# Thread: Planar line translation and rotations...

1. ## Planar line translation and rotations...

We have planar lines in the form of $X=tP+sQ$, where $P$ and $Q$ are two fixed different points and $s,$ $t$ are varying reals satisfying $s+t=1$. We need to find the formula for the images of the line $X=tP+sQ$ in the following three cases:
1. Under the translation by a vector B.
2. Under rotation about a pont C by 180 degrees.
3. Under rotation about a point C by 90 degrees.

2. ## Re: Planar line translation and rotations...

"Translation by vector B" just adds vector B: X= tP+ sQ+ B.

To "rotate through angle $\theta$ around the point C"
i) Subtract C so you are rotating around C.
ii) Multiply by the matrix $\begin{bmatrix}cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta)\end{bmatrix}$
iii) Add C to move back.

In particular, a rotation by 180 degrees, that matrix is $\begin{bmatrix}-1 & 0 \\ 0 & -1\end{bmatrix}$, just multiplying x and y by -1.
A rotation by 90 degrees is given by $\begin{bmatrix}0 & -1 \\ 1 & 0\end{bmatrix}$