Observations on Silhouette Sizes

This paper appears in issue Volume 9, Number 1.
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Abstract

Silhouettes have many applications in computer graphics such as non-photorealistic edge rendering, fur rendering, shadow volume creation, and anti-aliasing. The number of edges, s, in the silhouette of a model observed from a point is therefore useful in analyzing such algorithms.

This paper examines, from a theoretical viewpoint, a menagerie of objects with interesting silhouettes (including those with minimal and maximal silhouettes). It shows that the relationship between and s and the number of triangles in a model, f, is bounded above by s = O(f) and below by s = Ω(1), and that the expected value of s over all observation points at infinity is proportional to the sum of the dihedral angles.

In practice, the models used with silhouette-based rendering algorithms are triangle meshes that are manually constructed or result from scans of human-made objects. They consist of only surface geometry with few cracks; there is no internal detail like the engine under a car’s hood. Geometric and aesthetic constraints on these models appear to create an inherent relationship between f and s. Measurements of the actual silhouettes of real-world 3D models with polygon counts varied across six orders of magnitude show them to follow the relationship s ~ f 0.8. Furthermore, the expected value of s at infinity is a good approximation of the expected silhouette size for a viewer at a finite location.

Author Information

Morgan McGuire, Brown University, Dept of Computer Science, PO Box 1910, Providence, RI 02906 morgan@cs.brown.edu

Figures

Click on a thumbnail image below to view the full-resolution version.

Figure 8. Log-log graph of f vs. s data from Table 1. The dashed blue line is the f = s^0.5 trend for convex models and the thick red line is the f = 0.7s^0.8 trend observed over all data. The horizontal bar symbols (—) mark the max an min silhouette sizes observed overa ll view positions; the black diamonds (♦) are the average silhouette sizes. Blue squares () show the analytic expected silhouette size. All four metrics are quite close—the expected value is a good predictor.

Figure 9. Log-log graph of f vs. E[s∞] for 897 models downloaded from the Internet, used to establish an empirical order of growth. The dashed blue lines are (from bottom to top): f = s^0.5, f = s^0.75, and f = s. The data (except for some outliers due to malformed meshes) lies between the outermost lines, clustered around the thick red trendline f = 0.7s^0.8.

Data

The data used to generate Figures 8 and 9 are available as spreadsheets:

common-models.xls (in the text as Table 1)
all-models.xls

Links

A continually growing archive of three-dimensional model geometry is maintained at http://www.cs.brown.edu/research/graphics/games

Errata

The “Web Information” section of the text erroneously lists the page for this paper as http://www.acm.org/jgt/papers/McGuire03 rather than this page, http://www.acm.org/jgt/papers/McGuire04. For convenience, a redirect page has been created at the incorrect page to bring readers to this one.

BibTeX Entry

@article{McGuire04,
  author = "Morgan McGuire",
  title = "Observations on Silhouette Sizes",
  journal = "journal of graphics, gpu, and game tools",
  volume = "9",
  number = "1",
  pages = "1-12",
  year = "2004",
}