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DIAMETERS AND DIAMETRAL PLANES

Diameters of conics. A theorem of plane analytic geometry states that the midpoints of the parallel chords created by a system of parallel lines cutting through a conic in some specified direction all lie on a single straight line (i.e. that their locus is a straight line).. That straight line is called a diameter of the conic. For any given direction in which the system of parallel lines cuts the conic a separate diameter is generated. Any conic has infinitely many diameters. In the central conics ( the ellipses and hyperbolas), they form a pencil of lines through the center of the conic.

Diametral plane of a quartic surface. A theorem of solid analytic geometry states that the midpoints of the parallel chords created by a system of parallel lines cutting through a quadric surface in some specified direction all lie in a single plane (i.e. that their locus is a plane). This plane is called a diametral plane. For any given direction (λ, μ, ν) in which the system of parallel lines cuts through the quadric surface a separate diametral plane is generated.

Equation of the diametral plane conjugate to a system of parallel chords cutting a quadric surface in direction (l, m, n). The equation of the diametral plane conjugate to (i.e. associated with) a system of parallel chords cutting a quadric surface

f(x, y, z) = ax2 + by2 + cz2 + 2fyz + 2gxz + 2hxy + 2px + 2qy + 2rz + d = 0

in the direction (l, m, n) is:

(al + hm + gn)x + (hl + bm + fn)y + (gl + fm + cn) z + (pl + qm + rn) = 0