  Designing of touchless user interface is gaining popularity in various
contexts. Using such interfaces, users can interact with electronic devices
even when the hands are dirty or non-conductive. Also, user with partial
physical disability can interact with electronic devices using such systems.
Research in this direction has got major boost because of the emergence of
low-cost sensors such as Leap Motion, Kinect or RealSense devices. In this
paper, we propose a Leap Motion controller-based methodology to facilitate
rendering of 2D and 3D shapes on display devices. The proposed method tracks
finger movements while users perform natural gestures within the field of view
of the sensor. In the next phase, trajectories are analyzed to extract extended
Npen++ features in 3D. These features represent finger movements during the
gestures and they are fed to unidirectional left-to-right Hidden Markov Model
(HMM) for training. A one-to-one mapping between gestures and shapes is
proposed. Finally, shapes corresponding to these gestures are rendered over the
display using MuPad interface. We have created a dataset of 5400 samples
recorded by 10 volunteers. Our dataset contains 18 geometric and 18
non-geometric shapes such as "circle", "rectangle", "flower", "cone", "sphere"
etc. The proposed methodology achieves an accuracy of 92.87% when evaluated
using 5-fold cross validation method. Our experiments revel that the extended
3D features perform better than existing 3D features in the context of shape
representation and classification. The method can be used for developing useful
HCI applications for smart display devices.
