Geometric Tomography (Encyclopedia of Mathematics and its by Richard J. Gardner

By Richard J. Gardner

Geometric tomography offers with the retrieval of data a couple of geometric item from information relating its projections (shadows) on planes or cross-sections by way of planes. it's a geometric relative of automated tomography, which reconstructs a picture from X-rays of a human sufferer. the topic overlaps with convex geometry and employs many instruments from that quarter, together with a few formulation from critical geometry. It additionally has connections to discrete tomography, geometric probing in robotics and to stereology. This entire learn offers a rigorous therapy of the topic. even supposing essentially intended for researchers and graduate scholars in geometry and tomography, short introductions, compatible for complex undergraduates, are supplied to the fundamental thoughts. greater than 70 illustrations are used to elucidate the textual content. The publication additionally provides sixty six unsolved difficulties. each one bankruptcy ends with huge notes, ancient feedback, and a few biographies. This re-creation comprises various updates and enhancements, with a few three hundred new references bringing the complete to over 800.

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Geometric Tomography (Encyclopedia of Mathematics and its Applications)

Geometric tomography bargains with the retrieval of data a few geometric item from info pertaining to its projections (shadows) on planes or cross-sections by means of planes. it's a geometric relative of automatic tomography, which reconstructs a picture from X-rays of a human sufferer. the topic overlaps with convex geometry and employs many instruments from that quarter, together with a few formulation from necessary geometry.

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The results of the previous section show that a circle intersects a line in two distinct points, just one point, or not at all. We can therefore conclude that any two distinct circles intersect in two distinct points, just one point, or not at all. In particular, the circles 28 Circles C C C D D D Fig. 2. Three ways in which circles can intersect intersect in a single point exactly when L touches C, D at the same point: in that case we say that C, D touch at that point. e. have the same centre.

Here is the promised pay-off. 3 Suppose that every point on the line L lies on a conic Q. Then Q = L L for some line L . In particular, that is the case when L meets Q in more than two points. ) Proof We can assume L it is not parallel to the y-axis. 13) holds. For every x there is a unique value of y for which L(x, y) = 0, and hence Q(x, y) = 0. That means that J (x) = 0 for all x: since a non-zero quadratic has ≤ 2 roots, that means J is identically zero, so Q = L L . 1 tells us that every point on L lies on Q, so we reach the same conclusion.

That reduces our problem to that of finding a practical criterion for the origin to be a centre for the general conic Q(x, y) = ax 2 + 2hx y + by 2 + 2gx + 2 f y + c. 2 The origin is a centre for a general conic ( ) if and only if the coefficients of the linear terms x, y are both zero. Proof The origin is a centre of Q if and only if the following conics coincide. e. the coefficients of the linear terms x, y are zero Q(x, y) = ax 2 + 2hx y + by 2 + 2gx + 2 f y + c Q(−x, −y) = ax 2 + 2hx y + by 2 − 2gx − 2 f y + c.

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