In my recent post on the camera obscura, I discussed the optical illusion produced by so-called anamorphic images, i.e. images which only appear normal from a particular point of view. One can readily understand such images from the point of view of geometrical optics, but I thought I’d go a step further and show how a little geometry can be used to construct your own simple anamorphs. In this post we discuss the simplest form of anamorphic image — one constructed from piecewise planar images — and when my sanity returns I’ll contemplate doing posts on other, more complicated distortions.
Let me first show you the result, and then we’ll discuss how I arrived at it. I chose one of my favorite images, a picture of ‘Sue’ I took while at the Field Museum, chopped it up and made an anamorph out of it. From the ‘proper’ point of view, it looks as follows:
From this point of view, the T-rex picture looks almost normal, except for two issues: 1. The images are slightly misaligned, and 2. If you look carefully at the lower center picture, it is out of focus. Moving the camera to my other eye, this is what one sees from this perspective:
Aha! The images are in fact different sizes and at different distances from the observer; it is only from our ‘proper’ point of view that they line up to give us the correct T-rex image. A photograph from a long-view of the arrangement shows how different the individual images are in size and location:
In fact, the images stretch backwards over a 3-foot range. This is why the lower center image looks slightly out of focus: it is at a significantly more distant location than the other pieces.
This sort of arrangement is straightforward to design, but rather tricky to set up! Let’s look at the design first. I took a 6″ by 9″ image as my starting point. I decided upon an observation point 2′ away from the center of the original image. From a geometric perspective, a side view schematic is as follows:
An image at distance d from the observer of height y will look essentially the same as a scaled image at distance d’ and height y’. The relationship of the two heights is readily found by comparing the similar triangles (red and green) formed by the objects and the observer:
The ratio d’/d of the distances from the image to the observer should be the same as the ratio of the heights, i.e.
An identical argument holds for the size of the image in the horizontal direction:
If I break an image up into pieces, each piece should be scaled in size based on the ratio of d’/d. I chose my original image to be 6 inches by 9 inches (small enough to print on a piece of paper), and to be observed at a distance d = 2 feet. The image was broken into six pieces as shown below:
Each image was placed as listed in the table below, with the appropriate scalings listed beside them:
- Image 1: d = 2 ft, x’ = 3”, y’ = 3”
- Image 2: d = 2.5 ft, x’ = 3.75”, y’ = 3.75”
- Image 3: d = 3.5 ft, x’ = 5.25”, y’ = 5.25”
- Image 4: d = 4 ft, x’ = 6”, y’ = 6”
- Image 5: d = 5 ft, x’ = 7.5”, y’ = 7.5”
- Image 6: d = 3 ft, x’ = 4.5”, y’ = 4.5”
A ‘top-down’ view of the images would appear as follows (proportions adjusted for readability):
I displayed the entire mess on top of my dining room table. The final geometric concern is the height at which the image center should be placed above the table. Because image 5 is the largest, at 7.5 inches, the origin of the picture should be at 7.5 inches above the tabletop. In other words, the bottoms of images 1,2 and 3 should all be at 7.5 inches, the tops of images 4, 5 and 6 should be at 7.5 inches, and the observation point should be at 7.5 inches as well.
Practical headaches abound in creating the image in practice! Because the pieces cannot block each other, I had to hang them from a supporting structure. Proper alignment was also a challenge, as my crude measurements of the tabletop distances were somewhat off. In the end, I put the images in approximately the correct positions, and then ‘tweaked’ the positions until I got a good illusion. (This was the part that nearly cost me my sanity, by the way: you’ve been warned!)
I felt the illusion worked well enough for demonstration purposes; compare the picture above with the original here.
One final, optics related, comment: such an anamorphic projection works better the larger the object-to-image distance. The reason for this is the limited depth of focus of the eye (and corresponding depth of field), at least for objects at short ranges from the observer. In essence, though the eye can adjust to put any object in focus (though you might need Coke-bottle glasses to do it, like me), objects at significantly different distances cannot be simultaneously put into focus.
I’ll leave a detailed discussion of depth of focus, and more complicated anamorphs involving angled and curved surfaces, for a future post.
I just wanted to point out that it’s easy to increase the depth of field of the eye by constricting the pupil. When the pupil is constricted, the eye (or any “camera” for that matter) starts to work more and more like a pinhole camera, and pinhole cameras have almost infinite depth of field.
In practice what this means is using lots and lots of light. If the anamorphic planes were brightly illuminated, your eye (or your auto-focus camera) would automatically “stop down” to a narrower pupil size, giving you a larger depth of field.
(All of this is covered in the Wikipedia articles cited, but I thought it was worth emphasizing, given that this kind of grew out of your long-standing interest in pinhole cameras.)
Thank you so much for this, I am looking at some paintings that I believe may have used optical distortions, reflections, etc. I’m gathering possible explanations for effects that don’t quite add up.
Sandra: You’re welcome! I hope it helps!
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I am currently in the third year of studying Fine Art at Arts University Bournemouth. I’m hoping to make a sculpture with the same underlying idea as “Co-Founder” and “The Hurwitz Singularity”. I want to build up layers of segments of maps so that when viewed from the front it will appear to be one image. I have realised that in order to create the correct perspective the back layers will need to be bigger than the front layer- as in the Ponzo Illusion. I am just wondering if I would need to used a specific formula to work out how to stagger the sizes of the layers to create this illusion and thought you may be able to help.
Hi Hannah! For flat pieces, I simply used similar triangles, as in the post. An image at distance d with width y centered on the viewing axis will appear to be the same size at a distance d’ if the width y’ is taken to be y’ = yd’/d. The same ratio applies for the height of the image. If you break up the image into 3 pieces across, each of those pieces will have its size scaled by the same amount.
Feel free to send me an email if you want to talk about it in more detail. Basically, it’s all about geometry!
That’s great thanks for your help.