C∞ smooth freeform surfaces over hyperbolic domains

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C^∞ smooth freeform surfaces over hyperbolic domains

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ACM Symposium on Solid and Physical Modeling archive
2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling table of contents

San Francisco, California

SESSION: Short papers table of contents

Pages: 367-372

Year of Publication: 2009

ISBN:978-1-60558-711-0

Authors

Wei Zeng	Stony Brook University
Ying He	Nanyang Technological University
Jiazhi Xia	Nanyang Technological University
Xianfeng Gu	Stony Brook University
Hong Qin	Stony Brook University

Sponsor

: SIAM Activity Group on Geometric Design

Publisher

ACM New York, NY, USA

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Downloads (6 Weeks): 23, Downloads (12 Months): 43, Citation Count: 0

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ABSTRACT

Constructing smooth freeform surfaces of arbitrary topology with higher order continuity is one of the most fundamental problems in shape and solid modeling. This paper articulates a novel method to construct C^∞ smooth surfaces with negative Euler numbers based on hyperbolic geometry and discrete curvature flow. According to Riemann uniformization theorem, every surface with negative Euler number has a unique conformal Riemannian metric, which induces Gaussian curvature of --1 everywhere. Hence, the surface admits hyperbolic geometry. Such uniformization metric can be computed using the discrete curvature flow method: hyperbolic Ricci flow. Consequently, the basis function for each control point can be naturally defined over a hyperbolic disk, and through the use of partition-of-unity, we build a freeform surface directly over hyperbolic domains while having C^∞ property. The use of radial, exponential basis functions gives rise to a true meshless method for modeling freeform surfaces with greatest flexibilities, without worrying about control point connectivity. Our algorithm is general for arbitrary surfaces with negative Euler characteristic. Furthermore, it is C^∞ continuous everywhere across the entire hyperbolic domain without singularities. Our experimental results demonstrate the efficiency and efficacy of the proposed new approach for shape and solid modeling.

REFERENCES

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