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Mathematics LibreTexts

15.5: Projective transformations

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    23679
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    A bijection from the real projective plane to itself that sends lines to lines is called projective transformation.

    Note that any affine transformation defines a projective transformation on the corresponding real projective plane. We will call such projective transformations affine; these are projective transformations that send the ideal line to itself.

    The extended perspective projection discussed in the previous section provides another source of examples of projective transformations.

    Theorem \(\PageIndex{1}\)

    Given a line \(\ell\) in the real projective plane, there is a perspective projection that sends \(\ell\) to the ideal line.

    Moreover, a perspective transformation is either affine or, in a suitable coordinate system, it can be written as a composition of the extension of perspective projection

    \(\beta:(x,y) \mapsto (\dfrac{x}{y},\dfrac{1}{y})\)

    and an affine transformation.

    Proof

    We may choose an \((x,y)\)-coordinate system such that the line \(\ell\) is defined by equation \(y=0\). Then the extension of \(\beta\) gives the needed transformation.

    Fix a projective transformation \(\gamma\). If \(\gamma\) sends the ideal line to itself, then it has to be affine. It proves the theorem in this case.

    Suppose \(\gamma\) sends the ideal line to a line \(\ell\). Choose a perspective projection \(\beta\) as above. The composition \(\beta\circ\gamma\) sends the ideal line to itself. That is, \(\gamma=\beta\circ\gamma\) is affine. Note that \(\beta\) is self-inverse; therefore \(\alpha=\beta\circ \gamma\) — hence the result.

    Exercise \(\PageIndex{1}\)

    Let \(P\mapsto P'\) be (a) an affine transformation, (b) the perspective projection defined by \((x,y) \mapsto (\dfrac{x}{y},\dfrac{1}{y})\), or (c) arbitrary projective transformation. Suppose \(P_1,P_2,P_3,P_4\) lie on one line. Show that

    \(\dfrac{P_1P_2\cdot P_3P_4}{P_2 P_3 \cdot P_4 P_1}=\dfrac{P'_1P'_2\cdot P'_3P'_4}{P'_2P'_3\cdot P'_4P'_1};\)

    that is, each of these maps preserves cross ratio for quadruples of points on one line.

    Hint

    To prove (a), apply Proposition 14.3.1.

    To prove (b), suppose \(P_i = (x_i, y_i)\); show and use that

    \(\dfrac{P_1 P_2 \cdot P_3P_4}{P_2P_3 \cdot P_4P_1} = |\dfrac{(x_1 - x_2) \cdot (x_3 - x_4)}{(x_2 - x_3)\cdot (x_4 - x_1)}|\)

    if all \(P_i\) lie on a horizontal line \(y = b\), and

    \(\dfrac{P_1 P_2 \cdot P_3P_4}{P_2P_3 \cdot P_4P_1} = |\dfrac{(y_1 - y_2) \cdot (y_3 - y_4)}{(y_2 - y_3)\cdot (y_4 - y_1)}|\)

    otherwise. (See 20.8.4 for another proof.)

    To prove (c), apply (a), (b), and Theorem \(\PageIndex{1}\).