Instructional Unit The Conics By: Diana Brown Introduction 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 is a circle. If the cone is sliced on a slight angle, the curve is called an ellipse. If the slice is made parallel to the edge of the cone, the curve formed is called a parabola. If the slice is perpendicular to the base of the cone, the curve is one of two branches of a hyperbola. Mathematicians are interested in two branches of the hyperbola, formed by putting two cones together. Diagram of Conics as represented above:
Real world representations of conics We see cones around us every day: ice cream cones, traffic cones, and cone-shaped sushi Ellipse: The orbits of the planets are elliptical and the earth itself is an ellipsoid. A circle viewed from an angle looks like an ellipse. Parabola: 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. Hyperbola: The pattern of light cast on a wall by a lampshade is a hyperbola. More real world representations of conics 
Return to the Conics Homepage Return to EMAT 6690 Homepage Go to Day Two(Circles)
The answer to this mystery is in the modern mathematical idea of a boundary. The key point is that points on the the boundary of a set need not belong to the set. In this case, the plane cuts the cone into an upper half and the lower half. It doesn't matter which way the cone's oriented, so you can get any picture you like in your mind. The intersection of the plane with the cone is the boundary of both halves. Let's arbitrarily assign the boundary points to the lower half cone. Then the upper half cone has the same boundary, but does not include any of those points. The size of the boundary is exactly the same in each case -- there's only one boundary. That solves the problem. There is no "next" circle up that's either the same size or a different size. That's a false mental picture of what happens when the plane cuts the cone. If you like, you can think of it this way. When the plane cuts the cone, there are three point sets formed:
Now you can see that any horizontal slice of the upper half cone is smaller (or larger, depending on which way your cone is oriented) than the size of the boundary; and any horizontal slice of the lower cone is larger (or smaller, respectively) than the boundary circle. There is no paradox. You can do the same thing in the plane. Take the unit circle. It divides the plane into two regions, the inside and the outside of the circle. The inside disk has the unit circle as its boundary; and the outside shape (the plane minus the unit disk) has the unit circle as its boundary. The unit circle is the boundary of both regions. It doesn't matter how you assign the boundary points, all to the inner region or all to the outer region or some to one and some to the other. No matter how you assign the points, the two regions have the same boundary, namely the unit circle.
In order to continue enjoying our site, we ask that you confirm your identity as a human. Thank you very much for your cooperation. If it's parallel to the base, it's a circle. If it doesn't go through the base, it's an ellipse. If it's does, it's hyperbolic/parabolic. |
What is produced when you slice a cone with a plane that is parallel to the base of the cone?
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