Cylinder

• A cylinder is a surface generated by a moving straight line (generator) which is parallel to a fixed line and interescts a given surface or curve (guiding curve).

• The equation of the cylinder whose generators are parallel to the line \(\dfrac{x}{l}=\dfrac{y}{m}=\dfrac{z}{n}\) and base conic is \(a x^{2}+2 h x y+b y^{2}+2 g x+2 f y+c=0\), \(z=0\) is given by:

\(a(n x-l z)^{2}\)+\(2 h(n x-l z)(n y-m z)\)+\(b(n y-m z)^{2}\)+\(2 g n(n x-l z)\)+\(2 f n(n y-m z)\)+\(c n^{2}=0\)

• A right circular cylinder is a cylinder whose generator lines are at a constant distance (radius) from the fixed line (axis).

• The equation of the right circular cylinder with axis as the \(z-axis\) and radius \(a\) is given by \(x^{2}+y^{2}=a^{2}\)

• The equation of the right circular cylinder whose radius is \(a\) and axis the line \(\dfrac{x-x_{1}}{l}=\dfrac{y-y_{1}}{m}=\dfrac{z-z_{1}}{n}\) is given by:

\(\dfrac{\left[l\left(x-x_{1}\right)+m\left(y-y_{1}\right)+n\left(z-z_{1}\right)\right]^{2}}{l^{2}+m^{2}+n^{2}}\)+\(r^{2}\)=\(\left(x-x_{1}\right)^{2}\)+\(\left(y-y_{1}\right)^{2}\)+\(\left(z-z_{1}\right)^{2}\)

Cylinder