Shape control of swept surface with profiles
Introduction
The sweeping operator is a powerful function in geometric modeling for describing the shapes of complex surfaces or solids. The swept surface or volume can then be described by orbit sets which are formed by curves, surfaces or objects moving through a spine (also called as a space trajectory). The method is simple and efficient, and requires only the specification of the moving object and of the spine along which the object moves. The swept surface can be expressed as follows:Where r(u) is the spine or trajectory, c1(u,v) and c2(u,v) are the planar contours which can be deformed and twisted along the spine, and N and B are the unit vectors of a moving frame along the spine. The method has many good characteristics such as:
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Reduces the surface design to curve modification so that the design problem is simplified.
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Reduces the depth of the CSG feature tree in the geometric modeling system and reduces the operator steps in the product design so that the design efficiency is improved.
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The complex shape can be described intuitively in terms of a contour moving along a spine with twist and deformation.
The non-rational expression in Eq. (1) for the swept surface is usually approximated by a NURBS surface for compatibility with the data structures and algorithms in the geometric modeling systems. With a NURBS surface, the shape control technique, such as control point relocation and weight modification can be adopted. But a production designer or a user of the commercial CAD software may be unfamiliar with the mathematical basis of NURBS surfaces, so these surface modification techniques may be too professional, less intuitive, and difficult to master. Other methods can be used to control the shape of the swept surface using a profile or a transform matrix [1], [3], [6], [9]. But the shape deformation is limited to the scaling deformation of the contour on the local moving frame along the spine. Therefore, the ability to modify the swept surface shape is limited so cannot utilize all of the advantages of the sweep method and cannot provide an intuitive and convenient method for shape modification.
This paper presents some shape control methods. But many problems need to be resolved such as how to define the deformed point on the surface and how much of the region on the contour is deformed by the profiles (referred to as the deforming curve). These issues will be discussed in 2 Terms and definitions, 3 Local deformation of contours. Furthermore, some deformation rules introduced in Section 4 are used to determine the deformed regions on the contours according to the shape of each profile curve. Finally, several examples are given that have verified the robustness and efficiency of the method implemented in the commercial geometric modeling software gems 5.0 which was developed by the CAD Center of Tsinghua University.
Section snippets
Terms and definitions
The contour C(u,v) in Eq. (1) can be considered to be the contour C(u) deformed or twisted by the profile curve or by a twist angle along the spine. The profile can be reparameterized with the spine r(v) such that which is simply denoted as profile curve f(v). One effective method for constructing the swept surface [2], [6], [7] is to form the local moving frame at the selecting points on the spine and to set the intermediate contours on these local moving frames by
Local deformation of contours
From the unified NURBS expression for the swept surface, the shape of the swept surface can be changed by the relocation of its control points or by modifying its weights, but the procedure is not intuitive for users. The characteristics of the swept surface can be used to reduce the local deformation of the surface to deform the intermediate contours. The local region on the swept surface which is to be deformed is defined by the user. The following defines two types of deformation regions on
Deformation rules for the swept surface
For flexible and convenient control of a surface shape by the user, the surface modification region needs to intuitively anticipate by the user. This section discusses the fourth issue described in Section 2, that is, how to deform the contours with profiles and the size of the contour region deformed by the profiles. The different contours regions deformed by the profiles are used to propose five types of deformation rules for the swept surface.
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Bi-directional symmetric deformation. This rule
Sweep algorithm with deformation
If twist is considered along the spine, each intermediate contour is rotated about the tangential vector of the spine with twist angles equally divided according to the spine arc-length ratio. For NURBS curves, the sweep algorithm with deformation and twist can be described as follows:
Step 1. To maintain compatible between the spine and the profiles, simultaneously subdivide the spine and the profiles to obtain the split points on the spine and the profiles.
Step 2. Place the local moving frame
Examples
The following examples were implemented in the commercial CAD system gems 5.0, which was developed by the National CAD Center of Tsinghua University. gems 5.0 is a feature-based geometric modeling system with many advanced functions such as drawing with intelligent PDA, solid and surface modeling, assembly, 2D engineering drawing, FEA, sheet-metal and rendering. Many Chinese mechanical engineering factories and companies use the system. Example 1 The contour is a circle, the spine is a line segment, and
Conclusion
Flexible and convenient control of the surface shape by the user in a commercial geometric modeling system is presented using a simple, intuitive deformation method in which the shape control of the swept surface is reduced to modifying the shapes of profiles or the spine. Two user selectable functions, an intersection method and a parameter method, are proposed to define the deforming points on the profiles, the deformed points on each intermediate contour, the deforming distance and the
Acknowledgements
This work was supported by the National Natural Science Foundation of China (No: 69772019).
Guo-ping Wang is an associate professor in the Department of Computer Science and Technology at Peking University. He received his BS and MS degrees from Harbin Institute of Technology in 1987 and 1990, respectively, and PhD degree from the Fudan University in 1997, all in Mathematics. From 1997–1999, he was a postdoctoral researcher in Department of Computer Science and Technology in Tsinghua University at Beijing. His current research interests are in Virtual Reality, Computer Graphics and
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Guo-ping Wang is an associate professor in the Department of Computer Science and Technology at Peking University. He received his BS and MS degrees from Harbin Institute of Technology in 1987 and 1990, respectively, and PhD degree from the Fudan University in 1997, all in Mathematics. From 1997–1999, he was a postdoctoral researcher in Department of Computer Science and Technology in Tsinghua University at Beijing. His current research interests are in Virtual Reality, Computer Graphics and Computer-Aided Geometric Design.
Jia-guang Sun is a professor in the Department of Computer Science and Technology at Tsinghua University. He is also Director of National CAD Engineering Center at Tsinghua University and Academician of the Chinese Academy of Engineering. He received his BS degree in Computer Science from the Tsinghua University in 1970. From 1982–1986, he was a visiting scholar in UCLA. His current research interests are in Computer-Aided Geometric Design, Computer Graphics and Product Data Management.