Anamorphic Art

Cylindrical Mirror Anamorphoses

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. 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 (silvered mylar) into a cylinder, and such a sheet is provided with the book. If you have trouble getting hold of metallised plastic, it is worth buying the book with its 24 images.

Dover Book Publications
The image on the cover shows "Sancho Panaza on his donkey" restored in the mirror. The flat anamorphic image looks like this:
Sancho Panaza on his donkey
Note the circle, at the top centre. This is where you place the mirror. 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.
La Perspective pratique
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 an Anamorphic Cylindrical Mirror 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. 

Step 1

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.

Step 1
Step 2

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:

Steps 2 and 3
Step 3

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. 

Step 4

Draw a design or picture on the square grid by colouring in the cells of the grid. 

Step 4
Step 5

Using the labels on the two grids, identify the cell to colour on the circular grid and create the anamorphic version.

Step 5
Step 6

Place your cylindrical mirror on the circle and look into the mirror to see the image restored.