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Math Help - Planar line translation and rotations...

  1. #1
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    Question 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.
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  2. #2
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    Re: Planar line translation and rotations...

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

    To "rotate through angle [itex]\theta[/itex] 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}
    Last edited by HallsofIvy; August 7th 2014 at 02:43 PM.
    Thanks from dennydengler
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