## 3. Haptic Representation of an Object's Shape and Size by Using Haptic Devices  
A first issue to face when defining the integration of AR and haptics is how to make a sound and expressive haptic representation of a digital element or object. Ignoring the limits imposed by technology, ideally the haptic representation of that element should coincide with the \"physical\" characteristics of the object (i.e., size, volume, textures, weight, etc.). This kind of representation is specially adapted to mid-air devices and gloves, that is, those devices in which the user receives the feedback in their hand. For example, if a user is haptically inspecting a 3D model of a soda bottle, they should be able to perceive with their hands the physical features of the real bottle from which the 3D model was extracted. In reality, this interpretation presents several limitations. The most obvious issue is related with the accuracy and capabilities of the haptic device itself. In general mid-air devices deliver haptic feedback as points located over the own device. This happens in both ultrasonic (e.g., Ultrahaptics) and air-vortex-ring based devices. Hence, that point is the basic representation unit. Due to the technology in use to provide the feedback, points may not be perceived by the user as a sharp point. Instead, it may perceived more as an air field centered on that point with a diffuse radius. Figure 1 illustrates this issue. In the left side of the figure appears a \"sharp\" point with well-defined limits. In the right side, a situation similar to the one perceived with the Ultrahaptics is represented [33]. The objective is to combine several of these _fuzzy_ haptic points to generate the haptic representation of an object. We will assume that any digital object may be represented as a 3D model, i.e., as a representation by means of vertices and faces. Thus, an ideal figure such as a sphere (which by definition has no vertices) becomes a representation based on adjacent faces composing that sphere. Depending on the haptic experience to be provided due to the service goal, it is feasible to generate different representations of the same object. We classify the representations as follows:  
* High-fidelity representation: This type of representation seeks to give haptic feedback that resembles as close as possible the physical characteristics of the object to be represented. More specifically, the characteristics that we contemplate in this paper are volume, size, and shape, leaving aside more advanced ones like weight and textures due to technology limitations. Hence, the haptic device will need to reproduce the 3D model characteristics, mainly the vertices and the faces. This type of representation is useful in those situations in which it is required for the user to explore a model (e.g., to perceive its dimensions, to resize it, etc.). To build high-fidelity representations, it is feasible to use different strategies:
* _Vertex-based_ represents only the vertices that shape the model as scattered points. * _Edge-based_ represents only the edges between the devices of the model. These edges are defined by the faces. By definition, vertices are also represented, since they are present in the limit of the edges. * _Volume-based_: In this case, both vertices and edges are represented as are the surface of the faces as well as the interior of the figure. * Single-feature-based representation: In this case, we seek to approximate a 3D object to one or several haptic points, leaving aside fidelity. This is especially appropriate for those 3D models that are small enough and in which details would go unnoticed when using the high-fidelity representation. Instead of representing all the features that form the 3D model, it may be translated to a single point, calculated, for example, using the centroid of all its vertices. Figure 2 illustrates the issue in which the larger the 3D object is, the worse the adaptation will be using this type of representation.