A Differentiability of $\mathbf{T}_g$ with respect to $r$
In this appendix we study continuity and differentiability of \(\mathbf{T}_g\) (i.e. of \(\mathbf{R}_g,\mathbf{t}_g\)) with respect to \(r\). Variations in \(r\) imply a smooth variation on the domains \(\Omega_g\) being considered for the Procrustes analysis (domains are enlarged or reduced by boundary \(\partial \Omega_g\)). Regarding the continuous Procrustes problem definition in (Al-Aifari et al., 2013), we can define our analogous space-continuous squared Procrustes residual for \(\mathbf{X}_g\) and \(\mathbf{Y}_g\) as:
\[
d^2_{\mathcal{P}}(\mathbf{X}_g,\mathbf{Y}_g)=\begin{matrix}
\\ \min
\\ \mathbf{R}_g,\mathbf{t}_g
\end{matrix}\left(\int_{\Omega_g}\left \|\mathbf{R}_g\mathbf{x}+\mathbf{t}_g-\mathbf{y}\right\|^2 \mathrm{d} {\Omega}\right),
\tag{25}
\]
where \(\mathbf{x}\) and \(\mathbf{y}\) represent mapped points of the shapes' surface manifolds. For now on, as \(\mathbf{x}\) are mapped to \(\mathbf{y}\) with area preserving maps (Melzi et al., 2019), we will use \(\mathrm{d}\Omega_g\) to refer to the differential elements on both the current and the target shape manifolds.
The optimal translation, in this continuous formulation, is
\[
\mathbf{t}_g=\frac{\int_{\Omega_g}\mathbf{y}\mathrm{d}\Omega_g}{\int_{\Omega_g}\mathrm{d}\Omega_g}-\frac{\int_{\Omega_g}\mathbf{R}_g\mathbf{x}\mathrm{d}\Omega_g}{\int_{\Omega_g}\mathrm{d}\Omega_g}=\frac{\int_{\Omega_g} (\mathbf{y}-\mathbf{R}_g\mathbf{x})\mathrm{d}\Omega_g}{\int_{\Omega_g} \mathrm{d}\Omega_g},
\]
and can be differentiated with respect to \(r\) to obtain:
\[
\frac{\partial}{\partial r}\mathbf{t}_g=\frac{\partial}{\partial r}\left(\frac{\int_{\Omega_g} (\mathbf{y}-\mathbf{R}_g\mathbf{x})\mathrm{d}\Omega_g}{\int_{\Omega_g} \mathrm{d}\Omega_g} \right )
\\
=\frac{\partial}{\partial r}\left(\frac{1}{A} \right )\int_{\Omega_g} (\mathbf{y}-\mathbf{R}_g\mathbf{x})\mathrm{d}\Omega_g+\frac{1}{A}\frac{\partial}{\partial r}\left(\int_{\Omega_g} (\mathbf{y}-\mathbf{R}_g\mathbf{x})\mathrm{d}\Omega_g\right)
\\
=-\frac{1}{A^2}\frac{\partial A}{\partial r}\int_{\Omega_g} (\mathbf{y}-\mathbf{R}_g\mathbf{x})\mathrm{d}\Omega_g
\\
+\frac{1}{A}\left(
\int_{\Omega_g}\frac{\partial \mathbf{R}_g}{\partial r}\mathbf{x}\mathrm{d}\Omega_g
+\int_{\partial\Omega_g} (\mathbf{y}-\mathbf{R}_g\mathbf{x})\mathrm{d}(\partial\Omega_g)\right),
\tag{26}
\]
where \(A(r)\) is the area of the open domain \(\Omega_g\) and, in the last step, we applied the Leibniz's integral rule. Note that (26) depends on the existence of \(\frac{\partial \mathbf{R}_g}{\partial r}\). The optimal Procrustes rotation component \(\mathbf{R}_g\) can be obtained from:
\[
\mathbf{R}_g=\sqrt{\mathbf{M}_g\mathbf{M}_g^\intercal}\,\mathbf{M}_g^{-1},
\tag{27}
\]
where \(\mathbf{M}_g(r)\in\mathbb{R}^{3\times3}\) is equal to:
\[
\mathbf{M}_g(r)=\int_{\Omega_g}\left(\mathbf{x}-\bar{\mathbf{x}}_g\right)\left(\mathbf{y}-\bar{\mathbf{y}}_g\right)^\intercal\mathrm{d}\Omega_g.
\tag{28}
\]
In this continuous formulation \(\bar{\mathbf{x}}_g={\int_{\Omega_g}\mathbf{x}\mathrm{d}\Omega_g}/{\int_{\Omega_g}\mathrm{d}\Omega_g}\) and \(\bar{\mathbf{y}}_g={\int_{\Omega_g}\mathbf{y}\mathrm{d}\Omega_g}/{\int_{\Omega_g}\mathrm{d}\Omega_g}\). Equation (27), and therefore \(\mathbf{R}_g\), is continuous differentiable with respect to \(r\) given two conditions: \(\mathbf{M}_g\) being continuous differentiable and existence of \(\mathbf{M}^{-1}_g\) for \(r\in[r_0,R(t)]\). \(\mathbf{M}_g(r)\) is differentiable with respect to \(r, \forall \, r>0\):
\[
\frac{\partial}{\partial r}\mathbf{M}_g=
\int_{\Omega_g}\frac{\partial}{\partial r}\left(\left(\mathbf{x}-\bar{\mathbf{x}}_g\right)\left(\mathbf{y}-\bar{\mathbf{y}}_g\right)^\intercal\right)\mathrm{d}\Omega_g
\\
+
\int_{\partial\Omega_g}\left(\mathbf{x}-\bar{\mathbf{x}}_g\right)\left(\mathbf{y}-\bar{\mathbf{y}}_g\right)^\intercal\mathrm{d}(\partial\Omega_g).
\tag{29}
\]
On the other hand, \(\mathbf{M}_g^{-1}\) exists if there exist at least three non-aligned points \(\mathbf{x}\) and three non-aligned points \(\mathbf{y}\) in the domains \(\Omega_g\) of each manifold. This condition is certainly achieved for any 3D object (planar or volumetric) and \(r>0\).
