Imaginary line (mathematics)
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In geometry, an imaginary line is a straight line that only contains one real point. It can be proven that this point is the intersection point with the conjugated line.
It is a special case of an imaginary curve.
It can be proven that there exists no equation of the form ax + by + cz = 0 in which a, b and c are all real coefficients. However their do exists equations of the form ax + by + cz = 0, but at least one of the coefficients need be complex.
As follows, it can be proven that, if an equation of the form ax + by + cz = 0 in which a, b and c are all real coefficients, exist, the straight line is a real line, and it shall contain an infinite number of real points.
This property of straight lines in the complex projective plane is a direct consequence of the duality principle in projective geometry.
