Right circular cylinder with axis (x−α)/l=(y−β)/m=(z−γ)/n

A right circular cylinder is the surface traced by a line parallel to its axis at a constant distance r from the axis. So a point P(x, y, z) lies on it if its perpendicular distance from the axis equals r.

Let the axis pass through A(α, β, γ) with direction ratios (l, m, n).

Distance of P from the axis

Let AP = (x − α, y − β, z − γ). The length of the projection of AP on the axis is

AN = [l(x − α) + m(y − β) + n(z − γ)] / √(l² + m² + n²).

By Pythagoras in the right triangle APN, the perpendicular distance PN satisfies

PN² = AP² − AN²

Condition PN = r

(x−α)² + (y−β)² + (z−γ)²
   − [l(x−α) + m(y−β) + n(z−γ)]² / (l²+m²+n²) = r²

Equivalent form

Since |AP|²|d|² = (AP·d)² + |AP × d|² (Lagrange's identity), the same equation can be written as

[n(y−β) − m(z−γ)]² + [l(z−γ) − n(x−α)]² + [m(x−α) − l(y−β)]²
      = r²(l² + m² + n²)

(the left side is |AP × axis direction|², which is (distance × |direction|)²).

Answer: the required cylinder is the surface given by either form above.

Example: axis = z-axis (α = β = γ = 0, l = m = 0, n = 1) gives x² + y² = r², as expected.