In the papers [1–5] using synthetic methods and the facts from the conics theory in the Euclidean, pseudo-Euclidean, quasi-hyperbolic and hyperbolic planes, we have studied and proved numerous facts related to the cuspidal curves with emphasis on evolutes and Steiner’s deltoid. In this paper, the intention is to see what can be said about the curve of centers in some special pencil of conics and to show how, studying the center curves in the special pencil of conics a connection with Steiner’s deltoid curve has been found.

Stroke is a significant global health concern, ranking among the top causes of death and disability. Timely post-stroke recovery hinges on structured upper limb rehabilitation within the first 24–48 hours. Although virtual reality (VR) rehabilitation is promising, its accessibility is complex and cost-limited, especially in less developed regions. In this study, we developed and assessed nonimmersive VR rehabilitation games using the Kinect Xbox One and UNITY game engine, targeting upper limb movement sequences vital for post-stroke recovery. The developed games involved simple to progressively challenging movements, engaging the deltoid, biceps, and triceps muscles, and aligning with post-stroke rehabilitation criteria. We used peak-to-peak electromyography signal detection with a 20% threshold and three distinct feature extractions to represent muscle contractions during gameplay. Our cosine similarity analysis revealed the efficacy of games in meeting movement sequence criteria, benefiting mostly the shoulder motor and deltoid muscles (93% improvement) with more moderate improvements in the biceps (73%) and triceps (47%). Our results suggest that these games could contribute to upper limb stroke rehabilitation, particularly that of the deltoid muscle. This study highlights the potential of VR-based rehabilitation games in post-stroke upper limb recovery, emphasizing muscle engagement, target distances, and rehabilitation criteria in game design. Although healthy participants were involved here, future research involving post-stroke patients and the exploration of more immersive, multiplanar movements hold the potential for a more comprehensive rehabilitation experience. Our study contributes to advancing the common knowledge related to VR-based stroke rehabilitation, focusing on upper limb movement sequences with implications for therapeutic engagement and recovery.