## 3 Defining a Haptic Icon Set
### Haptic Icons  
Currently available general-purpose devices provide a natural interface to navigate through virtual 3D environments, usually offering up to 6-DoF position sensing. Proxy-based methods are among the preferred techniques for haptic rendering [18]. The exerted forces \(\mathbf{F}\) are proportional to the distance between a virtual proxy \(\mathbf{VP}\) (the virtual representation of the haptic device) and the haptic interface point \(\mathbf{HIP}\) (an unconstrained haptic position in the scene): \(\mathbf{F}=K(\mathbf{VP}-\mathbf{HIP})\), where \(K\) is the stiffness constant.  
As previously mentioned, haptic icons provide a powerful tool to display abstract information during the navigation whether to reinforce, complement, or substitute the visual channel. Haptic signals can be active or passive [43; 44].  
A haptic icon is passive if the observer does not have to move to perceive the signal (time-varying signals). In the other case, the signal can be considered active (space-varying signals). Active signals are more natural since users obtain haptic information through intentional exploration of the world. In contrast, the perception of the amplitude or the frequency of the signal might be affected by the speed of the user's movement. On the one hand, we want users to perceive the icon even when they are not moving the HIP, but on the other hand, we want users to be able to recognise the signal naturally. Therefore, we have chosen a hybrid approach (semi-passive). Our haptic icon is time-varying, but the signal is not directly transformed into force. Instead, our haptic icons change the virtual object's shape. In the simplest scenario, the signal is implemented as a proxy displacement. In this way, even if our signal is semi-passive, the haptic icons can be mapped to specific positions of the 3D space.  
Analogously to the work in [PERSON] et al. [32], our haptic icon is a periodic signal \(s(t,f,A,S)\) that changes over time \(t\), and the abstract information is encoded in its frequency \(f\), its amplitude \(A\), and its shape \(S\). The wave shapes implemented in our system correspond to different discriminable forms according to [PERSON] et al. These forms are the sine, sawtooth and square waves. To define the other two parameters, experimental evidence is needed. Although [PERSON] et al. carry out different experiments to calculate discriminable ranges of frequencies and amplitudes, their icons are calculated as a force (instead of being calculated as a position displacement) and they use a customised hardware design which is non-commercial and, consequently, hardly available for its use in other external applications. Therefore, the values of amplitude and frequency cannot be extrapolated to our system.  
Here, we propose the use of haptic icons, added as a one-dimensional time-varying signal that describes a modification in the shape of the virtual object to which the information is mapped. In the simplest case, the proxy \(\mathbf{VP}\) is constrained to a 3D surface and the force feedback exerted during a 3D exploration can be computed as: \(\mathbf{F}=K(\mathbf{VP}-\mathbf{HIP})+Ks(t,f,A,S)\mathbf{N}\), where \(\mathbf{N}\) is a unitary vector in the surface normal direction.  
In our experiments, to isolate the haptic stimuli and avoid bias, we have simplified our haptic rendering algorithm so that the proxy is constrained to a horizontal plane (xz-plane). Therefore, there will only be haptic feedback in the y-coordinate (vertical axis), and this force can be computed as \(\mathbf{F}=K(s(t,f,A,S)-HIP_{y})\mathbf{N}\), where \(HIP_{y}\) is the y-coordinate of \(\mathbf{HIP}\) and \(\mathbf{N}\) is equal to \((0,1,0)\).