The geometry of cylindrical mirror anamorphoses is really quite complex, but there are simple approximations which give images that are very close to being mathematically correct. As with conical mirror anamorphoses, the image is spread out in a circular region around the mirror. This is not surprising since a cylinder can be thought of as a cone whose tip is an infinite distance from the base. However, when you look at cylindrical mirror anamorphoses, you look sideways on to the cylinder, rather than from above. If the image goes all the way around the cylinder, then you have to move around it in order to see the different parts of the image, but normally the images are semicircular.
This type of anamorphosis was particularly popular as an optical toy. Dover books have reproduced a number of images from Victorian times in their book. Originally, the mirror would have been a solid cylinder, but now it is easier to make a piece of metallised plastic into cylinder, and such a sheet is provided with the book. If you have trouble getting hold of some metallised plastic sheet, it is worth buying the book with its 24 images.
The image on the cover shows “Sancho Panaza on his donkey” restored in the mirror. The flat anamorphic image looks like this:
Note the circle, at the top centre. This is where you place the mirror. (These images are used with permission of Dover Publications.)
The picture below is from Dubreuil’s “La Perspective pratique” published in 1679 and shows how to build up a square grid transformed anamorphically, to be restored by a cylindrical mirror.
The part at the top shows the grid marked with numbers to indicate where the cells appear in the distorted grid. The bottom part shows how the spacing for the curves are constructed. They are not equal, but get closer together towards the centre.
The curves almost look like circles, but in fact they are not. The way the grid is distorted depends on where the restored image appears to be and where the viewer’s eye point is for the correct restoration. It is possible to use a set of concentric circles. This gives images which are mathematically incorrect, but which look as if they are reconstructed correctly. You can then take a picture, place a regular grid over it and copy the corresponding parts of the grid.
The mathematics for obtaining the correct shape for the transformed grid is not simple because the mirror is curved. It relies on the optics of curved mirrors. Cylindrical mirror anamorphoses are the most common mirror ones because a cylindrical mirror is easier to make, and it is easy to recreate one if it is lost, since the radius of the cylinder is usually obvious from the picture. On many pictures it is often marked to show you where to place the mirror.
Making your own anamorphic cylindrical mirror images
There are a number of ways to make your own cylindrical mirror anamorphoses.
Constructing an image from circles
An exact, mathematically correct, anamorphic image is created for a cylindrical mirror with an image placed at a different position. This method does not involve anything more than an image on a square grid and a circular grid. It is a mapping, or a correspondence, between a cartesian set of a coordinates, and a polar set of coordinates. The following steps show a simple image formed just by filling in the cells of a grid.
Draw a grid (in this case a square one, but it does not have to be), and label its edges so that you can identify the corresponding cells in the circular grid. In order to make them less cluttered, not all edge points have been labelled in the following diagrams.
Draw a circle whose radius is equal to the radius of your cylindrical mirror. Then draw some half circles whose centre is the same as the one for the mirror circle. Finally draw some radii at angles of 22.5 degrees. This will give you a circular grid like this:
Label the circular grid as above. Note that the base of the square grid corresponds to the side of the circular grid which is closest to the mirror. Because images in mirrors are reversed back to front, you must take account of this in the labelling.
Draw a design or picture on the square grid by colouring in the cells of the grid.
Using the labels on the two grids, identify the cell to colour on the circular grid and create the anamorphic version.
Place your cylindrical mirror on the circle and look into the mirror to see the image restored.
Using the grid template
To make your own anamorphic images for a cylindrical mirror, go to the download post below and print out the grid. You will also need some metallised plastic sheet to make the mirror. Perhaps, the easiest place to get some is to buy the Dover book of the Magic Mirror.
There are also some examples of simple grid anamorphoses on the download page to give you some ideas.
Using Adobe Photoshop or The Gimp
Two software packages have the ability to take images and transform them to make a cylindrical anamorphic version. Essentially these calculate the cartesian rectangular to polar conversion that has been used above to create the circular grid.
Look for the Distort Filter that performs Polar Coordinates distortion. This description is for Adobe Photoshop® 4.0, but other versions are very similar.
Load your image:
Flip the image so that it will be correctly reflected in the mirror. Also add some border to the image.
Adding the border ensures that the image only become part of a circle. Also ensure that the final image is square, otherwise it will not be completely circular.
Apply the Polar coordinates distort filter to the image.
(Note: If you find that the options on the Filter menu are dimmed so as to be inaccessible, then you must make sure that the image is at least RGB quality by using the Mode option on the Image menu.)
Phillip Kent’s AnamorphMe software
You can get a free piece of software for creating cylindrical and conical mirror anamorphic images from Phillip Kent’s site at www.anamorphosis.com.
A machine for drawing cylindrical mirror anamorphoses
Creating cylindrical mirror anamorphoses is a slow business if you are drawing complicated images such as faces. Before the advent of computers it was an even longer process!! However, even then, some mathematicians and engineers tried to speed up the process using machines.
This one was invented by Jaques Leopold and published in M J Brisson’s “Dictionairre raisonné de physique” in 1781 and is a kind of etch-a-sketch for anamorphosis drawing.
Ready made images
The download post below also has some examples of images to view including the Maths Year 2000 Arithmakid logo. You will need a cylindrical mirror to see the images.
Restoring cylindrical mirror anamorphoses
The software methods used to create anamorphoses using Adobe Photoshop® and the Gimp above can also be used to reverse the process if you have an anamorphic image and want to see the original. The software also has a Polar to Rectangular option in the Distort filter.
There are examples of this technique in a post below.
Using the cylindrical mirror examples
To view the cylindrical mirror examples you need to make a mirror using a piece of aluminium coated thin plastic sheet. If you buy a book like the Dover book of the Magic Mirror, then one is provided. Most images show a circle where to place the mirror, which also gives you the idea of the diameter of the mirror. Because the human brain is very good at understanding visual information, you will get a reasonable image even if the diameter is not quite right and the mirror is not in the correct place.
Viewing images using software
One way to view cylindrical mirror anamorphoses is to use a ray tracing program to reconstruct an image. This can be thought of as using a program inside the computer which defines how rays of light behave in a scene. You have a virtual camera which looks at a scene. In this case a cylindrical mirror is placed on the anamorphic image. The results are often better than using a real mirror because virtual mirrors can be made perfect, whereas real mirrors can have tiny defects which could distort the image.
The best known such program is POV-Ray which stands for Persistence of Vision Ray Tracer. It is available free from www.povray.org. Povray takes a while to learn to use, although there is lots of useful information on the Povray site.
This is the image produced using this method of the anamorphic image which is of a man wheeling his stomach on a wheelbarrow.