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TECHNICAL PAPERS

# Constructive Heterogeneous Object Modeling Using Signed Approximate Real Distance Functions

[+] Author and Article Information
Pierre-Alain Fayolle

The University of Aizu, Department of Information Systems, Tsuruga, Ikki-Machi, Aizu-Wakamatsu City, Fukushima, 965-8580, Japand8052103@u-aizu.ac.jp

Hosei University, 3-7-2 Kajino-cho, Koganei-shi, Tokyo 184-8584, Japan

Benjamin Schmitt

Computer Graphics Research Institute and Hosei University, Digital Media Professional, Mitaka takagi Building, 1-15-5 Nakacho, Musashino-shi, Tokyo 180-0006, Japan

Nikolay Mirenkov

The University of Aizu, Department of Information Systems, Tsuruga, Ikki-Machi, Aizu-Wakamatsu City, Fukushima, 965-8580, Japan

J. Comput. Inf. Sci. Eng 6(3), 221-229 (Nov 25, 2005) (9 pages) doi:10.1115/1.2218366 History: Received May 16, 2004; Revised November 25, 2005

## Abstract

We introduce a smooth approximation of the $min∕max$ operations, called signed approximate real distance function (SARDF), for maintaining an approximate signed distance function in constructive shape modeling. We apply constructive distance-based shape modeling to design objects with heterogeneous material distribution in the constructive hypervolume model framework. The introduced distance approximation helps intuitively model material distributions parametrized by distances to so-called material features. The smoothness of the material functions, provided here by the smoothness of the defining function for the shape, helps to avoid undesirable singularities in the material distribution, like stress or concentrations. We illustrate application of the SARDF operations by two- and three-dimensional heterogeneous object modeling case studies.

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## Figures

Figure 1

The first quadrant is divided into two zones. The growing circular approximation is applied in zone I, whereas we introduce a fixed radius approximation with the bounding band in zone II.

Figure 2

Contour maps of the union of two ellipsoids for the three different union operations. The level of gray corresponds to range of approximate distances from points inside the solid to its surface. Left: SARDF union, middle: max: sharp corners indicate points of derivatives discontinuity, right: R union: contour map of the function clearly shows that the resulting function loses the distance like behavior quite close to the boundary even with exact distance functions used for the arguments.

Figure 3

A two-dimensional CAD part with three different material regions (blue: material 1, red: material 2, color gradient: functionally graded material)

Figure 4

Approximate distance map d1 from point X to the boundary of the region where only material 1 exists. Left: using R functions. Right: using SARDF operations.

Figure 5

Approximate distance map d2 from point X to the boundary of the region where only material 2 exists. Left: using R functions. Right: using SARDF operations.

Figure 6

A cross section parallel to x axis and the distribution of the materials in the cross section for the CAD part constructed with SARDF functions

Figure 7

Material distributions in the cross-section y=2 for materials 1 and 2 using: R functions in the constructive trees for the geometry of the solid and the material regions (right), min∕max in the constructive trees for the geometry of the solid and the material regions (left). The circled points correspond to points of C1 discontinuity of the material distributions.

Figure 8

Top left: the first material feature, top middle: the second material feature, with a zoom on one of the pins, on the right (top right), bottom: union of the two material features

Figure 9

Distribution of two materials. Blue color corresponds to material 1, red color to material 2. The color variation indicates the fraction of each material. Left: Two cross sections are made for x=0 and y=0 to show the material distribution. Right: A zoom is made to one of the pins with two additional cross sections.

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