Conical Mirror Anamorphoses

The geometry of conical mirror anamorphoses is very simple, and uses techniques similar to the ones used to put images onto a cone. There is only one type of conical mirror anamorphosis, where you look down on the mirror so that the image appears to be its base. The image is spread out in a circular region around the mirror.

The distortion is one of the most extreme ones of all anamorphoses and can be very hard to understand if you do not have the mirror. The abstract form of the image can also be aesthetic in its own right. This is an example of a square grid anamorphosis. The crossed lines are two diagonals of the square which fits in the circle at the base of the cone.

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. Although, conical mirror ones are quite common, because they are easy to construct, many may have been lost because the image is so unintelligible and if the mirror is lost then its exact shape is not easy to reconstruct.

This is an example of a construction of a conical mirror anamorphosis by Fernandino Galli da Bibiena from a book he published in 1731 on perspective and architectural design.


The image he is creating in his fig 2, starts out from a circular grid showing a face in the top right corner. It is not easy to see the face in fig 2. The eye looks in the mirror from point D and his use of numbers shows how the image is reversed.

Constructing the image

Each anamorphic image is created for a specific conical mirror, since the angle of the cone (and the viewing position) define the distortion. There is a description of how to make a conical mirror here.

In the download conical mirror examples in the post below (here), the images have been designed for a cone which has an apex angle of 60 degrees and a viewing position above the tip of three times the height of the cone. The description here uses a lower viewing position to make the diagram shorter.

Step 1:   The picture will not normally be small enough to fit in the base of the cone. Even if it were, then accuracy would suffer since the cone is quite small. When you have finished the drawing, you can reduce the picture on a photocopier. So decide on the enlargement ratio and draw a circle around the picture to correspond to the base of the cone. This (with only one point of the picture, point P) is the drawing at the left above.

Step 2: Using the enlarged radius of the cone base and the angle of the cone, construct a triangular cross section of the cone as shown in red in the diagram at the right above. Also draw the axis of the cone CD.

Step 3: Decide on where the eye will be positioned along the axis of the cone at point E.

Step 4: Choose a point P on the drawing. This point will become point Q on the anamorphic image.

Step 5: Draw a line joining P to the centre of the circle D and extend it outside the circle. Using a pair of compasses, transfer the length DP to the base of the cone.

Step 6: Working in the right diagram, join P to E to intersect the side of the cone cross section at point R.

Step 7: Erect a perpendicular RT at R and then draw line RQ so that angles ERT and TRQ, since the cone is the mirror. ER is the incident ray and RQ is the reflected ray and the angle of incidence is equal to the angle of reflection.

Step 8: Using a pair of compasses, transfer the length DQ to the picture as in the diagram at the left.

Step 9: Repeat for all points in the image.

A simpler construction and investigation

Galli da Bibiena’s diagram offers a simpler way to find the transformed point.

This diagram is a modification of the previous one and has the same lettering for the original important points.

Step 1: With centre C, the tip of the cone, and radius CE, draw a circle.

Step 2: Find point F such that angle CEF is the angle of the cone. Then join F to C and extend it to point Z. (See investigation below.)

Step 3: If point P corresponds to a point on the drawing, then construct its position as in the previous method. Join P to the eye-point E and intersect the side of the cone at R.

Step 4: Join F to R and extend the line to intersect the line DZ at point Q.

Step 5: Repeat for all points in the image.

Investigation

Investigate the mathematics of the simpler construction. Why is the angle ECF, the angle of the cone? Why is the point Z the farthest point that can be drawn?

Making your own anamorphic images from the grids

To make your own anamorphic images for a conical mirror, go to the download page and print out the images from your computer. You will also find instructions on how to make a mirror.

Ready made images

This post also has some examples of images to view including the Mathsyear 2000  logo. You will need to make a mirror, which is also described there.

An unusual anamorphosis

The following anamorphosis is a simulation of an eighteenth century image which appears in similar forms in many countries. It illustrates that the image is very difficult to interpret without the mirror.

By adding a conical mirror to the circle, the image in the mirror turns into clover leaf:


The person who originally thought of this deserves full marks for ingenuity.

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. 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.

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 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:

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 5
Using the labels on the two grids, identify the cell to colour on the circular grid and create the anamorphic version.

Step 6
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.
Step 1

Load your image:

Step 2

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.

Step 3
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.

Restoring Images on the Computer

This post describes some methods for using the computer to restore anamorphic images. They are not always perfect restorations, but can sometimes reveal the images in ways not easily seen by conventional optical means. In some cases gross errors can also occur if the wrong method is used.

Plane anamorphoses
There are methods using ray tracing programs like POV-Ray which allow you to look at the image from the correct point, but they are too complicated to describe here.

Many people, think that you can just squash the image, that is the reverse of stretching it, but to do so gives wildly incorrect results.


The picture above shows the effect of making the outer ellipse in the Scrots portrait of Edward VI circular. Although the face is recognisable, the text on the left (showing his age is 9) is nowhere near the shape of the text on the right (it is so small it appears blurred in this reproduction). Compare this with the correct restoration in the post above about plane anamorphoses.

Cylindrical Mirror anamorphoses

Most cylindrical mirror anamorphoses are approximated by a mapping from a square grid to a set of circles. This makes it easy to reverse the process with a computer. The result is not always perfect for two reasons. The anamorphosis may have been created using a more precise method than a simple set of circles. It may also have been created by eye and may not be perfect. However, the results are close enough to show the original image almost perfectly, and often to gain a better view than looking in the mirror.

Software to use

Adobe Photoshop has a filter for Polar Coordinates in the Distort options. The Gimp (which is a free program) has a similar filter.


The Wheelbarrow man example is from The Magic Mirror © Dover Books 1979 showing ” A fat man wheeling his stomach in a wheelbarrow”.  The restored image looks like this:

The image below  is an eighteenth century anamorphosis musing on the transitory nature of life mirrored as an illusion. The restored version is at the right.
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An anamorphic trick

The following trick shows how you can distort an image on a surface to create an anamorphosis.
It works best on any common currency rather than a photograph since the paper is ideal.

1. Take a note and find the face.

2. Make two strong, creased, folds though the corners of the mouth and the eyes like this.

3. Push the two folds together to make the gap between the two folds a concave curved surface.

4. Tilt the face so that you are looking from the chin up and see how happy you can make it.

5. Tilt it the other way, so that you are looking down the face. Misery now appears out of happiness.

The anamorphic effect is produced by the way in which you see the mouth and eyes distorted by the curvature.

Quick and Dirty Downloads

In order to use these templates, you will need a copy of the Adobe Acrobat reader.  You can download a copy from the Adobe Website at http://www.adobe.com.

These images use the simple non-perspective methods of creating anamorphosis.

Faces

A simple stretch download

This is a trio of prominent Victorians produced in a magazine at the end of the 19th century.
Can you recognise any of them?
The answer is given below.

Click here to download the file.

Some puzzle stretches

Can you read the messages in this one?

Click here to download the file.

A Vignola type of anamorphosis


When you print out this image, you will see a set of lines on each side. Fold across the page, alternately making a mountain fold and then a valley fold, so you have a zig-zag. Then look in each direction to bring the image back.

Click here to download the file.

Answer to faces
Oscar Wilde, William Ewart Gladstone and Alfred Lord Tennyson.

Cone Grids and Images to Download

In order to use these templates, you will need a copy of the Adobe Acrobat reader. You can download a copy from the Adobe Website at http://www.adobe.com.

Instructions

These images are all based on putting images onto cones. There are also some grids for you to create your own anamorphoses.

When you have printed the sectors, cut them out and put some glue on the tab. Do not bend the tab as this distorts the cone. Making a cone with the image inside is harder when the tab is inside since the tab gets in the way at the tip of the cone. You may find it easier to trim away some of the tab towards the tip of the cone. Because the cone is harder to make, most images are for viewing inside the cone.

To use the grids, fill in squares on the normal grid, and then copy them to the anamorphic sector. Then make up the cone. You may find it easier to find your way around if you make a grid without colouring in first.
Faces

The MathYear2000 logo inside a cone

Click here to download the file.
Socrates inside a cone

This is the image of Socrates from the Niceron plane anamorphosis example to be viewed from inside a cone.

Click here to download the file.

A grid for viewing inside a cone

This PDF page has a square grid plus a sector for viewing or creating your own anamorphosis.

Click here to download the file.

A grid for viewing outside a cone

This PDF page has a square grid plus a sector for viewing or creating your own cone anamorphosis.

Click here to download the file.

Cylinder Images to Download

In order to use this template, you will need a copy of the Adobe Acrobat reader. You can download a copy from the Adobe Website at www.adobe.com.

Instructions
The downloadable PDF contains three images which you will need to print out onto a sheet of A4.

Cut out the images and place them around a soft drinks can, securing with some glue or a piece of sticky tape. Look at the image straight on. Note how the curved lines become straight.

The first two images are an illusion within an illusion!

Click  here to download the file.