This paper presents an N-ary Gaussian Model for predicting endpoint distributions in pointing tasks across task scenarios. Built on the foundational principles of the Ternary Gaussian model series, our model framework allows researchers to define parameter constraints and automatically refine model combinations, eliminating the need for predefined equations based on data analysis. We utilize the Bayesian Information Criterion (BIC) for model selection, ensuring simplicity while maintaining predictive accuracy. We conducted a comparative analysis against published baselines across 7 diverse datasets, covering 1D, 2D, and 3D tasks, different input modalities, different display devices, and time-constrained scenarios, demonstrating the robustness and generalization of the N-ary Gaussian Model. The N-ary Gaussion model offers an automated solution for modeling pointing uncertainty, and also incorporates cross output device, input modality, and temporal constraint factors into spatial pointing uncertainty modeling for the first time.

Virtual reality (VR) allows users to observe and manipulate 3D geometry from multiple viewpoints. Most VR selection work, however, optimizes techniques for selecting entire objects. Selecting a single face on a polyhedron remains underexplored and is more challenging because the interaction must act on a local component while preserving the object's global structure. We introduce a design space tailored to this task with three dimensions: viewing strategy, disambiguation consistency, and interaction metaphor. Guided by this space, we design eight freehand techniques for polyhedral face selection. A within-subjects study with 16 participants evaluates these techniques across polyhedral complexity (two radii; face counts 4, 6, and 12). The results identify three complementary top techniques, reveal tradeoffs between viewing choices and geometric preservation, and yield concrete guidelines for matching techniques to target geometry and task demands. A follow-up study with complex, realistic models confirms the robustness and practical usability of the three techniques. Together, these contributions shift attention from whole object selection to precise component selection in VR and provide actionable methods for 3D modeling, assembly, and texturing.

For individuals in situations where their hands are occupied, no-handed touch interaction represents a significant yet underexplored area. In this paper, we investigate touch performance on screens using body parts (the nose, tongue, chin, elbow, and toe) rather than the hand. Experiments were conducted with 37 participants to analyze their touch interaction, where they selected targets using various body parts under different conditions. We employed the Dual-Gaussian model and Ternary-Gaussian model to describe the touch endpoints of these body parts. Based on the statistical criterion derived from the models, we proposed a no-handed assistive technology, which achieved up to a 24.4% improvement in tapping speed and a 58.47% increase in tapping accuracy. Additionally, this paper provides HCI developers with design guidelines for optimizing no-handed touch interaction interfaces.

Proximity-based feedback provides users with real-time guidance as they approach an interaction goal. This type of feedback is particularly useful for tasks that require guidance during the interaction process, such as selecting moving targets. This work explores proximity-based feedback types and modalities to improve the selection of moving targets in VR by leveraging three feedback types that combine visual, auditory, and haptic modalities. We evaluated the performance of these mechanisms through two user studies, analyzing both objective data (e.g., selection time, error rate) and subjective data (e.g., user experience, preferences) to explore the characteristics of feedback types across different modalities and to examine the roles of various modalities within multimodal combinations. Our findings suggest optimal selection mechanisms for developers and should be tailored to different goals: achieving user precision, enabling quick movement to a target, considering task duration, and enhancing entertainment value. We also discuss applications that correspond to these different perspectives.

This paper presents a Shape-Adaptive Ternary-Gaussian model for describing endpoint uncertainty when pointing at moving targets of arbitrary shapes. The basic idea of the model is to combine the uncertainty related to the target shape with the uncertainty caused by the target motion. First, we proposed a model to predict endpoint distribution on static targets based on a Dual-Space Decomposition (DUDE) algorithm. Then, we linearly combined a 2D Ternary-Gaussian model with the newly proposed DUDE-based model to make the 2D Ternary-Gaussian model adaptable to moving targets with random shapes. To verify the performance of our model, we compared it with the original 2D Ternary-Gaussian model and a recent proposed Inscribed Circle model in predicting endpoint distribution. The results show that the proposed model outperformed the two baseline models while maintaining good robustness across different shapes and moving speeds.

Modeling the endpoint uncertainty of moving target selection with crossing is essential to understand factors such as speed-accuracy trade-off and interaction efficiency in crossing-based user interfaces with dynamic contents. However, there have been few studies looking into this research topic in the HCI field. This paper presents a Quaternary-Gaussian model to quantitatively measure the endpoint uncertainty in crossing-based moving target selection. To validate this model, we conducted an experiment with discrete crossing tasks on five factors, i.e., initial distance, size, speed, orientation, and moving direction. Results showed that our model fit the data of μ and σ accurately with adjusted R2 of 0.883 and 0.920. We also demonstrated the validity of our model in predicting error rates in crossing-based moving target selection. We concluded with a set of implications for future designs.

Background Crossing-based target selection motion may attain less error rates and higher interactive speed in some cases. Most of the research in target selection fields are focused on the analysis of the interaction results. Additionally, as trajectories play a much more important role in crossing-based target selection compared to the other interactive techniques, an ideal model for trajectories can help computer designers make predictions about interaction results during the process of target selection rather than at the end of the whole process. Methods In this paper, a trajectory prediction model for crossing-based target selection tasks is proposed by taking the reference of a dynamic model theory. Results Simulation results demonstrate that our model performed well with regard to the prediction of trajectories, endpoints and hitting time for target-selection motion, and the average error of trajectories, endpoints and hitting time values were found to be 17.28%, 2.73mm and 11.50%, respectively.