Imagine an image on the base of a right circular cone. We want to transfer the image onto the slanted face of the cone so that when we look from the correct point, the image appears as it was originally drawn. The two cases of the image being on the inside and outside surface give the same result when viewed, but the images look different when drawn.
These are images painted on cones by John Sharp.
This is what you see if you look at the first cone painting from the side.
When you look directly above the tip of the cone, you see the Yin-Yang symbol
These two pictures show what you see if you look at the second cone painting from the side.
When you look directly above the tip of the cone, you see the eye of Horus.
Now, it is difficult to draw onto a cone, but we can make a cone from a sector of a circle by rolling it up. The problem is now to draw on this sector. Although this sounds complicated, geometrically, this is the easiest method for making anamorphic images. In the following descriptions, all references to cones are to right circular cones, that is cones formed from a circle with their apex a point perpendicularly above the centre. To paint on these wooden cones, the image was created on a sector and this was then traced onto the wood using carbon paper.
Making a cone
If a cone is sliced in half perpendicularly to the base, the cross section is a triangle. Suppose we start with a sector of a circle whose angle is a fraction n of 2. Suppose it has a radius R and that the radius of the base of the cone is r. The arc length of the sector 2R is equal to the circumference of the circle at the base of the cone (2r), so r = nR. In the examples shown here the sector angle is 120°, so n is 1/3. Knowing these two radii, it is easy to construct the triangular cross section.
In the following description the image is going to be placed on the inside surface of the cone.
Step 1 – Draw a circle around the picture or radius r and decide on an angle for the sector (e.g. 120° as here to give a factor of 1/3 for n), calculate the radius of the sector (R = r/n) and then construct the triangular cross section of the cone (CB = R and DB = r). Place the drawing inside the circle.
Step 2 – Decide on where the eye will be positioned along the axis of the cone at point E.
Step 3 – Choose a point P on the drawing. This point will become point Q on the sector which forms the cone. Imagine that the triangular cross-section is rotated until it lies along line DP. Then point P is on the radius DB as shown on the right. Join E and P and extend the line to intersect the side BC at point Q. We now know how far point Q is from the centre of the sector in the figure below.
Step 4 – To find the position of Q, on the sector below, draw an arc of a circle radius a (equal; to CQ above) and then measure the angle q in the base circle. Draw an angle which is /3 at the centre of the sector and intersect the arc to find the position of Q. It has to be a factor of 1/3 since a 360° rotation has to be compressed into the 120° sector.
Step 5 – Repeat for all points in the image.
Using a grid to speed up drawing
To transfer the picture on the base of the cone point by point to the sector is quite time consuming, but the steps are easy to follow. An easier way is the grid method the artist in Dürer’s woodcut is using, so a grid has been provided on the downloads post. The result is as follows:
The grid is interesting in its own right. Print the grid sector on the downloads page and make it into a cone. When you make it do not bend the tab, just put some glue on it and fit the edge of the tab to the other edge of the sector. This grid is for an image which you see at by looking into the cone.
There is another image on the downloads post which shows a grid meant to be looked at on the outside of the cone like the following picture.
Look along the axis of the cone and you will see a square grid. As you move your eye along the axis of the cone, there will be one place (about three times the height of the cone from the base) where the grid is perfect.
Note that all the lines are curved on the sector. Each curved line corresponds to a line on the grid which was the starting point on the base of the cone. Each line of the base grid defines a plane together with the eye position. This plane cuts the cone in a curve called a conic section. There are three types of conic section: ellipses, hyperbolae and parabolae. Which one you get when you cut a cone depends on the position of the plane. In these cases they are hyperbolae. Corresponding points on centres of the sides in the “outside” version are much closer to the centre.
Making your own anamorphic images from the grids
To make your own anamorphic images on the cones go to the downloads post and print out the images from your computer. You will find:
* A normal 11 by 11 grid to use as a master to create your drawings.
* An “outside” sector grid where you need to make sure the image appears on the outside of the cone.
* An “inside” sector grid where you need to make sure the image appears on the inside of the cone.
* These are available in PDF format.
Use copies of the normal grid and colour them in to make pictures like the example and then transfer your picture to a copy of the sector and make it into a cone to create your own anamorphic artwork.
Ready made images
The downloads page also has some cone sectors to cut out and view including the Maths Year 2000 logo.