Kuzee
Vector Algebra and Analytic Geometry of Space

Cylinders

The Elliptic Cylinder

Elliptic cylinder

An elliptic cylinder is the surface formed by moving an ellipse in the direction of a straight line parallel to the plane of the ellipse and perpendicular to it.

The Standard Equation

The standard equation of the elliptic cylinder

x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

where a,b>0a, b > 0.

The Geometric Properties

  • The generators: the straight lines parallel to the zz-axis
  • The directrix: the ellipse in the xyxy-plane
  • The sections: the sections by planes parallel to the xyxy-plane are ellipses

The Parabolic Cylinder

Parabolic cylinder

A parabolic cylinder is the surface formed by moving a parabola in the direction of a straight line parallel to the plane of the parabola and perpendicular to it.

The Standard Equation

The standard equation of the parabolic cylinder

y2=2pxy^2 = 2px

where p>0p > 0.

The Geometric Properties

  • The generators: the straight lines parallel to the zz-axis
  • The directrix: the parabola in the xyxy-plane
  • The sections: the sections by planes parallel to the xzxz-plane are parabolas

The Hyperbolic Cylinder

Hyperbolic cylinder

A hyperbolic cylinder is the surface formed by moving a hyperbola in the direction of a straight line parallel to the plane of the hyperbola and perpendicular to it.

The Standard Equation

The standard equation of the hyperbolic cylinder

x2a2−y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

where a,b>0a, b > 0.

The Geometric Properties

  • The generators: the straight lines parallel to the zz-axis
  • The directrix: the hyperbola in the xyxy-plane
  • The sections: the sections by planes parallel to the xzxz-plane are hyperbolas

The Properties of Cylinders

The Features of the Equations

The equation of a cylinder does not contain the variable in the generating direction:

  • the elliptic cylinder: the equation does not contain zz
  • the parabolic cylinder: the equation does not contain zz
  • the hyperbolic cylinder: the equation does not contain zz

Symmetry

  • the elliptic cylinder: symmetric about the yzyz and xzxz planes
  • the parabolic cylinder: symmetric about the xzxz plane
  • the hyperbolic cylinder: symmetric about the xzxz and yzyz planes

Infinity

Cylinders are all surfaces that extend infinitely, with no boundary.

Special Cases

The Circular Cylinder

When a=b=ra = b = r in the elliptic cylinder, it becomes a circular cylinder:

x2+y2=r2x^2 + y^2 = r^2

The Degenerate Cylinder

When aa or bb tends to infinity in the elliptic cylinder, it degenerates into a plane.

Exercises

Exercise 1

Determine the type of the surface x2+y2=4x^2 + y^2 = 4.

cylinderrecognising the equationthe circular cylinder

Approach:

Recognise the features of the equation of a cylinder.

Steps:

  1. The equation: x2+y2=4x^2 + y^2 = 4
  2. The equation does not contain zz, so this is a cylinder
  3. The section is the circle: x2+y2=4x^2 + y^2 = 4
  4. This is a circular cylinder

Answer: this is a circular cylinder, with its axis parallel to the zz-axis

Exercise 2

Determine the type of the surface y2=2xy^2 = 2x.

cylinderrecognising the equationthe parabolic cylinder

Approach:

Recognise the standard form of the parabolic cylinder.

Steps:

  1. The equation: y2=2xy^2 = 2x
  2. The equation does not contain zz
  3. This is the surface formed by moving the parabola y2=2xy^2 = 2x in the zz-direction
  4. This is a parabolic cylinder

Answer: this is a parabolic cylinder, opening towards the right

Summary

Symbols Used in This Article

SymbolTypePronunciation/ExplanationMeaning in This Article
x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1Elliptic cylinderelliptic cylinderThe standard equation of the elliptic cylinder
y2=2pxy^2 = 2pxParabolic cylinderparabolic cylinderThe standard equation of the parabolic cylinder
x2a2−y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1Hyperbolic cylinderhyperbolic cylinderThe standard equation of the hyperbolic cylinder
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