Conic Sections: A First Lesson

We begin by asking the question what is a conic section? Moreover what is a conic? A conic is simply two cones merged together at their tips to create a double cone shape. Imagine two ice cream cones touching at the bottom and facing away from each other. Here’s a picture to illustrate. As you can see the conic is really a simple shape. The radius of a section of the cone is simply equal to the length of the z-axis. In this case our equation is given by .  If you are not familiar with the z-axis then here’s a quick run down…The z axis is just the height of an object in the Euclidean Plane. Imagine if you placed a Rubik’s cube on a sheet of paper, obviously you can trace the bottom of the cube and give it x and y coordinates if you wanted to. But these traces would not be the cube or even an accurate description of what the cube looked like. You would need to introduce a new element or the z axis to finish off the description of the cube. So as I said before the z axis is just the height of an object or the measurement of the third dimension. With that said now we can turn our attention to what happens when we cut the cone with a plane, or in other words how can we divide the cone into two different pieces?

The first section is the easiest to describe because it cuts the cone parallel to the Euclidean Plane to create a circle.  As you can see the plane which is just the z plane cuts the cone to give a circle where the cone and the plane meet. The equation to create the plane is given by n = z, where n can be any real number. Remember the cone can go on expanding until infinity so it theory we can place a given plane with any real value (positive or negative) to slice the cone. Now we move onto the next section.

If we have a circle when the z plane is parallel to the Euclidean plane then what happens when we tilt the z plane slightly? The answer is we get a new conic called an ellipse. The ellipse is a fascinating part of many of our everyday lives and there is even another lesson devoted solely to the ellipse. Here are the pictures of the slice creating an ellipse.  Notice that the plane has a different slope than the conic. In general the conic goes on forever so in general as long as the plane has a slope of less than the conic and not the same we will get an ellipse. It turns out that if the slope is the same or more than the conic then we will get another section.

In fact if the slope of the plane is more than the conic (i.e. if the plane slices the conic on both ends) we have what is called a parabola. In this particular section the slope of the plane is 2, which is double the slope of the conic. Here are the pictures.  The last section exists when the slope of the plane is the same as the slope of the conic which is called a parabola. The parabola has the same shape as a hyperbola except with only one tail. Here are the pictures to demonstrate.  All of these different sections have fascinating properties, but for this two week exposition we will only focus on the wildcard of the bunch: the ellipse. Since we have an idea of what a conic section is and we’ve been exposed to Algebra II and Geometry we can explore the various intricate parts of the ellipse. So let’s go back to the homepage and start the lessons on the ellipse, and hopefully this will help us examine the other sections in the future as a class or by ourselves.