Let us admire the Renaissance painter at work. Imagine a side view of this scene in a plane equipped with an orthonormal coordinate system. As in the diagram below, we may picture the artist's eye at the origin and the master's canvas on the line with equation x = 1, parallel to the y-axis.
If the canvas is infinite, every point A in the plane can be projected onto a point A' on the canvas. We can even see that all the points lying on a single line through O are projected onto the same point on the canvas. Thus, every line in the plane through the origin corresponds to a unique point on the line with equation x = 1. Every line? Not quite! One still holds out: the y-axis. No matter—let us add a new point to the line representing our canvas, a point "at infinity", corresponding intuitively to where the y-axis and the line with equation x = 1 would meet. And so painters' work on perspective gives us a whole new perspective on the geometry of the plane!
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A new point… of view
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