This paper presents a unified framework for computing a surface to approximate a target shape defined by discrete data points. A signed point-to-surface distance function is defined, and its properties are investigated, especially, its second-order Taylor approximant is derived. The intercorrelations between the signed and the squared distance functions are clarified, and it is demonstrated that the squared distance function studied in the previous works is just the Type I squared distance function deduced from the signed distance function. It is also shown that surface approximations under different criteria and constraints can all be formulated as optimization problems with specified requirements on the residual errors represented by the signed distance functions, and that classical numerical optimization algorithms can be directly applied to solve them since the derivatives of the involved objective functions and constraint functions are all available. Examples of global cutter position optimization for flank milling of ruled surface with a cylindrical tool, which requires approximating the tool envelope surface to the point cloud on the design surface following the minimum zone criterion, are given to confirm the validity of the proposed approach.

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State Key Laboratory of Mechanical System and Vibration, School of Mechanical Engineering,

zhulm@sjtu.edu.cn
State Key Laboratory of Digital Manufacturing Equipment and Technology,
State Key Laboratory of Mechanical System and Vibration, School of Mechanical Engineering,
State Key Laboratory of Digital Manufacturing Equipment and Technology,

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December 2010

Research Papers

# Geometry of Signed Point-to-Surface Distance Function and Its Application to Surface Approximation

Li Min Zhu,

Li Min Zhu

State Key Laboratory of Mechanical System and Vibration, School of Mechanical Engineering,

zhulm@sjtu.edu.cn
Shanghai Jiao Tong University

, Shanghai 200240, P.R. China
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Xiao Ming Zhang,

Xiao Ming Zhang

State Key Laboratory of Digital Manufacturing Equipment and Technology,

Huazhong University of Science and Technology

, Wuhan 430074, P.R. China
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Han Ding,

Han Ding

State Key Laboratory of Mechanical System and Vibration, School of Mechanical Engineering,

Shanghai Jiao Tong University

, Shanghai 200240, P.R. China
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You Lun Xiong
State Key Laboratory of Digital Manufacturing Equipment and Technology,

You Lun Xiong

Huazhong University of Science and Technology

, Wuhan 430074, P.R. China
Search for other works by this author on:

Li Min Zhu

Shanghai Jiao Tong University

, Shanghai 200240, P.R. Chinazhulm@sjtu.edu.cn

Xiao Ming Zhang

Huazhong University of Science and Technology

, Wuhan 430074, P.R. China
Han Ding

Shanghai Jiao Tong University

, Shanghai 200240, P.R. China
You Lun Xiong

Huazhong University of Science and Technology

, Wuhan 430074, P.R. China*J. Comput. Inf. Sci. Eng*. Dec 2010, 10(4): 041003 (10 pages)

**Published Online:**November 23, 2010

Article history

Received:

July 16, 2009

Revised:

June 29, 2010

Online:

November 23, 2010

Published:

November 23, 2010

Citation

Zhu, L. M., Zhang, X. M., Ding, H., and Xiong, Y. L. (November 23, 2010). "Geometry of Signed Point-to-Surface Distance Function and Its Application to Surface Approximation." ASME. *J. Comput. Inf. Sci. Eng*. December 2010; 10(4): 041003. https://doi.org/10.1115/1.3510588

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