Saturday, May 2, 2015

Unizor - Geometry3D - Elements - Cones

Unizor - Creative Minds through Art of Mathematics - Math4Teens

Consider a plane α that we will call a base plane and a circle c with a center O and radius R on this plane. This circle will be used as a directrix of a conical surface we will construct.
Although non-circular directrix can be considered, we will unlikely deal with such. So, unless otherwise specified, a directrix is assumed to be a circle, conical surface in this case is classified as circular.
Also assume there is a point S outside this plane that we will use as an apex of our conical surface.

Let's construct a conical surface σ using circle c as a directrix and point S as an apex.

Cone is an object in solid geometry formed by a part of a conical surface σ between a base plane α and an apex S and a part of the base plane inside that conical surface - circle c on the base plane α.

The only vertex of a cone is its apex.
There are no edges of a cone.
Circle c is called a base of a cone.
A perpendicular from apex S to a base of a cone is its altitude or height.
If a perpendicular from apex S to a base of a cone falls into its center O, a cone is called a right cone.

In most cases we will be dealing with right circular cones calling them just cones unless otherwise specified.

Now let's imagine that we have another plane β parallel to base α and intersecting a cone's altitude somewhere between an apex and a base.
A geometric object consisting of a conical surface between planes α and β and parts of these planes lying inside the conical surface - a circle on the base plane α (we can call it "bottom") and a corresponding circle on the plane β (we can call it "top") - is called a truncated cone.
All the terminology of cones is the same for truncated cones. The altitude of a truncated cone is a distance between "top" and "bottom" bases along a common perpendicular.

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