John Sharp shows how geometry is used to create anamorphic art
Artists, like scientists and mathematicians, are always looking for new ways to explore the world and they do not always do so in conventional ways. Some artists have moved away from flat rectangular pictures that you have to view straight on. You must look at these "unconventional" pictures in a special way or from a particular angle; you cannot see what the artist intended unless you do so because the pictures will be distorted. The artist uses simple geometry that follows the rules of perspective, but in a non-standard way. This is called anamorphic art. The word comes from the Greek ana meaning back and morphe meaning form. The picture is only "back" in its correct form if you look at it from a particular viewpoint.
Anamorphic art shows how what you see is dictated by the point you are looking from. If you stand in front of a picture to look at it, you see it differently from how you would if you stood to one side of it. With the anamorphic portrait of Edward VI painted by William Scrots, which hangs in the National Portrait Gallery in London (see above) , you have to look at it from the right side in order to see Edward VI properly. A little gap in the frame is provided so that you can do so.
Painting a standard perspective picture is best thought of as looking through a window. In the woodcut by Albrecht Duerer (see far right) the artist paints directly on a sheet of glass through which he views his subject. Notice that the artist's eye is in front of a vertical pointer. He uses the pointer to ensure he always looks from the same place and so sees the objects he is drawing in the same position. You can see why he needs to do this with the following experiment. Close your left eye, hold up a finger in front of your face about a foot away and note what the finger points to. Open your left eye, close your right eye and you will see that the finger points to a different place. Providing the artist always looks through the glass from the tip of the pointer, he can also tell whether his subject has fidgeted and moved position!
The flat sheet of glass could be replaced by a curved one. If the artist still looked from the pointer he could not tell whether the painting was on flat or curved glass. But if he moved away from the pointer he would see a distorted image and it would be obvious that it was not on a flat surface. This is then an anamorphic painting because he can only see the painting correctly if it is viewed from the correct position.
Sometimes an artist makes a grid on the glass and then has a corresponding grid on a piece of paper. The grid on the glass helps him to draw on the paper exactly what he can see through the glass. He still has to use his fixed pointer, and also make sure he keeps his eye on the corresponding point on his paper grid. Otherwise he would lose his place on the paper and his drawing would go wrong.
A picture in a cone
There are many different types of anamorphic pictures. As we have seen, the picture may not be on a flat surface. Imagine you pasted a conventional picture on to the inside surface of a cone. When you looked into the cone the picture would appear very distorted, because natural perspective would be changing the position of points of the picture in relation to your eyes. To see the picture looking normal on the inside of the cone you need to apply a little geometry.
The following activity explores this type of anamorphic art where the picture is on the inside surface of a cone. It provides an interesting link between art and maths, involving the geometry of the cone.
Imagine a picture of an apple on the base of a right circular cone (a cone with a circle as base and its point vertically above the centre). We want to transfer the picture on to the slanted internal face of the cone, so that when we look into the cone, the picture appears as it was originally drawn. It is not easy to draw inside a cone. But we can take a sector of a circle, which we draw on first and then roll up to make the cone. The problem is how do we draw on this sector?
An easy way to do it is to use a pair of ready-made grids like the photocopiable ones on this page. Then follow the instructions.
To transfer the picture on the base of the cone to the sector without the readymade grids would be quite time consuming, but for older pupils the geometry involved can be followed in the easy steps shown in the box at the top right of this spread.
The activity can be used as a starting point for other ideas. Primary pupils can think about how objects are related and make visual links. This is a short link to maps and the simple use of co-ordinates. The way you transfer an image point by point is a starting point for many aspects of geometry all the way through secondary level and opens the way to a university course in non-Euclidean geometry. Questions such as "what curve does the line become on the cone?" can make a lesson on conic sections much more interesting.
PUT YOUR PICTURE INTO A CONE
Make a photocopy of the blank 11 x 11 squared grid and two copies of the sector with its distorted "cone grid" (which is also 11 by 11 but has been altered using a computer). Cut out both the cone grids.
Make a cone from one of your cone grids. Roll it up to form the cone with the grid on the inside. Just put glue on the tab and stick it to the outside of the cone. Do not fold the tab or it will distort the cone.
Hold the cone and close one eye. Look into the cone, moving it towards or away from your eye until the grid looks like a regular square.
Draw a picture on the square grid by filling in the squares. (Follow our example of a diagonal first to see how to do it.) Translate the squares one by one to the cone grid. Start with a square close to the edge or corner. We have put grid references on our example to make this easier. In our example start with square K1. Then fill in J2. You can find the locations on the cone grid by following the grid reference numbers and letters. Alternatively, you can trace your picture with your finger and work out one by one how each square sits in relation to its neighbours. For example, K1 sits corner to corner with J2. Move along the diagonal. When you come to F6 fill in the odd shape at the tip of the sector.
Then fill in G6 and G7 and the other squares around the centre one. Note that F7 is split down the middle so that you get half squares down the edges of the sectors. So you will need to fill in both halves.
Now move along the rest of the diagonal to end up at A11. Look and compare it with where A11 is on the original flat grid.
Roll up the cone grid to form the cone with the picture on the inside. Just put glue on the tab and stick it to the outside of the cone. Do not fold the tab or it will distort the cone.
Hold the cone and look into it with one eye, moving it towards or away from your eye until the grid becomes a square and you see the picture just as you drew it on the original flat grid.
HOW THE GEOMETRY WORKS
How the base radius and the sector radius relate If the 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 29. Suppose the sector circle has a radius R and that the radius of the base of the cone is r. The arc length of the sector 29nR is equal to the circumference of the circle at the base of the cone (29r), so r = nR. In the examples shown here the sector angle is 120x, so n is 13. Knowing these two radii, it is easy to construct the triangular cross section.
The following steps show how to convert a point on the picture to a point on the cone. To draw a complete picture you would have to do this for every point. If you drew your original picture on a squared grid you could choose a useful set of points on that grid to start building up your picture on the sector.
Draw a circle around the picture of radius r and decide on an angle for the sector (eg, 120o as here to give a factor of 13 for n), calculate the radius of the sector (R = rn) and then construct the triangular cross section of the cone (CA = R and DA = r). Place the drawing inside the circle.
Decide on where the eye will be positioned along the axis of the cone at point E.
Choose your first point on the drawing. In our diagram point P is the only one shown, but you will be choosing many more to complete the drawing. Point P will become point Q on the sector which forms the cone.
Imagine that the triangular cross-section is rotated until it lies along the line DP. Then point P is on the radius DB as shown on the diagram.
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.
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 in the base circle.
Draw an angle which is 13 at the centre of the sector and intersect the arc to find the position of Q. It has to be a factor of 13 since a 360o rotation has to be compresed into the 120o sector.
Repeat for all points in the picture.
WHERE TO FIND OUT MORE
What you would see if these were on the inside of a cone?
There is more on anamorphic art at www.anamorphic.com and on the Maths Year 2000 website at: www.mathsyear2000.org An exhibition is planned at the City Library and Arts Centre in Sunderland from October 22 to December 19, tel: 0191 514 1235.
John Sharp teaches art and geometry at the City Lit in London