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A different conical mirror anamorphosis

I discovered the following conical mirror anamorphosis in the Museum of Childhood in London.

insideconical

Although it is called a Perspectograph, there is no element of perspective in it.

It is unusual in that I know of no other version of the conical mirror anamorphosis which uses a strip of paper with a conical mirror. All the ones I have seen are a flat sheet on which the image lies.

This raises some interesting questions on the angle of the mirror used for conical mirror anamorphoses.

The following drawing is a typical example tracing the rays of reflection from the eyepoint N. The image KPQRL is what you see when the points 4321L on the anamorphic image are reflected. Note that the cone tip has an angle of approximately 60 degrees.

con_mir01The angle of the cone is important since as the cone tip angle decreases the point 4 (the image of the centre point K) moves farther away. The following diagram shows how the rays are deviated for an angle of 80, 90 and 120 degrees at the tip of the cone. Even at an angle of 80 degrees the plane image would be quite a long way away. For 90 degrees the ray corresponding to the centre of the base of the cone would be horizontal and so the plane anamorphic image would be infinite. If the tip angle is greater than 90 degrees, then the reflected rays would not not intersect the plane of the base of the cone.

conical_raysNow if the image is placed on the inside of a cylinder around the base of the cone, the reflected rays will intersect the cylinder and the angle of the cone does not cause a problem.

conical_cyls

How to create images for this anamorphosis

For a cone with the tip angle of 90 degrees, the distance of a point on the base of the cone to the edge of the cone is the same as the height of the reflected point on the cylinder as follows:

equal_reflect

There is no need to write a software program to perform the transformation of an image. The free image software The GIMP has a distort filter which will do this for you.

I have used square images, as in the following example.

portrait

Step 1. Open the image in The GIMP and choose the Filters menu then –> Distorts -> Polar Coordinates

Untick the box under the preview which says Polar coordinates. Leave the box Map from top tickedThis converts a circular image (polar coordinates) to one with cartesian mapping, so that it appears as a strip. The Map Backwards box should be ticked to mirror the image so that it appears correctly when viewed. The preview then shows this effect.

dist_p_coords

You can have some great fun with the grotesque results it produces.

This strip is the wrong shape for the anamorphosis.

Step 2. Go to the Image menu and choose Scale Image

scale

In this dialogue box, first uncheck the link (see arrow above) which makes the scaling have the same aspect ratio.

Since the cylinder needs to wrap around the base of the conical mirror and have the same height as the radius of the cone. So the strip must be 2π times as wide as its height. Use a calculator to determine the pixels you need.

When you accept and generate the image, it looks like this:

portrait_strip

This strip can then be printed and made into a cylinder and placed around the edge of a conical mirror with a 90 degree tip angle, that is to say with a radius equal to its height.

The following are images created using computer ray tracing with POVray. First a side view (POVray maps the image both inside and outside the cylinder) and then the view from above.

ConeMir_Cyl_portrait 45 side

POVray maps the image so that it is both inside and outside the cylinder. Looking from above into the mirror, the image is restored

ConeMir_Cyl_portrait 45

When I saw the following for the first time, I was astounded by how much a difference viewing in the mirror made and how good it was as a transformation.

The POVray code should be obvious to anyone used to coding this type of file. The scaled strip is mapped onto a cylinder and a conical mirror created inside. If the comments aren’t enough, use POVray’s help. This is a PDF which you can select the text and paste into POVray.

povray code cyl mir

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