By C. R. Wylie Jr., Mathematics
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Additional resources for Introduction to Projective Geometry
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.