This inner product allows to measure the distortion of angle and lengths in an ε-neighborhood around a point on the surface, which is induced by the embedding of the surface in the ambient Euclidean space

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2.3.1.6 The First Fundamental Form The first fundamental form describes the local parametrization of the surface. It is the inner product on the tangent space of a surface in three-dimensional Euclidean space, which is induced canonically from the dot product in R 3 . This inner product allows to measure the distortion of angle and lengths in an ε-neighborhood around a point on the surface, which is induced by the embedding of the surface in the ambient Euclidean space. Hence, it is also called as the “metric tensor“. The metric tensor can be derived as follows. Let ω0 = (u0,v0) ∈ Ω be the origin of the local coordinate system corresponding to a point p0 = f(x(u0,v0),y(u0,v0),z(u0,v0)) ∈ S. The function mapping f(ω) for a point ω in the neighborhood of ω0 can be written in terms of the local first order Taylor approximation as: f(ω) = f(ω0)+∇ f(ω0)(ω−ω0). (2.9) Let a,b ∈ R 2 be the two arbitrary vectors as shown in the Figure 2.5. The scalar product of mapping of these two vectors can be simplified using Eq. (2.9) as: h f(ω0 +a)− f(ω0), f(ω0 +b)− f(ω0)i ≈ h∇ f(ω0)a, f(ω0)bi ≈ a T ∇ f(ω0) T∇ f(ω0) | {z } First Fundamental Form b. (2.10) The first order derivative of f denoted as ∇ f is a Jacobian matrix of dimensional 3×2 computed on the Euclidean surface. Whereas, the first fundamental form represents an inner product matrix ∇ f T∇ f computed using the dot product of rows of the Jacobian matrix and hence

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4.3 Human Poses In this experiment, we consider human poses obtained using tracking software. A consumer stereo camera4 is placed in front of a test person, and the tracking software described in [10] is invoked in order to track the pose of the persons upper body. The recorded poses are represented by the human body end-effectors; the end-points of each bone of the skeleton. The placement of each end-effector is given spatial coordinates so that an entire pose with k end-effectors can be considered a point in R 3k . To simplify the representation, only the end-effectors of a subset of the skeleton are included, and, when two bones meet at a joint, their end-points are considered one end-effector. Figure 5 shows a human pose with 11 end-effectors marked by thick dots. −1 −1.5 0 −0.5 1 0.5 2 1.5 2.5 −0.5 0 0.5 0 0.5 1 1.5 2 2.5 Fig. 5. Camera output superimposed with tracking result (left) and a tracked pose with 11 end-effectors marked by thick dots (right). The fact that bones do not change length in short time spans gives rise to a constraint for each bone; the distance between the pair of end-effectors must be constant. We incorporate this into a pose model with b bones by restricting the allowed poses to the preimage F −1 (0) of the map F : R 3k → R b given by F i (x) = kei1 − ei2 k 2 − l 2 i , (12) where ei1 and ei2 denote the spatial coordinates of the end-effectors and li the constant length of the ith bone. In this way, the set of allowed poses constitute a 3k − b-dimensional implicitly represented manifold. We record 26 poses using the tracking setup, and, amongst those, we make 20 random choices of 8 poses and perform linearized PGA and exact PGA. For each experiment, τSvˆ , ˜τSvˆ , ρ, and σ are computed and plotted in Figure 6. The