Editing Cylinder (section)

==Cylindrical surfaces==
{{anchor|elliptic cylinder|parabolic cylinder|hyperbolic cylinder}}
In some areas of geometry and topology the term ''cylinder'' refers to what has been called a '''cylindrical surface'''. A cylinder is defined as a surface consisting of all the points on all the lines which are parallel to a given line and which pass through a fixed plane curve in a plane not parallel to the given line.{{sfn|Albert|2016|p=43}} Such cylinders have, at times, been referred to as ''{{dfn|generalized cylinders}}''. Through each point of a generalized cylinder there passes a unique line that is contained in the cylinder.{{sfn|Albert|2016|p=49}} Thus, this definition may be rephrased to say that a cylinder is any [[ruled surface]] spanned by a one-parameter family of parallel lines.

A cylinder having a right section that is an [[ellipse]], [[parabola]], or [[hyperbola]]  is called an '''elliptic cylinder''', '''parabolic cylinder''' and '''hyperbolic cylinder''', respectively. These are degenerate [[quadric surface]]s.<ref>{{citation |first1=David A. |last1=Brannan |first2=Matthew F. |last2=Esplen |first3=Jeremy J. |last3=Gray |title=Geometry |year=1999 |publisher=Cambridge University Press |isbn=978-0-521-59787-6 |page=34 }}</ref>
[[File:Zylinder-parabol-s.svg|thumb|120px|Parabolic cylinder]]
When the principal axes of a quadric are aligned with the reference frame (always possible for a quadric), a general equation of the quadric in three dimensions is given by
<math display=block>f(x,y,z)=Ax^2 + By^2 + C z^2 + Dx + Ey + Gz + H = 0,</math>
with the coefficients being [[real number]]s and not all of {{mvar|A}}, {{mvar|B}} and {{mvar|C}} being 0. If at least one variable does not appear in the equation, then the quadric is degenerate. If one variable is missing, we may assume by an appropriate [[rotation of axes]] that the variable {{mvar|z}} does not appear and the general equation of this type of degenerate quadric can be written as{{sfn|Albert|2016|p=74}}
<math display=block>A \left ( x + \frac{D}{2A} \right )^2 + B \left(y + \frac{E}{2B} \right)^2 = \rho,</math>
where
<math display=block>\rho = -H + \frac{D^2}{4A} + \frac{E^2}{4B}.</math>

=== Elliptic cylinder ===
If {{math|''AB'' > 0}} this is the equation of an ''elliptic cylinder''.{{sfn|Albert|2016|p=74}} Further simplification can be obtained by [[translation of axes]] and scalar multiplication. If <math>\rho</math> has the same sign as the coefficients {{mvar|A}} and {{mvar|B}}, then the equation of an elliptic cylinder may be rewritten in [[Cartesian coordinates]] as:
<math display=block>\left(\frac{x}{a}\right)^2+ \left(\frac{y}{b}\right)^2 = 1.</math>
This equation of an elliptic cylinder is a generalization of the equation of the ordinary, ''circular cylinder'' ({{math|1=''a'' = ''b''}}). Elliptic cylinders are also known as ''cylindroids'', but that name is ambiguous, as it can also refer to the [[Plücker conoid]].

If <math>\rho</math> has a different sign than the coefficients, we obtain the ''imaginary elliptic cylinders'':
<math display=block>\left(\frac{x}{a}\right)^2 + \left(\frac{y}{b}\right)^2 = -1,</math>
which have no real points on them. (<math>\rho = 0</math> gives a single real point.)

=== Hyperbolic cylinder ===
If {{mvar|A}} and {{mvar|B}} have different signs and <math>\rho \neq 0</math>, we obtain the ''hyperbolic cylinders'', whose equations may be rewritten as:
<math display=block>\left(\frac{x}{a}\right)^2 - \left(\frac{y}{b}\right)^2 = 1.</math>

=== Parabolic cylinder ===
Finally, if {{math|1=''AB'' = 0}} assume, [[without loss of generality]], that {{math|1=''B'' = 0}} and {{math|1=''A'' = 1}} to obtain the ''parabolic cylinders'' with equations that can be written as:{{sfn|Albert|2016|p=75}}
<math display=block> x^2 + 2 a y = 0 .</math>

[[File:(Texas Gulf Sulphur Company) (10428629273).jpg|thumb|In [[projective geometry]], a cylinder is simply a cone whose [[apex (geometry)|apex]] is at infinity, which corresponds visually to a cylinder in perspective appearing to be a cone towards the sky.]]