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SURFACE OF REVOLUTION, CYLINDRICAL SURFACE, CYLINDER, DIRECTRIX, GENERATRIX, CONICAL SURFACE, PYRAMIDAL SURFACE



Surface of revolution. A surface which can be generated by revolving a plane curve about an axis in its plane. Sections of a surface of revolution perpendicular to this axis are circles, called parallel circles or simply parallels: sections containing the axis are meridian sections, or simply meridians. The earth is a surface of revolution which can be generated by revolving a meridian about the line through the north and south poles. A surface of revolution can also be generated by a circle moving always perpendicular to a fixed line with its center on the fixed line and expanding or contracting so as to continually pass through a curve which always lies in a plane with the straight line.

                                                                                            James & James. Mathematics Dictionary.



Cylindrical surface. A cylindrical surface is generated by a straight line which moves along a fixed curve and remains parallel to a fixed straight line. The fixed curve is called the directrix of the surface and the moving line is called the generatrix (or generator) of the surface.


A cylindrical surface whose generatrix is parallel to one of the coordinate axes and whose directrix is a curve in the coordinate plane that is perpendicular to the generatrix, has the same equation as the directrix. For example, if the directrix is the ellipse


             ole.gif


in the x-y plane, the equation of the cylinder is

 

             ole1.gif  


                                  




Cylinder. (1) A cylindrical surface (2) Suppose we are given two parallel planes and two simple closed curves C1 and C2 in these planes for which lines joining corresponding points of C1 and C2 are parallel to a given line L. A cylinder is a closed surface consisting of two bases which are plane regions bounded by such curves C1 and C2 and a lateral surface which is the union of all line segments joining corresponding points of C1 and C2. Each of the curves C1 and C2 is a directrix of the cylinder and the line segments joining corresponding points of C1 and C2 are elements (or generators or rulings). The cylinder is circular or elliptic if a directrix is a circle or an ellipse, respectively. Sometimes a circular cylinder is defined to be a cylinder whose intersections with planes perpendicular to the elements are circles. The cylinder is a right cylinder or an oblique cylinder according as L is perpendicular to the planes or not perpendicular to the planes. The altitude of a cylinder is the perpendicular distance between the planes containing the bases and a right section is the intersection of the cylinder and a plane perpendicular to the elements that crosses the cylinder between the bases.


                                                                                    James & James. Mathematics Dictionary.



Conical surface. A surface which is the union of all lines that pass through a fixed point and intersect a fixed curve. The fixed point is the vertex, or apex, of the conical surface, the curve the directrix, and each of the lines is a generator or generatrix.,


                                                                                    James & James. Mathematics Dictionary.




Pyramidal surface. A surface generated by a line passing through a fixed point and moving along a broken line in a plane that does not contain the fixed point. It is a closed pyramidal surface if the broken line is a polygon.

                                                                                    James & James. Mathematics Dictionary.



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