Part of CG-01 — Straight Lines

General Second-Degree Equation as Pair of Lines

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The general equation ax2ax^2 + 2hxy + by2by^2 + 2gx + 2fy + c = 0 represents a pair of straight lines when Delta = abc + 2fgh - af2af^2 - bg2bg^2 - ch2ch^2 = 0. Here the coefficients follow the notation: a (x2x^2), 2h (xy), b (y2y^2), 2g (x), 2f (y), c (constant). The point of intersection is found by solving the partial derivatives: ax + hy + g = 0 and hx + by + f = 0. The angle between the lines is the same as the homogeneous case: tan(theta) = 2sqrt(h2h^2 - ab)/|a + b|. The lines are parallel when h2h^2 = ab and the Delta condition still holds (giving two parallel lines). The distance between parallel lines represented by this equation is 2sqrtg2acsqrt\frac{g^2 - ac}{sqrt}(a(a + b)).

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