Geometry Topology

Download Introduction to Projective Geometry by C. R. Wylie Jr., Mathematics PDF

By C. R. Wylie Jr., Mathematics

This lucid introductory textual content bargains either an analytic and an axiomatic method of airplane projective geometry. The analytic remedy builds and expands upon scholars' familiarity with easy aircraft analytic geometry and gives a well-motivated method of projective geometry. next chapters discover Euclidean and non-Euclidean geometry as specializations of the projective aircraft, revealing the life of an unlimited variety of geometries, each one Euclidean in nature yet characterised by means of a special set of distance- and angle-measurement formulas. Outstanding pedagogical beneficial properties comprise worked-through examples, introductions and summaries for every subject, and various theorems, proofs, and workouts that make stronger each one chapter's precepts. necessary indexes finish the textual content, in addition to solutions to all odd-numbered routines. as well as its price to undergraduate students of arithmetic, desktop technology, and secondary arithmetic schooling, this quantity presents a great reference for computing device technology professionals.

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Sample text

In the first place, it will illumine certain aspects of euclidean geometry and should therefore be helpful to those whose primary interest is the teaching of elementary geometry. Second, it will provide us with a specific model of the axiomatic system we shall introduce in Chap. 6, which will be useful in subsequent chapters for purposes of illustration and as a check on the consistency of our postulates. 2 Homogeneous Coordinates in the Euclidean Plane The most characteristic feature of the euclidean plane is probably its parallel property: through any point not on a given line there passes a unique line which is parallel to the given line.

In a perspective transformation, is every point in the picture plane the image of some point in the object plane? Does every point in the object plane have an image in the picture plane? Hint: Remember that the object plane extends on both sides of the picture plane. 9. In a perspective transformation, is a line always represented by a line? Is a nonsingular conic always represented by a nonsingular conic? 10. In a perspective transformation, are there any parallel lines which are not represented in the picture plane by lines meeting on the vanishing line ?

In fact, if this were the case, every point in the plane of our discussion would be invariant, which is clearly false. The only conclusion we can presently draw is that the image of a general point on any line through O is some other point on that same line. In other words, in the transformation in the object plane, a point and its image are always collinear with O. Or to put it still differently, each line on O is invariant as a whole but is not point-by-point invariant. If we are given a point, P, in the plane σ, we now know that its image, P′, is some point on the line OP.

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