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# 15.5: Projective transformations

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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 $$\PageIndex{1}$$.