Abstract In order to develop the tactile interface that allows users to touch a 3D shape by means of a continuous and free hand interaction, it is needed to conduct a preliminary study for evaluating the concept. In this chapter we will analyse how it is possible to render a 3D shape by means of a trajectory according to the designers needs. After that, we will study the ways to make possible the physical rendering of this trajectory by controlling the elastic behaviour of a continuous plastic strip. This analysis allows defining the degrees of freedom that have to be controlled in order to actuate the system and the different kinematics solutions. Various conceptual solutions for the actuations systems will be investigated, and the best solution according to the project goals will be chosen.

Cutting Plane and Geodesic Trajectory

Experimentally [1], it has been observed that designers slide their hands along a trajectory, while exploring a surface. For this reason, the strip has to be modelled so that the centre line can represent the correct trajectory.

In order to obtain the trajectory that the strip has to represent, we have used the cutting plane metaphor. To understand the concept we can consider a simple shape, such as a cylinder, as shown in Fig.4.1.

A. Mansutti et al., Tactile Display for Virtual 3D Shape Rendering, PoliMI SpringerBriefs, DOI 10.1007/978-3-319-48986-5_4

Fig. 4.1 Cutting plane metaphor

the strip is made of continuous material with a finite width, the trajectories that it can represent are only those that can be obtained by means of the envelope of the strip on the surface. Lets consider again a simple surface, as for instance a cylinder, and a planar surface, which represents the real strip (Fig.4.2). We can observe that if we try to envelop the strip onto the cylinder along a plane that is orthogonal to the axis of

Fig. 4.2 Behaviour of plastic strip representing the geodesic trajectory the cylinder (Fig. 4.2a), the centre line of the strip lie on a plane. Therefore, in order to represent this trajectory, it is sufficient to bend the strip without twisting it. On the other hand, if we impose an angle between these two surfaces, we can observe that the behaviour of the strip changes: the trajectory is three-dimensional and the strip has to bend and twists to respect the correct envelope onto the cylinder (Fig. 4.2b). Consequently, the physical strip is able to represent the trajectories obtained by the cutting plane metaphor only for limited cases, such as cylindrical trajectories. For the other cases the best approximation is the Geodesic Trajectory. It can be defined as the shortest path between two points in a curved space, for which the tangent vector field is parallel along this curve. In order to be able to analyse the behaviour of plastic strip representing the geodesic trajectory we have developed a concentrated parameter model that will be illustrated in the following Section.