Abstract This paper surveys mesh segmentation techniques and algorithms, with a focus on part‐based segmentation, that is, segmentation that divides a mesh (featuring a 3D object) into meaningful parts. Part‐based segmentation applies to a single object and also to a family of objects (i.e. co‐segmentation). However, we shall not address here chart‐based segmentation, though some mesh co‐segmentation methods employ such chart‐based segmentation in the initial step of their pipeline. Finally, the taxonomy proposed in this paper is new in the sense that one classifies each segmentation algorithm regarding the dimension (i.e. 1D, 2D and 3D) of the representation of object parts. The leading idea behind this survey is to identify the properties and limitations of the state‐of‐the‐art algorithms to shed light on the challenges for future work.

Previous work on implicit surfaces has almost exclusively focused on homogeneously 2-dimensional surfaces, regardless of whether they are manifold or not. Consequently, current geometric and graphical systems, and even commercial mathematical software packages, cannot generate non-homogeneous implicit surfaces such as, for example, the Steiner surface or the Whitney surface correctly. This paper overcomes these difficulties by introducing a new uniform space partitioning-based algorithm that generates a polygonization for implicit surfaces possibly with self-intersections and isolated 0-and 1-dimensional singularities.

Mathematical systems (e.g., Mathematica, Maple, Matlab, and DPGraph) easily plot planar algebraic curves implicitly defined by polynomial functions. However, these systems, and most algorithms found in the literature, cannot draw many implicit curves correctly; in particular, those with singularities (self-intersections, cusps, and isolated points). They do not detect sign-invariant components either, because they use numerical methods based on the Bolzano corollary, that is, they assume that the curve-describing function f flips sign somewhere in a line segment AB that crosses the curve, or f ( A )· f ( B ) < 0. To solve these problems, we have generalized the False Position (FP) method to determine two types of zeros: (i) crossing zeros and (ii) extremal zeros (local minima and maxima without function sign variation). We have called this method the Generalized False Position (GFP) method. It allows us to sample an implicit curve against the Binary Space Partitioning (BSP), say bisection lines, of a rectangular region of R 2 . Interestingly, the GFP method can also be used to determine isolated points of the curve. The result is a general algorithm for sampling and rendering planar implicit curves with topological guarantees.