Analysis of Geometric Transformations in Automatic Door System Response with a Motion Sensor

Automatic sliding doors are usually studied as electronic systems, yet their motion offers a concrete context for geometric transformations. This study analyses how translation, reflection, and dilation describe the motion of a two-panel automatic sliding door at the main entrance of a shopping mall in Cirebon, Indonesia. The door was observed in 12 sessions over four days, with approximately 10–15 opening–closing cycles recorded per session, and the panel positions and opening widths were documented on video and mapped onto a Cartesian plane. Each panel moved by a pure horizontal translation along its rail, and the translations of the two panels were reflection-symmetric about the vertical centre line of the doorway. The representative opening width was 30 cm for a single user, 75 cm for two to three users, and 100 cm for a crowd. Taking the single-user opening as the reference aperture, the larger openings correspond to dilation factors k = 2.50 and k = 3.33 about the centre of the door, while the panels themselves remain rigid. Because detection distance and the number of users varied together and the door’s control logic was not accessible, these factors describe the observed openings rather than the controller’s rule. The door thus illustrates a composition of isometries combined with a change of scale of the aperture and can serve as a contextual resource for teaching geometric transformations.