Tangential Distance Fields for Mesh Silhouette Analysis Matt Olson and Hao Zhang GrUVi Lab, School of Computing Science, Simon Fraser University, Canada {matto, haoz}@cs.sfu.ca Abstract We consider a tangent-space representation of surfaces which maps each point on a surface to the tangent plane of the surface at that point. Such representations are known to facilitate the solution of several visibility problems, in particular, those involving silhouette analysis. In this paper, we introduce a novel class of distance fields for a given surface defined by its tangent planes. At each point in space, we assign a scalar value which is a weighted sum of distances to these tangent planes. We call the resulting scalar field a tangential distance field or TDF. When applied to triangle mesh models, the tangent planes become supporting planes of the mesh triangles. The weighting scheme used to construct a TDF for a given mesh and the way the TDF is utilized can be closely tailored to a specific application. At the same time, the TDFs are continuous, lending themselves to standard optimization techniques, such as greedy local search, thus leading to efficient algorithms. In this paper, we use four applications to illustrate the benefit of using TDFs: multi-origin silhouette extraction in Hough space, silhouette-based view point selection, camera path planning, and light source placement. Categories and Subject Descriptors (according to ACM CCS): I.3.5 Computational Geometry and Object Mod- eling — Geometric algorithms, languages, and systems; Hierarchy and geometric transformations 1. Introduction The majority of geometric problems in computer graphics deal with surfaces. Typically, a surface is specified by points lying on it. To obtain a discrete model, a surface is first sampled and then the sample points are connected into a piecewise linear triangulation, a triangle mesh, which tes- sellates the surface. A lesser known surface representation is via the tangent planes associated with points of the sur- face. Such a tangent-space surface representation is some- times referred to as the Blaschke image in Laguerre geom- etry [PPS03]. It has also been referred to as the surface dual [Joh04, Mor02, SRKP04] in the literature. However, to avoid confusion with the classical geometric dual trans- form [PDB * 01], we do not adopt that terminology. When dealing with triangle meshes, the tangent planes are replaced by the supporting planes of the mesh triangles. Certain geometric properties are easier to deal with via tangent-space representations. For example, two faces on a mesh which are geodesically far away from each other can have the same or similar supporting planes, making them geometrically close in the tangent space. The importance of supporting planes can be realized when studying visibility since these very planes, when crossed by the viewpoint, trig- ger visibility events; see Figure 1 for an illustration. Tangent-space surface representations are not easy to vi- sualize or utilize directly. However, one may derive useful information from them to benefit specific applications. For example, with a designated origin, each point P on a sur- face can be mapped to a point H(P) which is the orthogonal projection of the origin onto the tangent plane at P; see Fig- ure 1(d) for an illustration in the 2D case. This is called the Hough transform [Hou59] and the image of the mapping is called a pedal surface [Joh04]. The Hough transform, or its close relative, the dual transform, has been exploited to fa- cilitate silhouette extraction [HZ00, PDB * 01, OZ06] or back face culling [KMGL96] over polygonal models. In this paper, we explore beyond low-level representations derived from tangent-space surface representations, such as a point cloud resulting from Hough transform. We introduce a novel class of scalar fields, constructed as a weighted sum of signed distances to the set of tangent planes or support- ing planes, in the case of triangle mesh models. Referred