First of all, what the heck is a friggin’ conic section? We will find this out by first defining the “cone”. What’s the first thing that comes to mind when thinking of a cone? Icecream? Thought so. Icecream cones are named so because they are – believe it or not – cones in the technical or geometric sense as well. So now that we have eliminated all doubt as far as what cones are, we can now proceed to the definition of the conic section.
A conic section is the line curve that’s produced when intersecting a right circular cone with a plane. If you’re wondering what a right circular cone is, it’s a cone with a circular base whose axis passes through said base at right angles to its plane. It’s a complicated way of explaining what your everyday icecream cone is shaped like, but is necessary as it contrasts from an “oblique cone” (see image).
This might be confusing if you imagine an icecream cone as you normally hold it facing up, with its point facing down. But imagine the cone with the wide end down and it’s easy to picture this. Now when you bring a plane – which is a flat, two-dimensional surface – into a cone, you will undoubtedly produce a cross-section where this plane intersects with the cone. This is the conic section. And there are four types of such conic sections.
In relation to a right circular cone:
- Parabola – the line curve you get when the plane is parallel to a straight line on its surface.
- Hyperbola – any line curve that is produced when the plane intersects both halves of a double cone.
- Ellipse – any line curve you get as long as the plane produces a closed curve.
- Circle – the line curve you get when the plane is parallel to the cone-base. This is a special case of an ellipse.
Remember that a hyperbola requires the intersecting of both halves of a double cone whereas the ellipse and parabola only require the intersecting of half (the circle is an ellipse). You may be wondering what happens when a plane is parallel to the axis and passes right through the center of the cone. This results in a special hyperbola called a degenerate hyperbola, and is simply two straight lines intersecting at the apex forming two “V”‘s meeting point-to-point.
There are a few facts you might find helpful in remembering these four conic sections:
- Circles and parabolas both retain the same shape but can vary in size. This means that like squares, they can be various sizes but are always the same shape.
- In contrast, ellipses and hyperbolas can be any of a wide range of shapes. Ellipses range from being nearly a parabola to nearly a circle, while hyperbolas can look very much like parabolas the closer they get to being parallel to the cone side.
- Circles and ellipses are closed line segments while parabolas and hyperbolas are open.
- The two arms of a parabola eventually become parallel to each other, while those of a hyperbola don’t, always making an angle relative to each other.
Man has known about parabolas to varying degrees since as early as the 3rd and 4th century BC. Galileo discovered that the flying path of a projectile forms a parabola, due to the uniform acceleration of gravity. Although this concept is extremely important in modern ballistics, real-life trajectories are influenced heavily by such factors as air resistance and variations in gravitational acceleration due to altitude.
Due to these and other critical factors, real-life trajectories more often resemble ellipses and circles, rather than parabolas. Other uses include devices such as reflectors, mirrors, satellite dishes, in fact, anything that reflects radiation of any kind, utilizes the shape of the parabola to achieve the desired results – which is usually the concentration of radiation onto a common focal point.