A Modified de Casteljau Subdivision that Supports Smooth Stitching with Hierarchically Organized Bicubic Bezier Patches

TL;DR

The paper tackles the long-standing problem of achieving $C^1$ or $G^1$ continuity when stitching tensor-product Bezier patches at extraordinary vertices, where conventional polynomial representations fail to guarantee smoothness. It introduces a simple yet effective modification of the de Casteljau subdivision that preserves boundary conditions and enables independent per-patch evaluation, producing hierarchically organized bicubic Bezier patches that are stitched with $G^1$ continuity at extraordinary vertices and $C^1$ continuity along edges emanating from them. The approach yields real-time interactive modeling of piecewise smooth manifolds with arbitrary topology and supports topology changes through vertex insertion, effectively converting quad meshes into smooth surfaces. The work provides a theoretical analysis of continuity constraints, a practical subdivision algorithm, and an implementation pathway that integrates with existing quad-remeshing strategies (e.g., Catmull-Clark or Doo-Sabin) to produce visually and geometrically high-quality results. This framework promises improved control for designers and real-time performance for interactive modeling, with potential extensions to higher-order continuity and more complex topologies.

Abstract

One of the theoretically intriguing problems in computer-aided geometric modeling comes from the stitching of the tensor product Bezier patches. When they share an extraordinary vertex, it is not possible to obtain continuity C1 or G1 along the edges emanating from that extraordinary vertex. Unfortunately, this stitching problem cannot be solved by using higher degree or rational polynomials. In this paper, we present a modified de Casteljau subdivision algorithm that can provide a solution to this problem. Our modified de Casteljau subdivision, when combined with topological modeling, provides a framework for interactive real-time modeling of piecewise smooth manifold meshes with arbitrary topology. The main advantage of the modified subdivision is that the continuity C1 on a given boundary edge does not depend on the positions of the control points on other boundary edges. The modified subdivision allows us to obtain the desired C1 continuity along the edges emanating from the extraordinary vertices along with the desired G1 continuity in the extraordinary vertices.

Figures (24)

• Figure 1: Two images from our interactive modeling session with modified de Casteljau subdivision. Users can visually experience that inserting an edge between two opposite faces of the cube opens a $G^1$ smooth hole by creating a 10-sided face.
• Figure 2: Comparison our method with AS&C-method akleman2017 around a three-valent extraordinary vertex. As it can be seen in this example, AS&C-method creates broken lines that cause $C^1$ discontinuity along the edges emanating from that extraordinary vertex while our method removes $C^1$ discontinuities by using hierarchically organized Bézier patches that are constructed by our modified de Casteljau subdivision.
• Figure 3: Comparison our method with AS&C-method akleman2017 around the complicated neighborhood of an extraordinary vertex with 10-valent. This extraordinary vertex comes from a ten-sided face that is obtained by combining two quadrilateral faces by inserting a single edge shown in Figure \ref{['fig_10SidedFace_6']}. The insert edge operation creates the hole shown here. As can be seen in this example, our method removes the discontinuity $C^1$ along the boundary edges emanating from the extraordinary vertex.
• Figure 4: The de Casteljau subdivision splits the original control polyhedron into four topologically identical copies. Here, the control points of the original $4 \times 4$ control polyhedron are shown in blue and the control points of one of the copies are shown with yellow control points. The positions of the control points of each of these copies produce the original Bézier formula.
• Figure 5: This figure shows the De-Casteljau kernels to produce each control.