- Consider the results when two simple mathematical objects are intersected.
- Be comfortable working with an infinite two-sided cone.
- Know the basic types of figures that result from intersecting a plane and a cone.
- Know some of the history of the study of conic sections.
Introduction: Intersections of Figures
Some of the best mathematical shapes come from intersecting two other important shapes. Two spheres intersect to form a circle:
Two planes intersect to form a line:
While simple and beautiful, for these two examples of intersections there isn't much else to investigate. For the spheres, no matter how we put them together, their intersection is always either nothing, a point (when they just touch), a circle of various sizes, or a sphere if they happen to be exactly the same size and coincide. Once we've exhausted this list, the inquiry is over. Planes are even simpler: the intersection of two distinct planes is either nothing (if the planes are parallel) or a line.
But some intersections yield more complex results. For instance a plane can intersect with a cube in numerous ways. Below a plane intersects a cube to form an equilateral triangle.
Here a plane intersects a cube and forms a regular hexagon.
- Describe all the types of shapes that can be produced by the intersection of a plane and a cube.
- What is the side-length of the regular hexagon that is produced in the above diagram when the cube has side-length 1?
Intersections with Cones
One class of intersections is of particular interest: The intersection of a plane and a cone. These intersections are called conic sections and the first person known to have studied them extensively is the Ancient Greek mathematician Menaechmus in the 3rd Century B. C. E. Part of his interest in the conic sections came from his work on a classic Greek problem called “doubling the cube,” and we will describe this problem and Menaechmus’ approach that uses conic sections in section three. The intersections of the cone and the plane are so rich that the resulting shapes have continued to be of interest and generate new ideas from Menaechmus' time until the present.
Before we really delve into what we mean by a plane and a cone, we can look at an intuitive example. Suppose by a cone we just mean an ice-cream cone. And by a plane we mean a piece of paper. Well, if you sliced through an upright ice-cream cone with a horizontal piece of paper you would find that the two objects intersect at a circle.
That's very nice. But unlike the intersection of two spheres, which also resulted in a circle, that's not all we get. If we tilt the paper (or the ice cream cone) things start to get tricky.
First things first, we better make sure we know what we mean by “plane” and “cone”. Let's use the simplest definitions possible. So by “plane”, we mean the infinitely thin flat geometric object that extends forever in all directions. Even though infinity is a tricky concept, this plane is in some sense simpler than one that ends arbitrarily. There is no boundary to think about with the infinite plane. And what do we mean by a cone? An ice-cream cone is a good start. In fact, it's very similar to how the ancient Greeks defined a cone, as a right triangle rotated about one if its legs.
Like we do with the infinite geometric plane, we want to idealize this object a bit too. As with the plane, to avoid having to deal with a boundary, let’s suppose it continues infinitely in the direction of its open end.
But the Greek mathematician Apollonius noticed that it helps even more to have it go to infinity in the other direction. This way, a cone can be thought of as an infinite collection of lines, and since geometric lines go on forever in both directions, a cone also extends to infinity in both directions. Here is a picture of what we will call a cone in this chapter (remember it extends to infinity in both directions).
A cone can be formally defined as a three-dimensional collection of lines, all forming an equal angle with a central line or axis. In the above picture, the central line is vertical.
- What other mathematical objects can be generated by a collection of lines?
Intersections of an Infinite Cone and Plane
Like with the ice cream cone and the piece of paper, some intersections of this infinite cone with an infinite plane yield a circle.
Other intersections yield something a bit less tidy. In the picture below, the plane is parallel to the axis of the cone.
But it turns out that the set of intersections of a cone and a plane forms a beautiful, mathematically consistent, set of shapes that have interesting properties. So in this chapter we embark on a study of these intersections called conic sections.
Note: Although we will not prove it here, it doesn’t matter if the cone is asymmetrical or tilted to one side (also called “oblique”). If you include “tilted cones” the same conic sections result. So, for simplicity’s sake let’s stick with non-tilted, or “right” cones and focus on what happens
Let's begin by doing a rough tally of the kinds of shapes it seems we can generate by slicing planes through our cone. First of all, we have the circle, as we discovered with the ice cream cone above. A circle is formed when the plane is perpendicular to the line in the middle of the cone.
If we tilt the plane a little, but not so much that it intersects both cones, we get something more oval shaped. This is called an ellipse. Later you will learn many of the fascinating properties of the ellipse.
If really tilt the plane more so that it only intersects one side of the cone, but we get a big infinite “U”.
This shape is called a parabola and like the ellipse it has a number of surprising properties.
Then if we tilt it even further, we intersect both sides of the cone and get two big “U's” going in opposite directions.
This pair of objects is called a hyperbola, and, like the parabola and ellipse, the hyperbola has a number of interesting properties that we will discuss.
Finally, in a few cases we get what are called “degenerate” conics. For instance, if we line up a vertical plane with the vertex of the cone, we get two crossing lines.
- Are there any other types of intersections between a plane and a cone besides the ones illustrated above?
- Is it possible for a plane to miss a cone entirely?
Applications and Importance
The intersections of cones and planes produce an interesting set of shapes, which we will study in the upcoming sections. We will also see numerous applications of these shapes to the physical world. Why is it that the intersection of a cone and a plane would produce so many applications? We don’t seem to see a lot of cones or planes in our daily life. But at closer inspection they are everywhere. Point sources of light, approximated by such sources as a flashlight or the sun, shine in cone-shaped array, so for instance the image of a flashlight against a slanted wall is an ellipse.
The geometric properties of conics, such as the focal property of the ellipse, turn out to have many physical ramifications, such as the design of telescopes. And, when we inspect the algebraic representations of conic sections, we will see that there are similarities with the law of gravity, which in turn has ramifications for planetary motion.
In summary, here are some of the ways that a plane can intersect a cone.
an “oval” called the ellipse
A big infinite “U” called a parabola.
Two big, infinite “U”s called a hyperbola.
Strange “degenerate” shapes like two crossing lines, as well as other examples you may have found in Review Question 1.
This array of shapes has a surprising amount of mathematical coherence, as well as a large number of interesting properties.
A three-dimensional collection of lines, all forming an equal angle with a central line or axis.
The points of intersection between a cone and a plane.