Parabola

Water fountains, as trajectories, form into a parabola. A trajectory subject to gravity will have a parabolic path because its downward speed increases with the square of time (acceleration) but moves horizontally at a nearly constant rate.

The beauty of the shape, has inspired architects, resulting in graceful and stable structures.

A parabola happens also to be the set of all points that are the same distance from a point (focus) and a line (directrix). Two good ways to get a feel for this property is to fold paper (you get the materials and instructions for folding the parabola in your Discovery Box subscription) and to play around with some toy models.

We can think of a parabola as a set of points moving in a straight line until they are equally distant from the focus and directrix.

When the directrix is at y = – 0.25, and the focus is at y = + 0.25, then y = x squared.

Click here for a toy model in which each point is like a little robot moving around at random until it is the same distance from the directrix and the focus. A parabola results.

Here’s our parabola model in Scratch. Here, we use an apple as the focus, balls as points, and bowties as lined up on the directrix. So the parabola is the set of balls (points) that are equally distant from a bow tie and an apple. So the balls should move until their y coordinates is the square of the x coordinate. But all the balls in this program do is follow their very simple rule: move until they are just as far from the focus as from the directrix. You will see a ball end up at x = 1, y = 1, another ball at x = 2, y = 4, and so on.

In your Discovery Box, you have the instructions and paper for folding to find the points on the parabola. Once you try the folding, and play with the models, you may want to learn more about the technical terms, and the mathematics.

Another way to get a parabola is to slice a cone at an angle.

Similarly, you get a hyperbola if you slice the cone straight down, parallel to cone’s center axis. You get an interesting 3D version of the hyperbola if you rotate it in a circle. You can make this out of straight sticks. That makes it a “ruled surface.”

This flexible model as a beautiful simplicity. So, it is easy to model this hyperboloid in Scratch.