Conics intersection
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[edit] Intersecting two conics
The solutions to a two second degree equations system in two variables may be seen as the coordinates of the intersections of two generic conic section. In particular two conics may possess none, two, four possibly coincident intersection points. The best method to locate these solutions is to exploits the homogeneous matrix representation of conic sections, i.e. a 3x3 symmetric matrix which depends on six parameters.
The procedure to locate the intersection points follows these steps:
- given the two conics C1 and C2 consider the pencil of conics given by their linear combination λC1 + μC2
- identify the homogeneous parameters (λ,μ) which corresponds to the degenerate conic of the pencil. This can be done by imposing that det(λC1 + μC2) = 0, which turns out to be the solution to a third degree equation.
- given the degenerate cone C0, identify the two, possibly coincident, lines constituting it
- intersects each identified line with one of the two original conic; this step can be done efficiently using the dual conic representation of C0
- the points of intersection will represent the solution to the initial equation system
