Concave Mirror Projection: How does it work?

 

Figure 1 – Kantian Box and Lasers
This is my experimental setup from the point of view of the two lasers set up in place of the two eyes of an observer. You can see the laser points where they have crossed over and hit the mirror (slightly separated above Kant's head), and where they intersect on the hidden image (the bright light on Kant's forehead).

The concave mirror projects an image to an illusory position in space in a way that is very convincing. It is a rare sort of image that seems to hover "in the air," independently of a screen or projection surface of any kind. I am always interested in how optical effects work, and in this case I suspected that the illusion exploited the principle of binocular convergence. When our two eyes have to turn inward to converge on an object that is close to us, our brain signals to consciousness that the object is close. To see the picture of Kant reflected in the concave mirror, the eyes follow paths of light that cross over at exactly the illusory location in front of the viewing-box. This point of convergence of the eyes is mistaken by the mind to be the location of the image.

 

Figure 2 – Point of Intersection
Click here to see a video of how I located the point of intersection.

To test this idea I set up the device and found the right viewing position for a convincing illusion. I then substituted two lasers for my eyes and directed the lasers to fix on a point on Kant's forehead (that is, on the inverted image of Kant, hidden in the box) (Figure 1). The path of light to the right eye is red and the path to the left eye is green. You can see how they have crossed when they reach the surface of the mirror. Then, by catching the laser points on a white card I could move the card to the point at which they converge and cross over (Figure 2). It was exactly at the position of the illusory location of the image. Just as I had suspected!

 

The principle of binocular convergence is being exploited to produce this illusion. While binocular convergence was understood in the 18^th century to be a powerful principle of depth perception, neither of the authors (Guyot 1769, Hooper 1774) who describe the concave mirror illusion mention this principle or give any explanation as to how the deception works. Theirs are incomplete examples of "rational recreations" which are ideally supposed to use pleasing illusions to explain the scientific principles behind them.

 

The illustration of the workings of the illusion that both authors give (Figure 3) is a side-on ray diagram that tells us very little (indeed, I think such diagrams often confuse more than illuminate). While the illusion is clearly enhanced by the use of both eyes when viewing, they do not mention that and depict only a single eye in their illustration. They appear not to know how it works.[1] I am interested in cases where artisanal knowledge is embedded in artefacts without being described in explanatory literature. This is not such a case since no-one appears to have conscious access to knowledge of the binocular logic of this illusion.

 

Figure 3 – Hooper's Diagram
The diagram shows how to set up the illusion, but not how it works (Hooper 1774: 134). EF is the concave mirror; C is an inverted image or object (in this case an object).

Technical Notes:

My mirror is 6 inches (15 cm) in diameter. Hooper (1774: 132) recommends that it be " at least ten inches in diameter [25 cm]." He also writes that " its distance from the partition [must be] equal to three-fourths of the distance of its center." Assuming the mirror is a portion of a sphere, its centre is the centre of that sphere. Its focal length is half that distance. My mirror has a focal length of 11 ³/₄ inches (30 cm) and a centre of 23 ¹/₂ inches (60 cm). The distance from the mirror to the image's point of projection (Hooper seems to be assuming that the image will be projected at the location of the partition) is 1 foot 11 inches (58.4 cm). This is more than Hooper's recommendation of 1 foot 5 ³/₄ inches (45 cm). There probably is a determinate relationship between focal length (30 cm), the distance from mirror to image (66 cm) and the distance to the projected image (58.4 cm). A ray diagram (which, to capture all the relevant angles, would best be in 3 dimensions) might help to explain why that relationship holds.

 

The observer can move further from and closer to the apparatus and the illusion persists at these different distances. The point of convergence of the eyes will be the same at every position, but the angle of convergence will vary. The further away, the closer to parallel the direction of the eyes will be and perhaps the weaker the illusion. The inter-ocular distance can vary and still the point of intersection remains the same. The viewer can also move side to side and up and down without the image changing its apparent location on Kant's shoulders. This stability, despite not being projected on any material surface, gives it an apparently "objective" quality. Perhaps for this reason Hooper calls it a "real apparition."

 

Footnotes:

References:

• Guyot, Edme-Gilles and Guillaume-Germain Guyot. 1769. Nouvelles Recréations Physiques et Mathematiques. Paris: Gueffier.
• Hooper, William. 1774. Rational Recreations. London: L. Davis.