15.5: Projective transformations
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.
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
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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.
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
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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 \(\PageIndex{1}\).