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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

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

Cylinder. (1) A cylindrical surface (2) Suppose we are given two parallel planes and two
simple closed curves C_{1} and C_{2} in these planes for which lines joining corresponding points of C_{1}
and C_{2} 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 C_{1} and C_{2} and a lateral surface which is the
union of all line segments joining corresponding points of C_{1} and C_{2}. Each of the curves C_{1} and
C_{2} is a directrix of the cylinder and the line segments joining corresponding points of C_{1} and C_{2}
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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