Mathematical Equations Associated with Spline Interpolation for Determining and Analyzing Tennis Shots Trajectory and Velocity

Source abstract

Aim: This study investigated the application of spline interpolation for analyzing tennis ball trajectories and velocities. The motivation stemmed from the growing role of technology in sports performance and the need for quantitative tools that enable precise technical evaluation. The objective was to demonstrate that mathematical equations derived from spline interpolation can accurately describe shots trajectories and velocities, providing useful feedback for coaches and athletes. Material and Methods: A 15-year-old male tennis player executed 10 shots, including forehand (flat and topspin), backhand (flat and topspin), and serves. Distances and ball heights were measured manually at fixed points (0, 5, 10 m, landing), with times recorded via stopwatch. Curve Expert Professional was used to generate polynomial and trigonometric functions for trajectory modeling. Results: The analysis identified differences in maximum height and velocity between shot types. Topspin shots showed higher arcs and lower velocities, while flat shots produced flatter trajectories and higher speeds. Serves revealed distinct parameters influenced by toss height and angle. Equations reflected these dynamics and allowed interpolation of intermediate points (see Table 1, Figure 1). Discussion: The findings confirmed the suitability of spline interpolation, consistent with prior research (Cross, 1999b; Elliot et al., 2003). Despite limitations related to manual measurement and a single participant, the method proved reliable. Broader applications could integrate motion capture and sensors. Conclusions: Spline interpolation represents a feasible and effective approach for shot analysis in tennis, offering coaches a practical, science-based tool for technical optimization. Article history: Received 2025 November 09; Revised 2026 January 08; Accepted 2026 January 09; Available online 2026 March 30; Available print 2026 April 30. AUTHOR CONTRIBUTIONSAndrei-Cătălin Brisc contributed with data collection, conceptualization, draft writing.Marius Alin Baciu and Alina Paula Apostu contributed with supervision, methodology design and manuscript review, Radu-Tiberiu Șerban contributed to the writing and formatting of the manuscript. All authors have read and agreed to the published version of the manuscript. CONFLICT OF INTERESTThe authors declare no conflict of interest. ACKNOWLEDGEMENTThe authors acknowledge the support of the Faculty of Physical Education and Sport, Babeș-Bolyai University, Cluj-Napoca.