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Research Papers

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

[+] Author and Article Information
Li Min Zhu1

State Key Laboratory of Mechanical System and Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, P.R. Chinazhulm@sjtu.edu.cn

Xiao Ming Zhang, You Lun Xiong

State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan 430074, P.R. China

Han Ding

State Key Laboratory of Mechanical System and Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, P.R. China

1

Corresponding author.

J. Comput. Inf. Sci. Eng 10(4), 041003 (Nov 23, 2010) (10 pages) doi:10.1115/1.3510588 History: Received July 16, 2009; Revised June 29, 2010; Published November 23, 2010; Online November 23, 2010

## Abstract

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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## Figures

Figure 1

Point-to-surface distance function

Figure 2

Singular cases

Figure 12

Reconstructed ruled surface

Figure 5

Distribution of the geometric errors before optimization

Figure 6

Distribution of the geometric errors after optimization

Figure 7

Surface model of a blade of an impeller

Figure 8

Interference between the tool envelope surface and the design surface before optimization

Figure 9

Interference between the tool envelope surface and the design surface after optimization

Figure 10

Synthetic point cloud

Figure 11

Root mean squared error versus the number of iterations

Figure 3

Relationships among the distance functions discussed

Figure 4

Tool axis trajectory surface represented as a ruled surface

## Errata

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