Back to Analytic Geometry
Analytic Geometry:
AG 008 Conic Sections:
By William E. Steinman:
Recall back in Study Notes 24 when we defined a cone thus:
A cone is a solid formed by joining with straight-line segments the points of a closed curve to a point outside the plane of the curve.
The point becomes the vertex.
The joining lines are called the elements.
Here is a cone.
Imagine this to be a transparent solid figure.
This particular cone is a right circular cone. A right circular cone is generated by rotating a right triangle around one of its legs as the axis.
Notice that a cone need not be a right circular cone. The base could be any closed curve and the point could be anywhere outside the plane of the curve.
Okay, that’s enough review. Now we can consider geometric forms, which we call conic sections.
A conic section is the intersection of a right circular cone and a plane, which generates one of a group of plane curves, including the circle, ellipse, hyperbola, and parabola. To be sure, a circle is just a special case of the ellipse.
To get a complete mental picture of these figures it is necessary to imagine two right circular cones placed tip to tip. See the drawing below.
The circle is produced when the plane that intersects the cone is perpendicular to the axis of the cone. The ellipse is produced when the plane that cuts through the cone is not at right angles to the axis of the cone.
The ellipse is quite important in astronomy. Many years ago, scientist discovered that celestial bodies like the planets follow elliptical orbits as they make their way around the sun. So, the mathematics that describes the ellipse is necessary to understand and predict the paths of these bodies.
A parabola is produced if the plane is parallel to a straight line drawn on the slanting surface of the cone from the tip of the cone to its base. In other words, if the plane is parallel to a straight line segment. The parabola is very important in science. Two areas of application are in astronomy and ballistics.
In astronomy the parabola is used in the building of telescopes. Think parabolic reflectors and Hubble space telescope. The parabola also describes the path followed by a celestial body if approaches and circles another heavier body without being captured.
In ballistics, the parabola very accurately describes the ideal path of a bullet, a mortar shell, and even an arrow shot from a bow. So, it is imperative to understand the mathematics that describe the parabolic shape. Our military folks use that math to know where a shell will land when it is fired from a cannon. When our president, Harry Truman was a artillery officer in WWI he understood and used this math to good effect. It’s been a while since we have had a president that smart.
A hyperbola is produced when the plane passes through both ends of the cone. Hence, a hyperbola has two distinct elements. When plotted, these elements do not touch and they face in opposite directions.
You may want to refer to this sketch as we go along.
Next time, we can take up the parabola.