The general equation ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 represents a pair of straight lines when Delta = abc + 2fgh - af^2 - bg^2 - ch^2 = 0. Here the coefficients follow the notation: a (x^2), 2h (xy), b (y^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(h^2 - ab)/|a + b|. The lines are parallel when h^2 = ab and the Delta condition still holds (giving two parallel lines). The distance between parallel lines represented by this equation is 2sqrt(g^2 - ac)/sqrt(a(a + b)).
Part of CG-01 — Straight Lines
General Second-Degree Equation as Pair of Lines
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