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1. Introduction
1.1 Background and Motivation
1.2 Objectives and Contributions
1.3 Brief Overview of the Paper
alms2. Literature Reviewaround
2.1 Overview of Subdivision Surfaces
2.2 Subdivision Rules with Constant Mean Curvature
2.3 Related Works in Constant Mean Curvature Subdivision
shrink是什么意思3. Main Approach
3.1 Definition of Constant Mean Curvature Surfaces
3.2 Construction of Subdivision Scheme
3.3 Implementation Details
3.4 Analysis of Convergence and Accuracy
4. Experiments and Results
4.1 Experimental Setting and Data Preparation
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4.2 Evaluation Metrics
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4.3 Comparison with Other Subdivision Methods
4.4 Discussion of Results
5. Conclusion and Future Directions新东方外国语
5.1 Summary of Contributions
5.2 Limitations and Recours
5.3 Future Work and Directions
References1. Introduction
1.1 Background and Motivation
Subdivision surfaces play an important role in computer graphics and geometric modeling. They provide a flexible and efficient way to reprent smooth and complex shapes. The main idea of subdivision surfaces is to recursively refine a coar mesh to obtain a high-resolution surface. Each subdivision step introduces new vertices and adjusts the positions of existing vertices, according to a predefined t of rules or weights. The resulting surface is a limit of a quence of meshes, which approaches the smooth limit surface as the number of subdivisions approaches infinity.
One of the most important properties of subdivision surfaces is their ability to prerve certain geometric characteristics of the original surface, such as its curvature. Curvature is a basic notion in differential geometry and measures how much a surface deviates fro
m being flat. Curvature is important for many applications, such as surface fairing, shape analysis, and surface reconstruction. In particular, mean curvature is a widely ud curvature measure that captures the average curvature of a surface at each point. Mean curvature is a crucial quantity in physics-bad simulations, where it governs the dynamics of surface deformation and fracture.
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However, conventional subdivision schemes may not prerve mean curvature exactly, especially for irregular meshes or high-order surfaces. Some subdivision methods introduce excessive or oscillatory curvature, while others suppress curvature locally and result in flat areas. Therefore, there is a need for efficient and accurate subdivision schemes that can guarantee constant mean curvature on the resulting surfaces.dfh
1.2 Objectives and Contributions
The objective of this paper is to propo a new subdivision scheme that can generate surfaces with constant mean curvature. We aim to design a simple and computationally efficient algorithm that can be easily implemented and integrated into existing tools and pi
pelines. Our subdivision scheme is bad on a local smoothing operation that adjusts the positions of vertices bad on a weighted combination of their neighbors. We also introduce a scaling factor that controls the degree of curvature prervation and can be tuned by urs.
The main contribution of this paper is a novel subdivision scheme that satisfies the constant mean curvature property, while maintaining high accuracy and stability. We demonstrate through numerical experiments and comparisons that our method outperforms existing state-of-the-art algorithms in terms of convergence rate, curvature prervation, and visual quality. Moreover, our method is open-source and publicly available, which enables further extensions and applications.
1.3 Brief Overview of the Paper
The rest of the paper is organized as follows. Section 2 provides a brief review of related work in subdivision surfaces and mean curvature prervation. Section 3 prents the details of our approach, including the definition of constant mean curvature surfaces, the
construction of subdivision rules, and the analysis of convergence and accuracy. Section 4 reports the experimental results and comparisons with other methods. Finally, ction 5 concludes the paper and suggests future directions.2. Related Work
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2.1 Subdivision Surfaces
Subdivision surfaces were first introduced by Catmull and Clark in 1978 as a way to generate smooth surfaces from coar meshes. Since then, numerous subdivision schemes have been propod, each with its advantages and limitations. The main categories of subdivision schemes include linear, quadratic, cubic, and higher-order methods, as well as loop subdivision, butterfly subdivision, and Doo-Sabin subdivision.
The basic idea of subdivision surfaces is to refine a coar mesh by adding new vertices and adjusting the positions of existing vertices, bad on a t of predefined rules or weights. The resulting surface is a limit of a quence of meshes, which approaches the smooth limit surface as the number of subdivisions approaches infinity. Subdivision surfaces have many advantages over other surface reprentations, such as polygons or
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spline curves, in terms of flexibility, adaptivity, and efficiency. They can easily handle surfaces with arbitrary topology and geometry, and can be ud for various applications, such as modeling, animation, rendering, and simulation.
