Optics 006 · Ray worlds, boundaries, and natural optics

Floating Image Mirascope

A close cutaway mirascope labels both confocal paraboloids, the central opening, the hidden point object, and the luminous real-image point. Color-separated path segments make the two reflections, middle collimation, aperture clipping, and free-space crossing readable at a glance.

Captured solid angle

\eta_{\Omega}

$0.50$

Real-image height

h_{\mathrm{image}}

$50\%$

Optical-path spread

\Delta\mathrm{OPL}

$0.00\pi$

Model regime $\text{valid model regime}$

Start with the essentials

Focus question: Where must two parabolic mirrors place a real image that can be seen but not touched?
One-sentence intuition: For the ideal confocal trace, all accepted rays share $\Delta\mathrm{OPL}=0$ . Aperture clipping changes captured solid angle and brightness, not the geometric image location.

Core mathematical model

Paraboloid profile

z=\frac{r^2}{4f}

The generated mirror mesh and its reflection normal use the same focal length.

Equal optical path

\mathrm{OPL}=\sum_j n_j\ell_j,\qquad \Delta\mathrm{OPL}=0

In the ideal air-filled confocal geometry, accepted ray paths from object to image have equal total length.

Collected solid-angle fraction

\eta_{\Omega}=\frac{\Omega_{\mathrm{accepted}}}{2\pi}

This hemisphere fraction quantifies aperture capture without pretending that all emitted directions reach the image.

Common difficulties

Treating the floating image as a hologram

Typical misconceptionThe image is a screen effect or a virtual image that exists only for one viewpoint.

Better mental modelThe traced rays physically converge in free space above the opening, forming a real image. Viewpoint changes alter which rays enter the eye, not the convergence point.

Run the experiment

01
Scene 1: Open mirascope
Follow one color sequence from the hidden object to the upper mirror, down the collimated middle segment, and from the lower mirror to the free-space image point.
What to observe: Every accepted path has one upper and one lower reflection before crossing the same real-image point.
02
Scene 2: Aperture clipping
Reduce the outer mirror aperture and distinguish it from the separately labeled central opening.
What to observe: Clipping removes high-angle rays and reduces collected solid angle, while image height remains tied to focal length.
03
Scene 3: Walk-around parallax
Orbit the camera around the walk-around scene while keeping the optical geometry fixed.
What to observe: The real image remains spatially fixed during camera orbit; only its visible ray subset and parallax change.