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# The fundamental theorems

In chapter two, we have seen that a projective transformation of the projective line is completely determined by the images of three distinct points. The higher dimensional analogue of this result makes uses of the notion of a projective frame.

Definition 3.1   An -tuple of points in is a projective frame if there exists a basis of such that for and . The basis is said to be adapted to the projective frame.

Exercise 3.1 (10)   Show that four points in form a projective frame if and only if no three of them lie on a common projective line.

Theorem 3.1 (First fundamental theorem)   If and are two projective frames in , there exists a unique projective transformation such that .

The proof depends on the following simple exercise.

Exercise 3.2 (10)   Let and be two basis of that are adapted to the same projective frame. Show that there exists a nonzero real number such that for all .

Exercise 3.3 (05)   Prove theorem 3.1.

Theorem 3.2 (Second fundamental theorem)   Let be a bijection. If sends projective lines to projective lines, then it is a projective tansformation.

In order to keep things simple, we shall prove this theorem only in the case . The reader is asked to extend the proof to arbitrary dimensions. We perform a series of reductions until we are left with a problem in affine geometry.

Exercise 3.4 (05)   Assume that a bijection that sends projective lines to projective lines, and fixes the points , and is the indentity. Show that this implies the second fundamental theorem.

From now on, we will consider the projective plane as the plane plus the line at infinity (see figure 1). We will also indentify the plane with . Equivalently, we shall use the chart and think of as the line at infinity.

Exercise 3.5 (15)   A bijection that sends projective lines to projective lines, and fixes the points , and satisfies the following properties:
1. The restriction of to is a bijection of .
2. As a bijection of , sends straight lines to straight lines and parallel lines to parallel lines.
3. sends the -axis, the -axis, and the diagonal unto themselves.
4. fixes the origin of and the point

Exercise 3.6 (05)   Show that fixes the point and .

We now define by the equation . It follows immediately from the previous exercises that , and .

Exercise 3.7 (15)   Use the construction given in figure 6 of the sum of two real numbers using parallel lines to show that .

Figure 6.

Exercise 3.8 (15)   Use the construction given in figure 7 of the multiplication of two real numbers using parallel lines to show that .

Figure 7.

As a consequence of these exercises, we have that is an automorphism of and, therefore, equal to the identity. If we define by the equation , then the same arguments prove that is the identity.

Now for the final touch:

Exercise 3.9 (10)   Use that sends parallel lines to parallel lines and fixes the and the axes point by point to conclude that is the identity.

Next: Bibliography Up: The Real Projective Plane Previous: Projective geometry vs. affine
Juan Carlos Alvarez 2001-01-22