Bisector of an imaginary angle may be real
Let's find the locus of points in R² defined by the homogeneous quadratic equation
|(1)||AX² + BXY + CY² = 0.|
|(2)||X/Y = (-B ± √B² - 4AC) / 2A.|
Denoting the two roots a = (-B + √B² - 4AC) / 2A and
|(3)||AX² + BXY + CY² = A(X - aY)(X - bY) = 0.|
Thus solving (1) is equivalent to solving
For a straight line X - aY = 0 that passes through the origin, a is the tangent of the angle between the line and the Y-axis; b in X - bY is interpreted similarly. If the two angles are α and β, the angle (up to a sign) between the two lines is
|(4)||tan(α - β)||= (tan α - tan β) / (1 + tan α · tan β)|
|= (a - b) / (1 + ab)|
|= √B² - 4AC / (C + A).|
The equation of the bisector between the two lines is
|(5)||2m / (1 - m²) = (a + b) / (1 - ab) = - B / (A - C).|
As an equation for m, (5) can be rewritten:
|(6)||Bm² - 2(A - C) m - B = 0.|
This is a quadratic equation in m with the discriminant equal to
Now the situation should give one a start. (1) defines a pair of lines. The lines, generally speaking, may be real or imaginary. In either case they appear to form an angle whose bisector is always a real line! What do we make out of this? Is the oddity real or imaginary?
The quandary admits a very simple and quite a natural answer. To visualize complex numbers that comprise two real quantities, one needs two real axes - a real plane. Since, in (1), both X and Y may come out complex, one may need two pairs of axes, i.e., a 4-dimensional space to visualize their mutual dependency. This is where the imaginary lines live. What we see in the real XY-plane is just a projection - often inadequate - of a 4-dimensional construct. From a 4D perspective, there is nothing strange in imaginary lines having real bisectors. This may happen even in 3D. Indeed, draw an angle and its bisector in the z = 0 plane (the xy-plane in 3D.) A rotation of the plane around the bisector line will move the two sides of the angle into the third dimension - invisible, so to speak, to the inhabitants of the plane - whereas the bisector itself will firmly remain in the plane.
By analogy, the explanation to the phenomenon of the imaginary angle that has a real bisector is that sitting in 3D we do not see the complete picture which is rather naturally 4D but only its cross-section. Sometimes, as in the case of an imaginary angle, the cross-section does not give a clue of what it is we are seeing. Sometimes, as in the case of a real angle, we get a better idea of a (complex) phenomenon. We may always wonder whether, in the latter case, we see a complete picture.
- G. Salmon, Treatise on Conic Sections, Chelsea Pub, 6e, 1960