# Measurement-Driven Synthesis of Female Digital Mannequin Using Convex Sub-Volumes
## 3 Materials and Methods
### Creation of the Convex Volume Approximation  
The basis for all measurement-driven mannequins is composition by a number of ellipsoids that are selectively placed, scaled, and oriented over a reference 3D model. This basis is called the Base Mannequin (\(BM\)) and coarsely approximates the shape of a female body given by a reference 3D model. Each other mannequin in this manuscript preserves the same topology of \(BM\), but differs in geometry (i.e., dimensions, position, and orientation of ellipsoids). It should be noted that \(BM\) is conformed both by topological and geometrical information, with the latter being the complement of the former and defined as follows:  
1. Topology: Set of ellipsoids and rules that define the composition of the 3D mannequin; 1. The mannequin is conformed by a collection of convex volumes (ellipsoids), without boolean union among them; 2. The collection has a finite number of ellipsoids; 3. Each ellipsoid (singleton or pairs) approximates a particular part or area of the body; 4. Pairs of ellipsoids correspond to parts of the body with reflection (symmetry) along the sagittal plane (i.e., left and right); 5. Singleton ellipsoids are symmetric with respect to the sagittal plane. 2. Geometry: Set of geometric properties \(S\) (position and orientation) and \(D\) (dimensions) that define each ellipsoid. **Approximation of the 3D Model with Ellipsoids**  
The process of creating the Base Mannequin starts by defining planar data points (see Figure 2) over the 3D model (see Table 2 for characteristics of the model). These data points serve as guidelines for the creation of ellipsoids and as future references on which some tailor measurements are performed (see Section 3.5). Following the definition of the planar data,a specific region of the body (mainly a principal muscle or area) is selected and traced by creating (placing, orienting, and scaling) one or two ellipsoids as needed (see Figure 3b). The intention is to fill up as much space as possible while making sure the ellipsoid(s) is (are) inscribed in the reference 3D model surface (see Figure 3). If two ellipsoids are used to approximate a part of the body (see Figure 3b), both will serve as the basis of a cushion in a post-processing step. The process exhibited in Figure 3 is replicated for other parts of the body, excluding the right arm, breast, and leg (see Figure 4a). The missing parts on the right hemisphere are obtained by mirroring the analogous ellipsoids of the left hemisphere through the sagittal plane (see Figure 4). This preserves proportions and body symmetry. The mirroring process is performed in such way that the SO(3) RHCCS property of each ellipsoid is maintained. It should be noted that symmetry is only obtained in this step of the research. Once kinematics are introduced to the model in future research, symmetry is lost. Take into consideration also that for this step, the shape alone is considered and the measurements of the Base Mannequin are irrelevant. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Name** & **Origin** & \
\begin{tabular}{c} **Number of** \\ **Faces** \\ \end{tabular} &
\begin{tabular}{c} **Number of** \\ **Vertex** \\ \end{tabular} & **Borders** & **Manifold** \\ \hline Reference 3D & [39] & 4984 & 4986 & 0 & True \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Characteristics of the input data of the reference 3D model. Figure 3: Approximation of the left thigh and hip with ellipsoids, front and lateral views. Red dotted line: surface boundary to approximate. (**a**) Approximation with one ellipsoid. (**b**) Improvement of the approximation with a second ellipsoid. Figure 2: Reference 3D model to be approximated by convex volumes. Planar data (red lines). **Creation of Cushions**  
As stated previously, some parts of the 3D model are approximated by pairs of ellipsoids (see Figure 5). Hence, some regions of the \(BM\), and thus the 3D mannequins, may present gaps or voids between the ellipsoids.