A Parametric-Space-Based Scan-Line Algorithm for Rendering Bicubic Surfaces

Abstract

In recent years hidden-surface-removal algorithms have been proposed to render curved surfaces, especially bicubic (or bipolynomial) patches. Most of these algorithms use the scan-line principle to determine the scene visibility. Lane, Carpenter, Whitted, and Blinn' proposed three different methods. The first algorithm is a subdivision method that stops when a desired degree of surface flatness is achieved. Then subpatches are approximated by polygons processed by a polygonoriented algorithm. Whitted's method computes the exact intersection of some curves in the patch with every scan plane and approximates the patch's intersection with the plane by a polygonal line defined by these points. The exact intersection curves are the cubics defining the boundaries of the patch, isoparametric curves in the patch, and cubic approximations of the silhouettes. Finally, Blinn proposes a method that is essentially a z-buffer algorithm in which numerical problems are simplified using the intersection of the scan planes with the boundaries and the silhouettes of the patch.Lane and Carpenter2 later improved and generalized their algorithm, and Schweitzer and Cobb3 proposed an algorithm similar to Whitted's method that offered higher quality silhouettes. Some methods compute curved-surface visibility in parametric space. These algorithms use different techniques employed in polygon-oriented methods. Strasser4 and Forest5 use a z-buffer algorithm; Griffiths'5'7 and Ohno's8 algorithms are based on depth comparison between points of a grid defined in the patch and some subpatches; Hornung et al.9 simply compute the boundaries of constant visibility areas in the patch.Griffiths'°has proposed a scan-line algorithm that does some work in the parametric plane. This algorithm obtains an approximation of the patch silhouettes in the parametric space. Griffiths' method approximates the intersection of the scan plane and the patch with chains of straight segments instead of considering the segments separately, as other algorithms do. In this way his method speeds up the visibility calculation considerably.We will present a new scan-line algorithm for curved surfaces that does most of the computations in the para-

A scanline algorithm for displaying trimmed surfaces by using Bézier clipping

“…In this method, silhouettes are extracted by using normal vectors stored in the grid cells. Pueyo and Brunet (1987) improved upon Griffiths' method; they proposed that curved surfaces are decomposed into grid cells beforehand, intersections of the restricted scanline and curved surfaces are calculated without interpolation (making use of the y-coordinates stored in the grid cells), and interpolation in parametric space is employed for intersections of the other scanlines. Silhouettes are detected by using the z-components of normal vectors stored in the grid points.…”

“…Even though the papers referred to here use scanline algorithms (Blinn 1978;Whitted 1978;Clark 1979;Lane et al 1980;Schweitzer and Cobb 1982;Griffiths 1984;Pueyo and Brunet 1987) the triangle formed by the scan segment and the light source is treated as a scan plane; the scanline algorithm mentioned before can be employed. That is, the shadow sections on the scan segment can be determined by using the intersection test between the scanline and curved surfaces when viewed from the light source.…”