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Parabolas

### Parabolas (old draft)

Let's pause for a moment and take a look at this shape we created using our string art tool.

Does this shape look familiar? What does it remind you of?

A bouncing ball captured with a stroboscopic flash at 25 images per second. Image: MichaelMaggs/Richard Bart

Water fountains project tiny droplets of water into the air. Image: Ali MoorePath of a motorcycle jumper. Image: Mr.Reid

When an projectile is launched it follows a path which is similar to the shape of our blade of grass.

These projectiles are following a path which was approximated by our blade of grassThese types of paths fascinated Galileo who was looking for a way to describe the paths of all projectiles. He was the first to show that there was an underlying pattern in all projectile paths. Here is an actual page from Galileo’s notebook where he is tracked the path of a ball thrown at different speeds.

Page from Galileo's notebook showing the path followed by a ball thrown at different speeds.When you plot the height of the ball over time you end up with a shape known to the Greeks as a parabola. A parabola is a mathematical description of this type of path. To warm up, we can think of a parabola as follows. Imagine we have an imaginary point (called a focus) and a horizontal line (called a directrix). A parabola is precisely the set of points which are equal distance from both the focus and the directrix. Using the interactive canvas below try and move your cursor so that the two circles line up -- that point is the same distance from both the vertex and the directrix.

When you hit a point which is equal distance from the vertex and directrix a green dot is placed on the canvas. If you continue doing this you will find these green dots trace out the path of a parabola. Here we have represented the shape of our grass using an abstract definition involving invisible points and lines. Earlier, we were using three points to define a curved blade of grass. Now we see that we can also use a single point and a line to control the same curve.

In the interactive canvas below you can change the distance between the vertex and directrix. Notice this allows you to define any type of parabola, both thick and thin, using a single control!

In the next tutorial we will dive deeper into the mathematics of parabolas.