# Cylinders

### Prerequisites : Coordinate Geometry I and Coordinate Geometry II

A cylinder is defined as a surface generated by a straight line (the generator) parallel to a fixed line, called the axis of the cylinder and satisfying another geometrical condition. The condition can be

I) intersecting a curve (the guiding curve or the generatrix) or

II) touching a give surface

A right circular cylinder is the one, whose guiding curve is a circle.

• ### Equation of Cylinder, whose Axis is Parallel to a Coordinate Axis

We have seen earlier that a curve in 3D is represented as a union of 2 surfaces, say $S=0$ and $P=0$. $P$ corresponds to a planar surface. If a cylinder’s guiding curve is $S=0, P=0$ and the axis of cylinder is parallel to $Z$ axis, we eliminate the variable $z$ from both the equations $S=0$ and $P=0$ to get the equation of the cylinder. (For $X$, eliminate $x$, for $Y$, eliminate $y$).

• ### Equation of Cylinder, whose Axis is NOT Parallel to a Coordinate Axis

If the axis of cylinder is not parallel to either of the coordinate axes, we proceed as follows:

1) Let $(x_1,y_1,z_1)$ be a point on the cylinder.

2) We then write the equation of generator using the direction ratios of axis.

3) We then make use of $P=0$ to get $x,y$ and $z$.

4) To get the equation of cylinder, we put these expressions of $x,y$ and $z$ in $S=0$ and drop the suffices of $latex x_1,y_1$ and \$z_1\$.

• ### Equation of Right Circular Cylinder

The section of a right circular cylinder perpendicular to the axis is a circle. To get the equation of the right circular cylinder, with equation of axis and radius given, we make use of the Pythagoras’ theorem. See the figure below :

• ### Enveloping Cylinder

Rays of sun intersecting the earth form an enveloping cylinder, which is always right circular. In this case, the radius of sphere becomes the radius of the cylinder. Rest of the part is as explained in the previous section.