A Survey on Non-rigid 3D Shape Analysis <s> Metrics that capture physical deformations <s> Preface Chapter 1. Three-dimensional differential geometry: 1.1. Curvilinear coordinates, 1.2. Metric tensor, 1.3. Volume, areas, and lengths in curvilinear coordinates, 1.4. Covariant derivatives of a vector field, 1.5. Necessary conditions satisfied by the metric tensor the Riemann curvature tensor, 1.6. Existence of an immersion defined on an open set in R3 with a prescribed metric tensor, 1.7. Uniqueness up to isometries of immersions with the same metric tensor, 1.8. Continuity of an immersion as a function of its metric tensor Chapter 2. Differential geometry of surfaces: 2.1. Curvilinear coordinates on a surface, 2.2. First fundamental form, 2.3. Areas and lengths on a surface, 2.4. Second fundamental form curvature on a surface, 2.5. Principal curvatures Gaussian curvature, 2.6. Covariant derivatives of a vector field defined on a surface the Gauss and Weingarten formulas, 2.7. Necessary conditions satisfied by the first and second fundamental forms: the Gauss and Codazzi-Mainardi equations Gauss' theorema egregium, 2.8. Existence of a surface with prescribed first and second fundamental forms, 2.9. Uniqueness up to proper isometries of surfaces with the same fundamental forms, 2.10.Continuity of a surface as a function of its fundamental forms Chapter 3. Applications to three-dimensional elasticity in curvilinear coordinates: 3.1. The equations of nonlinear elasticity in Cartesian coordinates, 3.2. Principle of virtual work in curvilinear coordinates, 3.3. Equations of equilibrium in curvilinear coordinates covariant derivatives of a tensor field, 3.4. Constitutive equation in curvilinear coordinates, 3.5. The equations of nonlinear elasticity in curvilinear coordinates, 3.6. The equations of linearized elasticity in curvilinear coordinates, 3.7. A fundamental lemma of J.L. Lions, 3.8. Korn's inequalities in curvilinear coordinates, 3.9. Existence and uniqueness theorems in linearizedelasticity in curvilinear coordinates Chapter 4. Applications to shell theory: 4.1. The nonlinear Koiter shell equations, 4.2. The linear Koiter shell equations, 4.3. Korn's inequality on a surface, 4.4. Existence and uniqueness theorems for the linear Koiter shell equations covariant derivatives of a tensor field defined on a surface, 4.5. A brief review of linear shell theories References Index. <s> BIB001 </s> A Survey on Non-rigid 3D Shape Analysis <s> Metrics that capture physical deformations <s> In this paper we define a new methodology for shape analysis of parameterized surfaces, where the main issues are: (1) choice of metric for shape comparisons and (2) invariance to reparameterization. We begin by defining a general elastic metric on the space of parameterized surfaces. The main advantages of this metric are twofold. First, it provides a natural interpretation of elastic shape deformations that are being quantified. Second, this metric is invariant under the action of the reparameterization group. We also introduce a novel representation of surfaces termed square root normal fields or SRNFs. This representation is convenient for shape analysis because, under this representation, a reduced version of the general elastic metric becomes the simple $\ensuremath{\mathbb{L}^2}$ metric. Thus, this transformation greatly simplifies the implementation of our framework. We validate our approach using multiple shape analysis examples for quadrilateral and spherical surfaces. We also compare the current results with those of Kurtek et al. [1]. We show that the proposed method results in more natural shape matchings, and furthermore, has some theoretical advantages over previous methods. <s> BIB002
Instead of using the L 2 metric on C f , one would like to explicitly capture the type of deformations that one needs to apply to f 1 in order to align it onto f 2 . Such deformations can be of two types; bending and stretching. This requires redefining Eqn. (5) in terms of an energy function that penalizes these deformations: where E s and E b are respectively the stretching and bending energies. The parameters α s and β s ∈ R + control the contribution of each of the terms to the energy function. Below we discuss two important pieces of work that implement this model. The first one (section 3.2.1) is inspired by the elasticity theory in physics BIB001 where surfaces are treated as thin shells, i.e. a thin three-dimensional material of thickness δ. Stretching in this case is caused by inlayer (tangential) shear or compression while bending is caused by friction due to transversal shear  . The second one (Section 3.2.2) showed that with specific choices of the bending and stretching terms as well as their weights, the deformation model of Eqn. (11) reduces to an L 2 metric in the space of Square Root Normal Fields (SRNF), a special representation of surfaces BIB002 .