  Shape is an important physical property of natural and manmade 3D objects
that characterizes their external appearances. Understanding differences
between shapes and modeling the variability within and across shape classes,
hereinafter referred to as \emph{shape analysis}, are fundamental problems to
many applications, ranging from computer vision and computer graphics to
biology and medicine. This chapter provides an overview of some of the recent
techniques that studied the shape of 3D objects that undergo non-rigid
deformations including bending and stretching. Recent surveys that covered some
aspects such classification, retrieval, recognition, and rigid or nonrigid
registration, focused on methods that use shape descriptors. Descriptors,
however, provide abstract representations that do not enable the exploration of
shape variability. In this chapter, we focus on recent techniques that treated
the shape of 3D objects as points in some high dimensional space where paths
describe deformations. Equipping the space with a suitable metric enables the
quantification of the range of deformations of a given shape, which in turn
enables (1) comparing and classifying 3D objects based on their shape, (2)
computing smooth deformations, i.e. geodesics, between pairs of objects, and
(3) modeling and exploring continuous shape variability in a collection of 3D
models. This article surveys and classifies recent developments in this field,
outlines fundamental issues, discusses their potential applications in computer
vision and graphics, and highlights opportunities for future research. Our
primary goal is to bridge the gap between various techniques that have been
often independently proposed by different communities including mathematics and
statistics, computer vision and graphics, and medical image analysis.
