A hybrid sweep is generated by two or more transformation on a plane curve For simultaneous rotation & translation,

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

$\text { Where, } n 1, n 2, n 3$ are translation factor.

Q. A line segment $P(t)$ is defined by points $P_{0}(0,0,0), P_{1}(0,3,0)$ if is translated 10 units along. X-axis and simultaneously rotated about X-axis through 2$\pi$ . Find point on the surface at $t=0.25$ and $\theta=135^{\circ}$

Solution: Equation of Line,

$\begin{aligned} P(t) &=(1-t) P_{0}+t P_{1} \\ P_{x}(t) &=(1-t) 0+t(0) \\ &=0 \\ P_{y}(t) &=(1-t) 0+t(3) \\ &=3 t \\ P_{x}(t) &=0 \end{aligned}$

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

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

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

At $t=0.25, \theta=2 \pi s=135^{\circ}$

$P(s, t)=\left[\begin{array}{ccc}{3.75} & {-0.53} & {0.53}\end{array}\right]$