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Stretchy pictures

By David, 26 September 2013 Activity

Stretch, squeeze, and bend! Try this maths trick to transform an image.

You will need

paper, ruler, pens and pencils, eraser.

What you will need.

  • Paper
  • Pencil
  • Red pen
  • Blue or black pen
  • Ruler

What to do

  1. Draw a square in red pen with 12 cm sides.

    Every 2 cm across your square, draw a red line going up and down the whole height of the square.

    Every 2 cm up your square, draw a red line going across the whole width of the square. You should end up with a square made up of little squares, 6 across and 6 up and down.

    A 6 by 6 grid of squares, and a 6 by 6 collection of long skinny boxes that are twider at  the top and bottom.

    Draw a square grid and a stretched one.

  2. In lead pencil, draw a wonky grid that’s 6 boxes by 6 boxes. You can stretch it, squeeze it, make it bend in curves, or lean over! Just remember, it’s got to end up being a grid of 6 boxes up and down, and 6 boxes left to right. Download an example here.
  3. Draw a picture in the big red square. It can be of anything you want, but it’s easiest if it’s simple. Try to put at least a bit in most of the little squares. If you can’t think what to draw, try drawing a house with a chimney.
  4. Look at the top left box in the red square. Draw what’s in that box inside the top left box of the wonky pencil grid. If the box has been stretched, remember to stretch everything inside the box the same way. Try doing it in pencil first, and when you’re happy, you can go over it with a pen.
    A picture of a house in the 6 by 6 square grid. The same picture is being drawn in the stretched grid - only half the boxes have been filled.

    Copy the picture across, square by square.

  5. Now copy the next red box into the corresponding wonky box. Keep copying the boxes until you have a finished drawing.
  6. When you’ve finished drawing, you can erase the pencil boxes so no-one will know how you made the picture!

What’s happening?

The procedure you are performing is called a geometric transformation (or transform). This is a very useful mathematical operation.

A picture of a house, and a stretched picture of a hous that's long and thin, and wider at the top and bottom.

The finished drawing.

A transformation changes shapes. There are many different types of transformations, including reflections, rotations and making a picture bigger or smaller (scaling). Transformations change shapes into different shapes. But transformations don’t completely change the shapes you start with. Transformations don’t remove lines, and they don’t create lines. Every line in the first picture is in the final result. Because of this, every transformation is reversible. There is another transformation that will take your stretched picture and return it to normal.

Transformations don’t have to go from a flat image to a flat image. A map of the world is a transformation of the surface of Earth. It is a lot smaller, and it is stretched so that it can be drawn on a piece of paper, but every country in the world will be on your map – none are created, and none are destroyed.

A picture of a painting, and a second picture of a painting, stretched tall on the left side, and short o nthe right side. It looks like it is hanging on a wall.

This process can be used to make things look 3D.

Applications

Transformations are very useful for security cameras. If a camera takes a photo of a number plate, it will often be at a strange angle. This means the image of the number plate will be stretched and very hard to read. If you use the right transformation, you can stretch the number plate back into a rectangle, and read the letters on it.

Transformations are also used in computer graphics. In order to make something look like it’s a 3D object, it needs to change shape when you move around it. To do this, programmers make objects out of triangles called polygons. They then draw a picture onto the surface of the polygon, so it looks like a tree, or a house, or whatever they want. In order for it to look 3D, the computer must stretch the polygon when you move around it. The computer uses a transformation called ‘the perspective projection’ to stretch the polygons. New computers can calculate millions of these transformations every second to create realistic animated movies or computer games.

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