Intersection of Lines With Quadrics
In projective space, a straight line may intersect a quadric at zero, one, or two points, or may be entirely contained in it. The line defined by two distinct points and is the set of points of the form where are arbitrary scalars from, not both zero. This generic point lies on if and only if, which is equivalent to
The number of intersections depends on the three coefficients, and . If all three of are zero, any pair satisfies the equation, so the line is entirely contained in . Otherwise, the line has zero, one, or two distinct intersections depending on whether is negative, zero, or positive, respectively.
Read more about this topic: Quadric (projective Geometry)
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