In Chapters 1 and 2 we considered parallel X-rays, that is, X-rays taken from infinity, of bounded Lebesgue measurable sets. In this chapter and the next one we turn our attention to X-rays emanating from finite points. This corresponds to the “fan-beam” X-rays of great importance in medicine; in fact, modern CAT scanners use only this type of X-ray.

It seems that even basic results on point X-rays of planar convex bodies require some knowledge of measure theory, so in this chapter, this is assumed from the outset. Two types of point X-ray and the related chordal symmetrals are introduced in the first section, in the context of measurable sets in En. The second section contains some lemmas also useful in the next chapter. These could be skipped at first, and read when encountered again in Section 5.3, which deals only with point X-rays of planar convex bodies.

An X-ray (or directed X-ray) of a planar convex body at a point gives the lengths of all the intersections of the body with lines through the point (or rays issuing from the point, respectively). The main result here is Theorem 5.3.8, stating that any set of four points in general position in the plane have the property that the X-rays at these points will distinguish between any two convex bodies. For directed X-rays, three noncollinear points will suffice for this purpose, as Theorem 5.3.6 demonstrates.