# Real-Time Physical Prototyping Tool Design Based on Shape-Changing Display
## 3 Shape-Changing Display for Real-Time Prototyping
### Tessellation  
3D modeling and texture mapping on spherical displays are usually performed by substituting 2D rectangular plane coordinates with latitude and longitude values. However, the ability to express a certain shape depends on the density of the vertices that match the actuating pins on an SCD. It is necessary to arrange the actuators uniformly over the entire surface to construct various shapes. With conventional modeling methods, the density of the vertex increases as the number of poles of the spherical coordinate (latitude \(\pm\) 90\({}^{\circ}\)) increases, and the physical size of the actuators causes interference between them. Therefore, an icosahedron-based icosphere model was used for the closed-surface SCD. The icosphere achieves a high resolution by finely dividing the icosahedron surface. For example, the \(2V\) icosphere is obtained by dividing all sides of the icosahedron into two. As shown in Figure 1, the \(n\)V icosphere has \(n^{2}\) triangles by dividing the sides of the icosahedron by \(n\). As the resolution of the icosphere increases in a stepwise manner, the total number of vertices, edges, and faces increases exponentially. In this case, the number of faces for the \(n\)V resolution is \(n^{2}\)\(\times\) 20, and the number of edges is \(n^{2}\)\(\times\) 30. The relationship between the vertex and face to determine the number of vertices indicates five faces at 12 vertices, constituting the icosahedron, and six faces at the other vertices. Therefore, the number of vertices is ((number of faces) \(\times\) 3 + 1)/6, that is, \(n^{2}\)\(\times\) 10 + 2 in the \(n\)V resolution. This is equal to the number of actuating pins required to create an icosphere of the resolution in an SCD. In the SCD, an actuator is placed at each vertex, and the shape is deformed by pushing or pulling the surface in the radial direction of the surface (Figure 2). We proposed the concept of dynamic range to standardize the degree of deformation. The dynamic range represented the ratio of the volume when the SCD was in the initial state (minimum size) to the volume of the fully actuated (maximum size) SCD. The volume ratio of the SCD was equal to the ratio of the cubed radius of the spherical guide structure, \(R\), at each size; therefore, the equation for calculating the dynamic range is as follows:  
\[DynamicRange=\frac{\left(R+ actuator_{maxHeight}\right)^{3}}{\left(R+ actuator_{minHeight}\right)^{3}} \tag{1}\]  
Figure 1: Number of vertices, edges, and faces according to the resolution of the icosphere. Figure 2: Example of transforming a shape by adjusting the vertex value in the icosphere model: (**a**) inputting the target shape, (**b**) setting the icosphere model, (**c**) actuating each vertex, (**d**) reconstructing it in the form of the target, and (**e**–**h**) iterated reconstruction of the new target shapes. This type of actuator was used in the same environment with the same _moving distance_, and the representation of the volume change could improve as the size decreased (Figure 3). Implementing smaller actuators could minimize the size of the SCD, whereas longer actuating pins could maximize the size, ultimately increasing the dynamic range. Assuming that (\(n^{2}\)\(\times\) 10 + 2) actuators were attached to the spherical guide structure to achieve the target resolution of the \(nV\) icosphere, the size of the spherical guide structure could be calculated according to the actuator size (Figure 4). The total area required when (\(n^{2}\)\(\times\) 10 + 2) circles were tightly arranged in a hexagonal form was \(\left(5\sqrt{3}\,n^{2}+\sqrt{3}\,\right)r^{2}\), where \(r\) is the radius of the actuator. The radius of the spherical guide structure, \(R\), could be calculated as  
\[R=\sqrt{\frac{5\sqrt{3}n^{2}+\sqrt{3}}{\pi}}r \tag{2}\]  
Figure 4: Process of arranging vertices according to the position of the limited actuating pins. Figure 3: Assuming that the actuating pins have hexagonal packing, the dynamic range, that is, the rate of change of the total volume, can be calculated.