Ein Punkt P Koordinatensystem K0:

    P0 = (x0) = x0*i0 + x0*j0
         (y0)

Der Punkt P Koordinatensystem K1:

    P1 = (x1) = x1*i1 + y1*j1
         (y1)

Der Punkt P Koordinatensystem K2:

    P2 = (x2) = x2*i2 + y2*j2
         (y2)

Darstellung von Quellsystem K0 im Zielsystem K1:

    m0 = (dx1) = dx1*i1 + dy1*j1            d=distance
         (dy1)

    i0 = (fx1) = fx1*i1 + sy1*j1            f=längsfaktor (Skalieren)
         (sy1)                              s=querfaktor  (Scheren)

    j0 = (sx1) = sx1*i1 + fy1*j1
         (fy1)

Transformation der Koordinaten für Punkt P:

    P1 = (x1) = m0 + x0*i0 + y0*j0 = (dx1) + x0*(fx1) + y0*(sx1) = (dx1 + x0*fx1 + y0*sx1)
         (y1)                        (dy1)      (sy1)      (fy1)   (dy1 + x0*sy1 + y0*fy1)

=>  P1 = (x1) = (dx1 + x0*fx1 + y0*sx1)     => Transformationsmatrix = (dx,fx,sx, dy,sy,fy)
         (y1) = (dy1 + x0*sy1 + y0*fy1)

Darstellung von Quellsystem K1 im Zielsystem K2:

    m1 = (dx2) = dx2*i2 + dy2*j2
         (dy2)

    i1 = (fx2) = fx2*i2 + sy2*j2
         (sy2)

    j1 = (sx2) = sx2*i2 + fy2*j2
         (fy2)

Transformation der Koordinaten für Punkt P:

    P2 = (x2) = m1 + x1*i2 + y1*j2
         (y2)

=>  P2 = (x2) = (dx2 + x1*fx2 + y1*sx2)
         (y2) = (dy2 + x1*sy2 + y1*fy2)

Kombination der Transformationen K0 -> K1 -> K2:

x1 und y1 von oben einsetzen:
=>  P2 = (x2) = (dx2 + (dx1 + x0*fx1 + y0*sx1)*fx2 + (dy1 + x0*sy1 + y0*fy1)*sx2)
         (y2) = (dy2 + (dx1 + x0*fx1 + y0*sx1)*sy2 + (dy1 + x0*sy1 + y0*fy1)*fy2)

ausmultiplizieren:
=>  P2 = (x2) = (dx2 + dx1*fx2 + x0*fx1*fx2 + y0*sx1*fx2 + dy1*sx2 + x0*sy1*sx2 + y0*fy1*sx2)
         (y2) = (dy2 + dx1*sy2 + x0*fx1*sy2 + y0*sx1*sy2 + dy1*fy2 + x0*sy1*fy2 + y0*fy1*fy2)

x0 und y0 ausklammern:
=>  P2 = (x2) = (dx2 + dx1*fx2 + dy1*sx2 + x0*(fx1*fx2 + sy1*sx2) + y0*(sx1*fx2 + fy1*sx2))
         (y2) = (dy2 + dx1*sy2 + dy1*fy2 + x0*(fx1*sy2 + sy1*fy2) + y0*(sx1*sy2 + fy1*fy2))

=>  dx12 = dx2 + dx1*fx2 + dy1*sx2    fx12 = fx1*fx2 + sy1*sx2    sx12 = sx1*fx2 + fy1*sx2
    dy12 = dy2 + dx1*sy2 + dy1*fy2    sy12 = fx1*sy2 + sy1*fy2    fy12 = sx1*sy2 + fy1*fy2

Inverse Transformation K1 -> K0:

    P0 = (x0) =
         (y0)

    P1 = (x1) = (dx1 + x0*fx1 + y0*sx1)
         (y1) = (dy1 + x0*sy1 + y0*fy1)

=>  x1 = dx1 + x0*fx1 + y0*sx1  <=>  x0 = (x1 - dx1 - y0*sx1) / fx1
    y1 = dy1 + x0*sy1 + y0*fy1  <=>  x0 = (y1 - dy1 - y0*fy1) / sy1

=>  (x1 - dx1 - y0*sx1) / fx1 = (y1 - dy1 - y0*fy1) / sy1
nach y0 auflösen:
    x1*sy1 - dx1*sy1 - y0*sx1*sy1 = y1*fx1 - dy1*fx1 - y0*fy1*fx1
    y0*fy1*fx1 - y0*sx1*sy1 = y1*fx1 - dy1*fx1 - x1*sy1 + dx1*sy1
    y0 * (fy1*fx1 - sx1*sy1) = y1*fx1 - dy1*fx1 - x1*sy1 + dx1*sy1
    y0 = (y1*fx1 - dy1*fx1 - x1*sy1 + dx1*sy1) / (fy1*fx1 - sx1*sy1)
    quot = 1 / (fy1*fx1 - sx1*sy1)
    y0 = (y1*fx1 - dy1*fx1 - x1*sy1 + dx1*sy1) * quot
    y0 = y1*fx1*quot - dy1*fx1*quot - x1*sy1*quot + dx1*sy1*quot
    y0 = (dx1*sy1 - dy1*fx1)*quot + y1 * fx1*quot - x1 * sy1*quot


=>  x1 = dx1 + x0*fx1 + y0*sx1  <=>  y0 = (x1 - dx1 - x0*fx1) / sx1
    y1 = dy1 + x0*sy1 + y0*fy1  <=>  y0 = (y1 - dy1 - x0*sy1) / fy1

=>  (x1 - dx1 - x0*fx1) / sx1 = (y1 - dy1 - x0*sy1) / fy1
nach x0 auflösen:
    x1*fy1 - dx1*fy1 - x0*fx1*fy1 = y1*sx1 - dy1*sx1 - x0*sy1*sx1
    x0*sy1*sx1 - x0*fx1*fy1 = y1*sx1 - dy1*sx1 - x1*fy1 + dx1*fy1
    x0 = (y1*sx1 - dy1*sx1 - x1*fy1 + dx1*fy1) / (sy1*sx1 - fx1*fy1)
    x0 = -y1*sx1*quot + dy1*sx1*quot + x1*fy1*quot - dx1*fy1*quot
    x0 = (dy1*sx1 - dx1*fy1)*quot - y1 * sx1*quot + x1 * fy1*quot


=>  P0 = (x0) = ((dy1*sx1 - dx1*fy1)*quot + x1 * fy1*quot - y1 * sx1*quot)
         (y0)   ((dx1*sy1 - dy1*fx1)*quot - x1 * sy1*quot + y1 * fx1*quot)

=>  quot = 1 / (fy1*fx1 - sx1*sy1)
    dx' = (dy1*sx1 - dx1*fy1)*quot   fx' =  fy1*quot   sx' = -sx1*quot
    dy' = (dx1*sy1 - dy1*fx1)*quot   sy' = -sy1*quot   fy' =  fx1*quot