This paper presents a new approach to rendering triangular algebraic free form surfaces. A hierarchical subdivision of the surface with associated tight bounding volumes provides for quick identification of the surface regions likely to be hit by a ray. For each leaf of the hierarchy an approximation to the corresponding surface region is stored. The approximation is used to compute a good star...

We present a method to efficiently construct and render a smooth surface for approximation of large functional scattered data. Using a subdivision surface framework and techniques from terrain rendering, the resulting surface can be explored from any viewpoint while maintaining high surface fairness and interactive frame rates. We show the approximation error to be sufficiently small for severa...

We show that parametric context-sensitive L-systems with affine geometry interpretation provide a succinct description of some of the most fundamental algorithms of geometric modeling of curves. Examples include the Lane-Riesenfeld algorithm for generating B-splines, the de Casteljau algorithm for generating Bézier curves, and their extensions to rational curves. Our results generalize the prev...

Fitting curve and surface by least-regression is quite common inmany scientific fields. It, however cannot properly handle noisy data with impulsive noises and outliers. In this article, we study l1-regression and its associated reweighted least squares for data restoration. Unlike most existing work, we propose the l1-regression based subdivision schemes to handle this problem. In addition, we...

This paper proposes four methods to approximate circular arcs using quartic Bezier curves. Barycentric coordinates of two/three combination of control points are used to obtain an optimal approximation. Interior control points of quartic Bezier curves are found by satisfying G data from given circular arcs. The maximum errors between circular arcs and approximated curves are calculated using Ha...

1 Piecewise Polynomials 2 1.1 Barycentric and Bernstein-Bézier Bases . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 B-Spline Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Trimmed Freeform Patches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Implicit Algebraic Surface Patches . . . . . . . . . . . . . . . . . . . . . ....

Lower bounds on the generation of smooth bi-cubic surfaces imply that geometrically smooth (G) constructions need to satisfy conditions on the connectivity and layout. In particular, quadrilateral meshes of arbitrary topology can not in general be covered with G-connected Bézier patches of bi-degree 3 using the layout proposed in [ASC17]. This paper analyzes whether the pre-refinement of the in...

In order to achieve a smooth surface while rendering a triangle-based mesh, we need to eliminate the mismatch between the smoothness of the shading and the non-smoothness of the geometry that is particularly visible at silhouettes. To eliminate these artifacts, we substitute the geometry of a cubic triangular Bézier curved patch for a triangular flat geometry. Meanwhile a subdivision algorithm ...

Fitting a sparse surface to approximate vast dense data is of interest for many applications: reverse engineering, recognition and compression, etc. The present work provides an approach to fit a Loop subdivision surface to a dense triangular mesh of arbitrary topology, whilst preserving and aligning the original features. The natural ridge-joined connectivity of umbilics and ridge-crossings is...

We introduce a novel method to convert trimmed NURBS surfaces to untrimmed subdivision surfaces with Bézier edge conditions. We take a NURBS surface and its trimming curves as input, from this we automatically compute a base mesh, the limit surface of which fits the trimmed NURBS surface to a specified tolerance. We first construct the topology of the base mesh by performing a cross-field based...