![]() ![]() For example, the LINE connecting the POINTS (1, 2, 1) and (0, 1, 0) is simply (1, 2, 1) x (0, 1, 0) =. If you think of the triples we are dealing with as vectors, then it turns out that both operations amount to finding a vector perpendicular to the two given vectors such a perpendicular vector can be computed by the well-known cross product. To do geometry with our strange new POINTS and LINES, we need to be able to find the LINE connecting two POINTS, and the POINT of intersection of two LINES. But there is no need to panic: this ambiguity is of the same type as appears in fractions, happily dealt with in primary school for example, the fractions 2/3, 4/6, and 14/21 are all the one and same number. The same is true for LINES - for example, and represent the same LINE. So, each of these triples represent this same POINT. To illustrate, notice that (1, 2, 1), (3, 6, 3) and (-5, -10, -5) are all points on the same POINT. There is an aspect of these so-called homogeneous coordinates that takes some getting used to. Notice that the POINT ( a, b, c) is contained in the LINE exactly if ax + by + cz = 0. the plane) with equation 2 x + 4 y + 5 z = 0. ![]() the line) connecting (0, 0, 0) with (1, 4, 5) and, represents the LINE (i.e. For example, (1, 4, 5) represents the POINT (i.e. Homogeneous coordinates of POINTS and LINESÄ«oth POINTS and LINES can be represented as triples of numbers, not all zero: ( x, y, z) for a POINT and for a plane. Remember that the POINTS and LINES of the real projective plane are just the lines and planes of Euclidean xyz-space that pass through (0, 0, 0). ![]()
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