A Developer's Guide to Silhouette Algorithms for Polygonal Models

part of artistic expression (for example, any pencil or pen-and-ink drawing), scientific illustrations (medical or technical), or entertainment graphics (such as in comics). Hence, computer graphics researchers have extensively studied the automatic generation of such lines. In particular, the area of nonphotorealistic rendering has focused on two main directions of research in this respect: the generation of hatching that conveys illumination as well as texture in an image and the computation of outlines and silhouettes. Silhouettes play an important role in shape recognition because they provide one of the main cues for figure-to-ground distinction. However, since silhouettes are view dependent, they need to be determined for every frame of an animation. Finding an efficient way to accomplish this is nontrivial. Indeed, a variety of different algorithms exist that compute silhouettes for geometric objects. This article provides a guideline for developers who need to choose between one of these algorithms for his or her application. Here, we restrict ourselves to discussing only those algorithms that apply to polygonal models, because these are the most commonly used object representations in modern computer graphics. (For an algorithm to compute the silhouette for free-form surfaces see, for example, Elber and Cohen.) Thus, we can use all algorithms discussed here to take a polygonal mesh as input and compute the visible part of the silhouette as output. Some algorithms, however, can also help compute the silhouette only, without additional visibility culling. The silhouette’s representation might vary depending on the algorithm class—that is, the silhouette might take the form of a pixel matrix or a set of analytic stroke descriptions. Definitions and terminology The silhouette S of a free-form object is typically defined as the set of points on the object’s surface where the surface normal is perpendicular to the vector from the viewpoint. Mathematically, this means that the dot product of the normal ni with the view vector at a surface vertex P’s position pi is zero: S = {P : 0 = ni ⋅ (pi − c)}, with c being the center of projection. In case of orthographic projection (pi − c) is exchanged with the view direction vector v. Unfortunately, for polygonal objects this definition can’t be applied because normals are only well defined for polygons and not for arbitrary points on the surface. Hence, typically no points exist on the surface where the normal is perpendicular to the viewing direction. However, we can find silhouettes along those edges in a polygonal model that lie on the border between changes of surface visibility. Thus, we define the silhouette edges of a polygonal model as edges in the mesh that share a frontand a back-facing polygon. Some authors refer to the contour rather than the silhouette. We define a contour as the subset of the silhouette that separates the object from the background. Also, the subset of the silhouette, which in 2D divides one portion of the object from another one of the same object, is not part of the contour. Those lines are sometimes called internal silhouettes. Other lines significant in the context of silhouettes are creases, borders, and self-intersections. Creases are edges that should always be drawn, for instance the edges of a cube. Typically, we can identify a crease by comparing the angle between its two adjacent polygons with a certain threshold. Border lines only appear in models where the mesh is not closed and are those edges with only one adjacent polygon. Lastly, self-intersection lines are where two polygons of a model intersect. They aren’t necessarily part of the model’s edges but are important for shape recognition. In the literature, these three additional line categories together with the silhouette lines are often called feature lines. Some silhouette detection methods make no algorithmic distinction between these types of lines. This is because you must determine Survey