Self-Image Multiplicity in a Concave Cylindrical Mirror

TL;DR

The paper addresses how an observer placed in front of a concave half-cylinder mirror can see multiple self-images, a phenomenon often omitted from textbooks. The authors develop a two-dimensional comeback/visibility framework and validate it with a three-dimensional experiment using a concave half-cylinder and a webcam, deriving explicit spatial partition rules with $n_{max}(θ) = floor(π/(π-2θ))$ and $r(n,α) = -R [ sin α tan( nα/(n-1) ) + cos α ]$. The eight tested positions agree with the predicted odd counts $N = 1,3,5,...$, confirming the predicted boundary curves. The work provides an accessible classroom demonstration and suggests broader relevance to geometric acoustics and wave-based ray tracing.

Abstract

Concave mirrors are fundamental optical elements, yet some easily observed behaviors are rarely addressed in standard textbooks, such as the formation of multiple reflected images. Here we investigate self-imaging -- where the observer is also the observed object -- using a concave cylindrical mirror. We predict the number of self-images visible from different observation points and classify space into regions by image count. We then test these predictions with an inexpensive stainless-steel concave cylindrical mirror commonly found in teaching labs. This activity links geometrical optics principles to direct observation and provides a ready-to-use classroom demonstration and student exercise.

Figures (6)

• Figure 1: Experimental setup for observing self-images in a concave mirror. Green arrows mark the light source. (A) Concave cylindrical mirror. (B) Webcam observer with a small marker (a UV-glow sticker) placed near the lens to co-locate the source and observer. (C) A 24"$\times$36" poster marks the predicted regions for different self-image multiplicities; placing the webcam in these regions yields the corresponding number of glowing dots (see Fig. \ref{['fig04']}).
• Figure 2: Light-ray paths for self-image formation. The observer/light source is the red point. (A1) A ray leaves the observer, undergoes multiple reflections in the concave mirror, and returns to the observer. (A2) The corresponding image location is found from a narrow ray bundle by locating where the outgoing bundle converges. (B1) Rays may undergo many reflections; the number of reflections defines the self-image order$n=1,2,3,...$. (B2) Reverse path of the higher-order ray in (A1).
• Figure 3: 2D theory for the ray return condition. In (A), the cyan region shows all positions in space where a $n=1$ light-ray path exists. In (B), we introduce the notation for analyzing higher-order $n>1$ light-ray paths: we represent the observer position $S$ in polar coordinates $(r,\varphi)$ with the origin at the center $O$ of the mirror’s circular arc. We consider positions along the the reflective symmetry axis $\Delta$ ($\varphi=0$) in (B1), and off-axis ($\varphi \neq 0$) in (B2).
• Figure 4: Spatial partition by self-image multiplicity for a semicircular concave mirror. The half-opening angle of this mirror is $\theta = \pi/2$. Regions of the two-dimensional space are colored according to the number of self-images $N=1,3,5,...$ visible to the observer.
• Figure 5: Imaging experiment with our concave cylindrical mirror. We position the webcam at eight locations (marked in green, i.e. 1, 2, 3, ..., 8) and keep its optical axis fixed along the x-direction, then capture images in total darkness. The UV-glow sticker can be charged with a UV light source before imaging if needed. We show the captured images at these locations on the sides of this figure; the observed self-image multiplicity matches our theoretical prediction.