CALCULUS Understanding Its Concepts and Methods

Cylinder

A cylinder is a special case of a quadric surface. Geometrically, a cylinder is the surface generated by a line that moves so that it is always parallel to a fixed line and always intersects a fixed curve. Any position of the generating line is an element of the cylinder and the fixed curve that all of these elements intersect is the directrix curve.

If the equation of a surface does not contain one of the variables (*x*,
*y*, or *
z*)
then the surface is a cylinder with elements parallel to the axis of that
variable and having the curve of intersection by a plane orthogonal to the
axis of that variable as directrix curve. A cylinder with axis parallel to the
*z*-axis
can be described by parametric equations of the form

A cylinder is
circular if the traces in planes perpendicular
to the axis of the cylinder are circles, in particular, the directrix curve is
an circle. The standard form and parametric form of an equation for a circular
cylinder with elements parallel to the
*z*-axis
are as follows:

A cylinder is
elliptic if the traces in planes perpendicular
to the axis of the cylinder are ellipses, in particular, the directrix curve
is an ellipse. The standard form and parametric form of an equation for an
elliptic cylinder with elements parallel to the
*z*-axis
are as follows:

A cylinder is parabolic if the traces in
planes perpendicular to the axis are parabolas, in particular, the directrix
curve is a parabola. The standard form and parametric form of an equation for
an elliptic cylinder with elements parallel to the
*z*-axis
are as
follows:

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Copyright © 2006 Darel Hardy, Fred Richman, Carol Walker, Robert Wisner. All rights reserved. Except upon the express prior permission in writing, from the authors, no part of this work may be reproduced, transcribed, stored electronically, or transmitted in any form by any method.