Junhyeok Lee

Virtual reality (VR) has emerged as a complex technological domain that demands high levels of realism and interactivity. At the core of this immersive experience lies a broad spectrum of mathematical modeling techniques. This survey explores how mathematical foundations support and enhance key VR components, including physical simulations, 3D spatial analysis, rendering pipelines, and user interactions. We review differential equations and numerical integration methods (e.g., Euler, Verlet, Runge–Kutta (RK4)) used to simulate dynamic environments, as well as geometric transformations and coordinate systems that enable seamless motion and viewpoint control. The paper also examines the mathematical underpinnings of real-time rendering processes and interaction models involving collision detection and feedback prediction. In addition, recent developments such as physics-informed neural networks, differentiable rendering, and neural scene representations are presented as emerging trends bridging classical mathematics and data-driven approaches. By organizing these elements into a coherent mathematical framework, this work aims to provide researchers and developers with a comprehensive reference for applying mathematical techniques in VR systems. The paper concludes by outlining the open challenges in balancing accuracy and performance and proposes future directions for integrating advanced mathematics into next-generation VR experiences.

The memory palace, a mnemonic technique from ancient Greece, remains widely used for memory competitions and learning. This study introduces a procedurally generated virtual memory palace combined with the Worlds-in-Miniature (WIM) interface in virtual reality. WIM enhances spatial perception by providing a miniaturized visualization of the memory palace and enabling instant movement between locations. Focusing on recognition memory, an experiment with 40 participants was conducted to measure memory performance under conditions with and without the WIM interface. Each participant memorized 30 target images within a procedurally generated memory palace and completed a recognition task distinguishing targets from non-targets. Data included immediate recognition performance and follow-up assessments at 7 and 14 days to evaluate long-term retention. The results showed that the WIM interface significantly improved recognition memory, particularly among participants with lower initial success rates. By providing spatial structuring, WIM facilitated stronger memory associations, leading to enhanced recall accuracy and sustained long-term retention. This method demonstrates the potential of VR memory palaces as effective tools for supporting memorization-based learning in education and training contexts.

As VR technology advances and network speeds rise, social VR platforms are gaining traction. These platforms enable multiple users to socialize and collaborate within a shared virtual environment using avatars. Virtual reality, with its ability to augment visual information, offers distinct advantages for collaboration over traditional methods. Prior research has shown that merely sharing another person’s viewpoint can significantly boost collaborative efficiency. This paper presents an innovative non-verbal communication technique designed to enhance the sharing of visual information. By employing virtual cameras, our method captures where participants are focusing and what they are interacting with, then displays these data above their avatars. The direction of the virtual camera is automatically controlled by considering the user’s gaze direction, the position of the object the user is interacting with, and the positions of other objects around that object. The automatic adjustment of these virtual cameras and the display of captured images are symmetrically conducted for all participants engaged in the virtual environment. This approach is especially beneficial in collaborative settings, where multiple users work together on a shared structure of multiple objects. We validated the effectiveness of our proposed technique through an experiment with 20 participants tasked with collaboratively building structures using block assembly.