It is a surface generated by translating a planer curve a certain distance along a specified direction.

Translation matrix in 3D transformation in parametric form (parameter‘s’) is,

$P(s)=\left[\begin{array}{cccc}{1} & {0} & {0} & {n_{1} s} \\ {0} & {1} & {0} & {n_{2} s} \\ {0} & {0} & {1} & {n_{3} s} \\ {0} & {0} & {0} & {1}\end{array}\right]$

Where, n1,n2,n3 are translation factor in x,y and z direction respectively.

A translational sweep is generated by translating a curve P(t) in specified direction.

$P(s, t)=P(s) P(t)$

Q. A curve $P(t)=[3 \cos 2 \pi t \quad 2 \sin 2 \pi t \quad 0]$ is translated by 10 points along z-direction, calculate the equation of surface.

Solution:

$P(s, t)=P(s) P(t)$

$\quad \quad \quad =\left[\begin{array}{cccc}{1} & {0} & {0} & {n_{1} S} \\ {0} & {1} & {0} & {n_{2} S} \\ {0} & {0} & {1} & {n_{3} S} \\ {0} & {0} & {0} & {1}\end{array}\right]\left[\begin{array}{c}{P_{x}(t)} \\ {P_{y}(t)} \\ {P_{z}(t)} \\ {1}\end{array}\right]$

Here, $n_{1}=0, n_{2}=0, n_{3}=10$

$P(s, t)=\left[\begin{array}{cccc}{1} & {0} & {0} & {10 s} \\ {0} & {1} & {0} & {0} \\ {0} & {0} & {1} & {0} \\ {0} & {0} & {0} & {1}\end{array}\right]\left[\begin{array}{c}{3 \cos 2 \pi t} \\ {2 \sin 2 \pi t} \\ {0} \\ {1}\end{array}\right]$

$\quad \quad \quad=\left[\begin{array}{c}{3 \cos 2 \pi t+10 s} \\ {2 \sin 2 \pi t} \\ {0} \\ {1}\end{array}\right]$

$P(s, t)=[3 \cos 2 \pi t+10 s \quad 2 \sin 2 \pi t \quad 0]$