Line Sweep Surface:

Let the line segment in the xy-plane and parallel to the y-axis defined by end points P₁ [0 0 0] and P₂ [0 3 0 ]. Calculate the point at t = 0.5, s = 0.5 on the sweep surface produced by simultaneously translating the line 10 units along with the x-axis and rotating it through 2π around the x-axis.

The sweep transformation matrix is a translation followed by a rotation specified by :

The parametric equation of line segment is following

P (u) = P₁ + (P₂ - P₁) t = [0  0  0  1] + [0 - 0   3 - 0     0 - 0   1 - 1] t

= [0   3t   0   1]

The sweep surface is specified by

Q (t, s) = [P (t)] [T (s)]

And

= [5  - 1.5  0  1]