Monday, April 27, 2015
Unizor - Geometry3D - Elements - Cylindrical Surface
Unizor - Creative Minds through Art of Mathematics - Math4Teens
The subject of this lecture is to introduce a concept of cylindrical surface in three-dimensional space.
Let's assume that we have a curve c in the three-dimensional space and, separately, a straight line d.
A curve 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 parallel to the given straight line d. The set of all points of all these lines we have constructed form a surface that we call cylindrical.
Let's introduce some terminology related to cylindrical surfaces.
A curve c determines the points through which we draw lines parallel to d. This curve c is called directrix because it directs the position of each line we draw.
Those parallel lines that we draw form the cylindrical surface and are sometimes called rulings.
The straight line d, to which all rulings must be parallel, is called generatrix because it helps to generate the surface.
Let's exemplify this construction without any rigorous proof of characteristics of constructed objects.
If a directrix c is a straight line not lying on the same plane with a generatrix d, the result of our construction will be some plane parallel to a generatrix d.
If a directrix c is a circle lying in a plane perpendicular to a generatrix d, the result of our construction will be infinitely long right circular cylinder with an axis parallel to a generatrix d and perpendicular to a plane where a directrix c lies.
If, instead of a circle we will choose a polygon, the resulting surface would be an infinitely long prism with edges parallel to a generatrix d and perpendicular to a plane where a directrix c lies.
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 cylindrical surface is that, if it's made of paper, it can be flattened on a flat plane without stretching or cuts (which is not the case with a spherical surface that we will introduce in subsequent lectures).
Another viewpoint to a cylindrical surface is that it can be considered as a surface formed by a line moving parallel to a straight line called generatrix along a curve called directrix.