) and (d)). Similar to texture mapping, the final tessellation results largely depend on the surface parameterization. Yang et al. [25] presented an algorithm to improve the Bézier surface parameterization based on Möbius transformations [26], [25], which can change only the distribution of iso-parametric curves, but not their shape. To obtain more uniform iso-parametric curves, a rational bilinear reparameterization algorithm was also presented in [25]. However, only the uniformity of iso-parametric curves was considered therein. Furthermore, the rational bilinear reparameterization coefficients are determined by a trivial interpolation method, which is only suitable for a special surface case. Different from the curve cases, a surface parameterization with only uniform iso-parametric curves is usually not enough for CAD applications. Besides the uniformity of iso-parametric curves, orthogonality is also considered as an important factor in these applications [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]. A surface parameterization with uniform and orthogonal iso-parametric curves not only preserves the appearance of texture, but also avoids degenerate elements for the tessellation application. From our point of view, the lack of satisfying parameterizations with both uniform and orthogonal iso-parametric curves is the bottleneck for Bézier surface rendering and tessellation algorithms to achieve better quality results.
In this paper, we first study the differential geometry of rational Bézier surfaces and try to find out the surfaces with uniform and orthogonal iso-parametric curves. We conclude that the only rational Bézier surface with uniform iso-parametric curves is a bilinear surface, and the only rational Bézier surface with both uniform and orthogonal iso-parametric curves is a rectangle. As a general surface has no parameterizations with completely uniform and orthogonal iso-parametric curves, we then present an optimization algorithm to improve the uniformity and orthogonality of iso-parametric curves based on the rational bilinear reparameterizations. In the optimization procedure, a nonlinear energy measuring the uniformity and orthogonality deviations is formulated. To minimize the uniformity and orthogonality deviations, a discrete version of the formulated energy is presented and then minimized numerically. The examples indicate that, in practice, the algorithm produces significantly more uniform and orthogonal iso-parametric curves across the rational Bézier surfaces.
The paper is organized as follows. Section 2 discusses the rational Bézier surfaces with uniform and orthogonal iso-parametric curves and shows how to use the rational bilinear reparameterizations to achieve more uniform and orthogonal iso-parametric curves across the rational Bézier surfaces. In Section 3, three examples are given to show the performance of our algorithm and Section 4 concludes the paper.
Section snippets
An optimization method for minimizing uniformity and orthogonality energies
In this section, we first try to find the surfaces with uniform and orthogonal iso-parametric curves. Then an optimization algorithm based on rational bilinear reparameterizations is presented to improve the uniformity and orthogonality of iso-parametric curves for general rational Bézier surfaces.
Experimental results
Our algorithm is implemented on a PC with an Intel 3.06 GHZ CPU, 2G Memory and Microsoft Visual Studio 2008. To show the performance of our algorithm, some examples are given below. In all examples, we perform the optimization using 11×11 sample points, and three different orthogonality weights .
Fig. 6 illustrates the optimization results of a rational Bézier surface of degree 3×3 shown in Fig. 5. To compare different results, each surface is illustrated using three methods:
Conclusions and future work
In this paper, we conclude that the only rational Bézier surfaces with uniform iso-parametric curves are bilinear surfaces, and the only rational Bézier surfaces with uniform and orthogonal iso-parametric curves are rectangles. Moreover, to improve the uniformity and orthogonality of iso-parametric curves for general rational Bézier surfaces, an optimization algorithm using the rational bilinear reparameterizations is presented. The coefficients of the rational bilinear reparameterization are
Acknowledgments
The authors thank the anonymous reviewers for their valuable suggestions. This work was supported by the China national natural science foundation (61202146, 61272243, 61070093 and U1035004), Shandong province outstanding young scientist research award fund (BS2012DX014 and BS2009DX026), independent innovation foundation of Shandong university, IIFSDU (11150072614024) and the ERC starting grant 257453 COSYM.
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