Elsevier

Computer-Aided Design

Volume 38, Issue 6, June 2006, Pages 641-660
Computer-Aided Design

Interpolating G1 Bézier surfaces over irregular curve networks for ship hull design

https://doi.org/10.1016/j.cad.2006.02.005Get rights and content

Abstract

We propose a local method of constructing piecewise G1 Bézier patches to span an irregular curve network, without modifying the given curves at odd- and 4-valent node points. Topologically irregular regions of the network are approximated by implicit surfaces, which are used to generate split curves, which subdivide the regions into triangular and/or rectangular sub-regions. The subdivided regions are then interpolated with Bézier patches. We analyze various singular cases of the G1 condition that is to be met by the interpolation and propose a new G1 continuity condition using linear and quartic scalar weight functions. Using this condition, a curve network can be interpolated without modification at 4-valent nodes with two collinear tangent vectors, even in the presence of singularities. We demonstrate our approach in a ship hull.

Introduction

A curve network is often used to design a complicated 3D free-form shape. Creating the curve network is easier than directly manipulating surfaces and it also enables more intuitive modelling. For that reason, among others, curve networks have been used in applications such as car body design, airplane fuselage design and ship hull design. After designing a 3D shape using a curve network, smooth surfaces are generated by interpolating (or ‘filling in’) the curve network with appropriate patches. In a ship design, a hull is usually designed with a curve network that consists of several cross-sectional curves and characteristic curves lying along the hull (see Fig. 1). One of the most important objectives in designing the hull of a ship is to obtain the maximum speed with a given propulsion power by reducing water resistance. Unlike small vessels such as yachts, large commercial ships have complicated freeform shapes in their fore-body, such as a bulbous bow, which is intended to reduce water resistance. Therefore, the curve network of the hull of a ship may include several topologically irregular regions (i.e. three-, five- and six-sided regions) and extraordinary vertices with valencies 3, 4 and 5, around the fore-body. Such curve networks are usually designed with non-uniform B-spline curves.

Given the irregular curve network produced by a ship designer, we need to generate a surface that satisfies the following conditions:

Tensor-product patches: A hull surface should consist of tensor-product surfaces such as Bézier and B-spline patches, which can be used in the shipbuilding CAD/CAM systems.

Interpolation of given curve network: Since, the curve network embodies the satisfaction of the design constraints, a hull surface should interpolate the curve network exactly. If that is mathematically impossible, the curve network should be interpolated as closely as possible.

Fair hull form surfaces: A hull should be smooth. At least G1 or tangent-plane continuity should be achieved.

Shipbuilding is an on-demand production industry, and a different design is often required for each ship that is ordered. The construction of hull surfaces from curve networks should therefore be as automatic as possible.

Constructing G1 patches from a given curve network has been a generally important problem in computer graphics and CAD/CAM, and a lot of solutions have been presented. Existing works can be classified into two categories according to the type of patches used: tensor-product surfaces such as Bézier and B-spline patches [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11] and subdivision surfaces [12], [13], [14], which have received considerable attention from computer graphics community. Methods of interpolating curve networks can be also separated into single patch approaches that interpolate a whole curve network with a single patch and piecewise patch approaches that use a composite surface for interpolation of a curve network. Most single patch approaches use subdivision surfaces [15], [16], which can easily represent arbitrary topologies, while piecewise patch approaches usually adopt tensor-product patches.

Most piecewise patch approaches are based on the following framework:

  • (1)

    Generate initial patches in each region of the given curve network, or assume that the initial patches are supplied with the curve network.

  • (2)

    Modify the initial patches by applying a G1 continuity condition to generate final G1 patches. Existing piecewise patch approaches assume that a given curve network only has three- and four-sided regions. But in practice, it is necessary to generate G1 patches from topologically irregular curve networks.

We will present a local method of constructing G1 Bézier patches from an irregular curve network. The advantages of our method are as follows:

Irregular curve network: Our method can handle a topologically irregular curve network represented in terms of nth degree Bézier curves, with three-, four-, five-, six-sided regions including T-junctions and vertices with valencies of 3, 4 and 5.

Automatic subdivision scheme: All irregular regions of the given curve network are subdivided into three- and/or four-sided sub-regions, each of which can be spanned by a Bézier patch. This requires additional information beyond the curve network.

Quality initial patches: We generate fair initial Bézier patches from implicit surfaces, and use these to determine the inner control points of the final G1 Bézier patches.

Local interpolation scheme: Our method can generate a patched surface that interpolates the given curve network keeping given curves unchanged. Peters [6] and Liu [9] have presented similar algorithms for curve network with 4-valent nodes, but they are global schemes while ours is a completely local scheme. For this, a new G1 continuity condition with linear and quartic scalar weight functions is proposed and used for constructing G1 Bézier surfaces that interpolate the given curve network.

The rest of this paper is structured as follows. In Section 2, we start with a brief survey of previous work on the generation of G1 patches from curve network and describe their limitations. In Section 3, we present an overview of our construction algorithm for G1 patches. Section 4 describes our subdivision scheme for topologically irregular regions using split curves generated from implicit surfaces. Section 5 describes the method that we use for generating initial Bézier patches using implicit surfaces. In Section 6, a G1 continuity condition with linear and quartic degree scalar weight functions is proposed. Singular cases of this G1 continuity condition are analyzed and we show how to avoid the singularities by using additional degree of freedom. We analyze the errors of our surface construction algorithm in Section 7. In Section 8, we verify our approach on a ship hull design. Section 9 concludes this paper.

Section snippets

Previous work

In this section, we give a brief survey of the work related to the G1 interpolation of a given curve network. Bézier [1] first proposed a method of connecting two adjacent Bézier patches with G1 continuity and it was developed by Farin [2] and Sarraga [3], who proposed a sufficient condition for G1 continuity, in terms of the control points of adjacent Bézier patches. Based on Farin's work, Piper [4] proved that biquartic Bézier patches are the simplest that can interpolate a given cubic Bézier

Overview

Our goal is to generate tensor-product G1-continuous surfaces that exactly interpolate a given curve network (or within a tolerance acceptable in the shipbuilding industry). In this study, we use piecewise Bézier patches. Fig. 2 shows the stages in our algorithm, which are explained as follows:

  • (1)

    Given an irregular curve network, we subdivide all pentagonal and hexagonal regions into triangular and/or rectangular sub-regions, and regions bounded by a T-junction are also split into three triangular

Subdivision of irregular curve network

In this section, we describe a method of subdividing topologically irregular regions of a given curve network into three- and/or four-sided regions, to allow the irregular curve network to be interpolated with piecewise Bézier patches.

Initial surface generation

An initial approximation of piecewise Bézier patches is needed to determine the off-boundary points of the final patches so that they meet the G1 continuity condition. In this section, we will summarize various methods for generating patches from boundary curves and propose a method of generating a fair net of initial Bézier patches using implicit surfaces that interpolate the boundary curves approximately.

G1 surface construction

After subdividing all irregular regions on the original curve network, we need to generate initial piecewise Bézier patches in all three- and four-sided regions of the subdivided curve network, and apply G1 continuity conditions to them.

Analysis of G1 surface construction algorithm

We will now analyze the difference between the original curve network and the G1 Bézier surfaces generated by our method, and also verify the G1 continuity of these surfaces.

Results

We will now describe the application of the techniques we have described to a simple curve network, and then to an actual ship hull.

Summary and future work

We have presented a method of constructing a smooth patched surface that interpolates a given irregular curve network. Our construction algorithm has the following features:

  • It can deal with a topologically irregular curve network consisting of piecewise Bézier boundary curves of degree n (up to six-sided region). A network that is not Bézier form, or which is incompatible with tensor-product patches, can be approximated by Bézier curves of common degree to a specified tolerance.

  • If nodes on a

Acknowledgements

This work was supported by grant no. R01-2002-000-00061-0 and R01-2005-000-11257-0 from the Basic Research Program of the Korea Science and Engineering Foundation.

Doo-Yeoun Cho is a post doctoral researcher in the Dept. of Naval Architecture and Ocean Engineering of Seoul National University, where he also received his BS in 1997, MS in 1999 and PhD in 2005. His research interests include CAD/CAM, geometric modeling and computer graphics.

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    Doo-Yeoun Cho is a post doctoral researcher in the Dept. of Naval Architecture and Ocean Engineering of Seoul National University, where he also received his BS in 1997, MS in 1999 and PhD in 2005. His research interests include CAD/CAM, geometric modeling and computer graphics.

    Kyu-Yeul Lee is a professor in the Dept. of Naval Architecture and Ocean Engineering, and in the Research Institute of Marine Systems Engineering of Seoul National University, Korea. He received his BS in 1971 at Seoul National University, and his MS in 1975 and PhD in 1982 at Technical University of Hannover, Germany, all in Naval Architecture. His research interests include geometric modeling, design automation, optimization, and CAD in shipbuilding.

    Tae-Wan Kim is an associate professor in the Dept. of Naval Architecture and Ocean Engineering, and in the Research Institute of Marine Systems Engineering of Seoul National University, Korea. He received a BS in Industrial Engineering from Hanyang University, Korea in 1985, MS and PhD in Computer Science from Arizona State University, USA in 1993 and 1996. From 1996 to 1999 he worked as a software engineer at EDS Inc. (formerly SDRC), USA where he was involved in developing the I-DEAS CAD/CAM system. His research interests include geometric modeling, NURBS curves and surfaces, and CAD/CAM.

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