General equation of conicoid

  • A general second degree equation in three variables is
    ax2 + by2 + cz2 + 2fyz + 2gzx + 2gxy + 2ux + 2vy + 2wz + d = 0

  • A conicoid or quadric surface in the three dimensional coordinate system is the set S of points (x, y, z) ∈ R3 that satisfies a general second degree equation in three variables.

  • When a = b = c = 1 and f = g = h = 0 the equation represents a sphere.

  • When u = v = w = d = 0 then the equation represents a cone.

  • When a = b = 1, h = 0 and z = k then the equation represents a right circular cylinder.

  • When x = k or y = k and a = b = 1, h = 0, then again the equation represents a cylinder.

  • A conicoid remains a conicoid under the translation or rotation of axes.


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