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.
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: