Unizor - Geometry3D - Elements - Conical Surface

Unizor - Creative Minds through Art of Mathematics - Math4Teens

The subject of this lecture is to introduce a concept of a conical surface in three-dimensional space.

Let's assume that we have a curve c in the three-dimensional space and, separately, a fixed point S outside it.
A curve c does not necessarily is a "flat" one (that is, we do not require all its points to lie on the same plane).
Now imagine that through each point on this curve c we construct a straight line connecting it to a fixed point S outside it. The set of all points of all these lines forms a surface that we call conical.

Let's introduce some terminology related to conical surfaces.

A curve c determines the points through which we draw lines connecting these points to point S. This curve c is called directrix because it directs the position of each line we draw.

The lines that we draw from each point of curve c through a fixed point S form the conical surface and are sometimes called rulings.

The fixed point S connected with each point on a directrix c with rulings, is called an apex or a vertex (a more general term applicable also to other points of geometrical objects).

If a directrix is a closed curve lying on some plane, the area it encompasses on a plane may be referred to as a base of a conical surface.

Let's exemplify this construction without any rigorous proof of characteristics of constructed objects.

If directrix c is a straight line, the result of our construction will be a plane containing both directrix c and apex S.

If directrix c is a circle and apex S is a center of this circle, the result of our construction will be a plane this circle belongs to.

If directrix c is a circle and apex S is positioned outside of a plane containing this circle such that a perpendicular from a apex onto this plane falls in the center of a circle, the result of our construction will be an infinitely long cone propagating to infinity on both sides of a apex.

If, instead of a circle, we choose as a directrix a polygon lying on a plane and apex S is positioned outside of a plane containing this polygon such that a perpendicular from a apex onto this plane falls inside a polygon, the resulting surface would be an infinitely long conical surface, part of which between the polygon and the apex forms a pyramid.

Notice, we did not define yet concepts or perpendicularity and parallelism between planes and straight lines. In this introductory lecture we just assume that students have intuitive understanding of these concept.


Interesting property of any conical surface is that, if it's made of paper, it can be flattened on a flat plane without stretching (which is not the case with a spherical surface that we will introduce in subsequent lectures).

Another viewpoint to a conical surface is that it can be considered as a surface formed by a line passing through a fixed apex and a point moving along a directrix.