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What is a dilation, or scaling around a point?

    When scaling a plane around a point, the result is a plane of a different size but the same shape.

    It is highly-recommended to draw everything out.

    SCALING FROM A CENTERPOINT

    Scaling is commonly done about ##(0,0)##, in which case it looks like this:

    and you can see that the ##z##-axis (into/out of the screen, or in the ##hatk## direction) would pass through the exact centers of both triangles ##DeltaABC## and ##DeltaA’B’C’##.

    In this scenario, it is clearer what has happened to each point. ##A(“-2,-2”)## has been scaled outwards by a factor of ##2## to achieve ##A'(“-4,-4”)##, ##B(“1,1”)## has been scaled outwards by a factor of ##2## to achieve ##B'(“2,2”)##, and ##C(“0,2”)## has been scaled outwards by a factor of ##2## to achieve ##C'(“0,4”)##.

    In general, scaling about a central point is given as:

    ##color(blue)(P'(x,y) = P(kx,ky) ” where ” k in RR ” and ” k > 0)##

    SCALING OFF-CENTER

    Scaling might also be done from an arbitrary non-central point. For instance, scaling a triangular plane ##D_k## outwards from point ##O## by non-zero scalar ##k in RR## where ##k = (OA’)/(OA) = (OB’)/(OB) = (OC’)/(OC)## gives the transformation

    for which ##A##, ##B##, and ##C## were points on the green triangle. In this case, the scaling shrunk ##DeltaABC## into ##DeltaA’B’C’## by a factor of ##k## where ##0 < k < 1##.

    You can see that ##DeltaABC## and ##DeltaA’B’C’## expand along the axes that pass through points ##O##, as well as ##A##, ##B##, ##C##, ##A’##, ##B’##, or ##C’##.

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