When a straight line moves in passing constantly through a certain point , the ruled surface it sweeps is called a conical surface or a generalized cone. Formally and more generally, a conical surface is a ruled surface with the given condition:
there is a point on , such that any ruling on passes through .
By definition, it is readily seen that this point is unique, for otherwise, all rulings of that pass through as well as another point must all coincide, concluding that must be nothing more than a straight line, contradicting the fact that is a surface. The point is commonly known as the apex of the conical surface .
No plane can be a conical surface, because one can always find a line (ruling) on the plane not passing through .
A conical surface is, strictly speaking, not a (regular) surface in the sense of a differentiable manifold. This is because of the differentiability at the apex breaks down. In fact, no neighborhood of the apex is diffeomorphic to . Nevertheless, it is easy to see that any two points on a conical surface can be joined by a simple continuous curve.
Any plane passing through the apex of a conical surface must contain at least two lines through the apex such that . In fact, it can be shown that there is a plane with the above property such that .
Given a conical surface , if there is a plane in such that , then it can be shown that is planar (that is, lies on a plane). This shows that if is a non-planar surface, it must have non-empty intersection with any plane in !
Given a plane not passing through the apex of a non-planar conical surface , the intersection of and is non-empty, as guaranteed by the previous remark. Let be the intersection of and . It is not hard to see that is necessarily a curve. The curve may be bounded or unbounded, and it may have disjoint components. If there is a plane not passing through the apex of a conical surface , such that its intersection with is a bounded, connected closed loop, then we call this surface a closed cone, or cone for short. Intuitively, it can be pictured as a surface swept out by a moving straight line that returns to its starting position. Any plane not passing through the apex of a closed cone intersects each ruling of at exactly one point.
The solid bounded by a closed cone , a plane not passing through the apex and the apex itself is called a solid cone. The portion of the surface of the cone belonging to the conical surface is called the or the of the solid cone and the portion belonging to the plane is the base of the solid cone.
The intersections of a solid cone and the planes parallel to the base plane are similar. The perpendicular of the apex and the base plane is the of the cone. The volume () of the cone equals to the third of product of the base area () and the ():
The formula can be derived directly by observing that, if we were to take another parallel slice of the solid cone, the area of the base from the slice is directly proportional to the square of the corresponding height . Integrating with respect to , where is between and gives us the above formula.
If the base of a cone is a polygon, the cone is called a pyramid (which is a polyhedron). The mantle of a pyramid consists of triangular faces having as common vertex the apex . A pyramid is if its base is a regular polygon and the of its base. A tetrahedron is an example of a regular pyramid. Note that, in a regular pyramid, all of the faces are isosceles. Colloquially, “pyramid” typically refers to a regular square pyramid.
If the base is a circle, the cone is called circular. If its of the base circle, the circular cone is . Colloquially, “cone” typically refers to a right circular cone.
In any cone, the line segment of a ruling between the base plane and the apex is a of the cone. All are equally long only in a right circular cone. If in this case, the of the side line is and the radius of the base circle , then the area of the mantle of the right circular cone equals .
|Date of creation||2013-03-22 15:29:54|
|Last modified on||2013-03-22 15:29:54|
|Last modified by||pahio (2872)|
|Synonym||circular double cone|
|Synonym||right circular double cone|
|Defines||right circular cone|