Conics and Cubics: A Concrete Introduction to Algebraic by Robert Bix (auth.)

By Robert Bix (auth.)

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By Robert Bix (auth.)

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Extra resources for Conics and Cubics: A Concrete Introduction to Algebraic Curves

Sample text

Because t is raised to the power i + j + (d - i - j) = d in every term, we can factor out t d and obtain t) = td '" i j z d-i-j = tdF( X,y,z. ) F( ..... JeijXY ° ° ° that F(ta,tb,tc) = if and only if F(a,b,c) = for any t:F and any point (a,b,c). In other words, if one choice of homogeneous coordinates for a point satisfies the equation F = 0, they all do. In homogeneous coordinates, an algebraic curve-or, simply, a curveis a homogeneous polynomial F(x,y,z). We imagine that the curve consists of all points in the projective plane that satisfy the equation F(x,y,z) = 0, where points corresponding to repeated factors of Fare repeated as many times as the factor.

Because we can use a transformation to interchange the coordinates of A, we can assume that the last coordinate is nonzero. Because the coordinates of A are homogeneous, we can divide them all by the last one, so that we have A = (r,s,l) for numbers rand s. Then the transformation x' = x - rz, y' =y -sz, z' =z, maps A to (0,0, 1). Following this with the transformation that interchanges the first and third coordinates gives a transformation that maps A to (1,0,0). Let Bl be the image of B under this transformation.

The line y = -x/3 in the Euclidean plane. The line x = 2 in the Euclidean plane. The line y = 4 in the Euclidean plane. The line y = x + 2 in the Euclidean plane. The line at infinity. 5. In each part of this exercise, two lines in the projective plane are given in homogeneous coordinates in the form of (2). 1). Find homogeneous coordinates for P in one of the forms in (1). If P is a point of the Euclidean plane, find its usual (x,y) coordinates. If P lies at infinity, find the slope of the lines in the Euclidean plane that contain P.

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