Dual-Object Shadow Analysis in Plane Mirrors: Integrating Real and Virtual Light Sources for Simplified Optics

Abstract

The study of shadows is a central topic in geometrical optics, but traditional approaches usually consider only a real light source and a real object. This works for simple cases, yet becomes difficult to apply when a plane mirror is introduced, since both real and virtual elements must be taken into account. To address this challenge, I propose an alternative geometric method that provides a systematic way to analyze shadow formation in mirror systems. The method treats four elements within a single ray-tracing framework: a real source (S1), its virtual counterpart (S2), a real object (O1), and its virtual image (O2). By examining how rays from S1 and S2 are blocked by O1 and O2, the method predicts where shadow regions will appear. The analysis shows that the real and virtual objects always form complementary shadows on opposite sides, although the exact location of these shadows (left, right, upper, or lower) depends on the position of the light source relative to the mirror–object setup. Virtual objects and sources are not physical entities, but by using them as geometric constructs the method gives consistent predictions and resolves common ambiguities in conventional diagrams. Beyond theoretical interest, the framework offers practical value for optics education and for producing clear, accurate visualizations of mirror-based shadow phenomena.