Definition of a ConeDate: 07/13/2000 at 22:15:00 From: Joshua Romanowski Subject: Parallel planes in cones I'm having trouble seeing how a right circular cone cut parallel to the axis of symmetry reveals a hyperbola. Shouldn't it be a parabola? Thanks. Date: 07/13/2000 at 23:00:43 From: Doctor Peterson Subject: Re: Parallel planes in cones Hi, Joshua. One thing that may be confusing you is that the "cone" we have in mind when we talk about conics is a double cone made of two identical cones end to end, such as you would form if you held a long thin stick at the middle and moved one end in a circle. A "vertical" slice parallel to its axis will go through both parts of the cone, producing the two parts of a hyperbola. It's only when you cut parallel to the slant of the cone that you get a parabola, a single infinite curve rather than one finite curve (the ellipse) or two infinite curves (the hyperbola.) A discussion and illustrations can be found in Xah Lee's Special Plane Curves pages: http://xahlee.org/SpecialPlaneCurves_dir/ConicSections_dir/conicSections.html Any cut steeper than the slope of the cone will give a hyperbola, not just a cut parallel to the axis; what amazes me is that such a hyperbola is symmetrical, even though the two parts seem so different due to the slant. If I've missed your objection to the curve being a hyperbola, tell me more about your thinking, so we can clarify the subject. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 07/14/2000 at 22:53:59 From: maximus_gold Subject: Re: Parallel planes in cones Doctor Peterson, Thank you very much for the reply. It makes much more sense now. However, what is bugging me is the assumption that the term implies two cones end to end. As you recall, the question stated a "right circular cone." Doesn't this imply just one side? What am I missing? Thanks in advance. Josh Date: 07/15/2000 at 21:05:39 From: Doctor Peterson Subject: Re: Parallel planes in cones Hi, Josh. There are a couple of ways to answer. First, even if all you get is one half of the hyperbola, it's still (part of) a hyperbola, not a parabola; so this aspect of the definition doesn't really make a lot of difference. Second, however, the term "right circular cone," or "cone" in general, is used in a couple of different ways. When we talk about volumes, we refer to a finite object bounded by a cone and a plane perpendicular to the cone's axis. That's what you usually picture as a cone at an elementary level. But the cone we refer to when we discuss conics is the surface generated by constructing all lines that pass through a circle and a point not in its plane; that's the "double cone" I described. In other words, what you think of as a cone is actually only part of one half of a full cone. Math terminology, I'm afraid, is not entirely consistent, especially where a term is used in different fields, or in both elementary and advanced math. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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