Introduction to Projective Geometry

Juan Carlos Álvarez Paiva

Projective transformations

In order to study the invariants of perspective in more detail, we simplify the situation by considering a class of transformations from the real line onto itself. These transformations can be described as follows: think of the real line inside the plane, draw a second line m and two points P and Q as in the figure below. If X is a point on the real line, its image T(X) is given by the intersection of the real line with the line passing through the point Q and the intersection of XP and m (click on the image). If this line is parallel to the real line we say that the image of X is the point at infinity.

Click here for animation

By moving the line m or the points P and Q we obtain many different transformations. The reader can experiment and get a feeling for this type of transformations by using the applet.

A toy version of Alberti's problem is : what are the invariants of these type of transformations? .

A preliminary remark that seems almost trivial, but is actually very important is that any quantity that is invariant under these transformations is invariant under their composition. Let us agree to call all possible transformations that can be obtained by the construction above or by composition of such transformations projective transformations.

To see how a projective transformation may move things around, we first consider two lines with three points on each. It is not hard to see that by combining two perspectives it is possible to transform the three points on the first line to the three points on the second line. The construction is given by the figure below. If you click on it, you will have a step-by-step construction.

Click here for animation

Applying this construction twice to come back to the original line, we see that a projective transformation can take three prescribed points to three prescribed points. In fact, even more is true:

First fundamental theorem. If A,B,C and A',B',C' are two triples of distinct points lying on a line, there is a one and only one projective transformation that sends A to A', B to B', and C to C'.

The downside of this result is that there can be no invariant of projective transformations involving just three points. Let us now try with four points.

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