Mathematical Curves

Mathematics is more than numbers and abstract ideas; it is a real and interesting part of our lives. In engineering and science, mathematics is part of any explanation of the way things work. We will begin our exploration of mathematical curves by looking at a group of four curves called the conics: the circle, the ellipse, the parabola, and the hyperbola.

The Conics

We see cones around us every day: ice cream cones, traffic cones, and cone-shaped sushi. There are four mathematical curves that can be found in a cone, and these curves are called The Conics, or The Conic Sections. The four conic sections, circle, ellipse, parabola, hyperbola can be seen as slices of a cone. If the cone is sliced parallel to the base, the resulting curve (in red, below) is a circle.

If the cone is sliced on a slight angle, the curve is called an ellipse, as shown below. The orbits of the planets are elliptical and the earth itself is an ellipsoid. A circle viewed from an angle looks like an ellipse.

If the slice is made parallel to the edge of the cone, the curve formed is called a parabola, an open curve whose sides do not meet, as shown above. The parabola is the path followed by a thrown ball or by a spout of water in a fountain. Upside down parabolas are seen in some suspension bridges.

If the slice is perpendicular to the base of the cone, the curve is one of two branches of a hyperbola, as shown above. Mathematicians are interested in two branches of the hyperbola, formed by putting two cones together. The pattern of light cast on a wall by a lampshade is a hyperbola.

There are many websites on the internet with fascinating information about the conics. One of the best websites can be seen by clicking on the link below:

http://britton.disted.camosun.bc.ca/jbconics.htm

You will find many references to this wonderful website in the following pages.

Using a flashlight you can see all four of the conics. The beam of light coming from the flashlight is a cone. Point this cone straight at the wall, and you will see a circle; then tilt the cone of light, our circle becomes an ellipse as the angle changes. As the angle changes more, we see a parabola, and last, one branch of a hyperbola.

Four curves that you've often seen. Watch for them... you'll find them everywhere.

Now let's explore these four curves in greater detail. Click on the link below to begin our exploration with the circle.

Go to The Circle